CPT-P49-2006
R-MATRICES IN RIME
Oleg Ogievetsky∗
Centre de Physique Théorique†, Luminy, 13288 Marseille, France
Todor Popov
Institute for Nuclear Research and Nuclear Energy,
Bulgarian Academy of Sciences, Sofia, BG-1784, Bulgaria
Abstract
We replace the ice Ansatz on matrix solutions of the Yang–Baxter equation by a weaker condition
which we call rime. Rime solutions include the standard Drinfeld–Jimbo R-matrix. Solutions of
the Yang–Baxter equation within the rime Ansatz which are maximally different from the standard
one we call strict rime. A strict rime non-unitary solution is parameterized by a projective vector
~φ. We show that in the finite dimension this solution transforms to the Cremmer–Gervais R-matrix
by a change of basis with a matrix containing symmetric functions in the components of ~φ. A strict
unitary solution (the rime Ansatz is well adapted for taking a unitary limit) in the finite dimension
is shown to be equivalent to a quantization of a classical ”boundary” r-matrix of Gerstenhaber and
Giaquinto. We analyze the structure of the elementary rime blocks and find, as a by-product, that
all non-standard R-matrices of GL(1|1)-type can be uniformly described in a rime form.
We discuss then connections of the classical rime solutions with the Bézout operators. The Bézout
operators satisfy the (non-)homogeneous associative classical Yang–Baxter equation which is related
to the Rota–Baxter operators. We calculate the Rota–Baxter operators corresponding to the Bézout
operators.
We classify the rime Poisson brackets: they form a 3-dimensional pencil. A normal form of each
individual member of the pencil depends on the discriminant of a certain quadratic polynomial.
We also classify orderable quadratic rime associative algebras
For the standard Drinfeld–Jimbo solution, there is a choice of the multiparameters, for which it
can be non-trivially rimed. However, not every Belavin–Drinfeld triple admits a choice of the
multiparameters for which it can be rimed. We give a minimal example.
∗On leave of absence from P.N. Lebedev Physical Institute, Theoretical Department, Leninsky prospekt 53, 119991
Moscow, Russia
†Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix–Marseille I, Aix–Marseille II et du Sud
Toulon – Var; laboratoire affilié à la FRUMAM (FR 2291)
http://arxiv.org/abs/0704.1947v3
Contents
1 From ice to rime 3
2 Rime Yang–Baxter solutions 6
2.1 Non-unitary rime R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Unitary rime R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Rime and Cremmer–Gervais R-matrices 9
4 Classical rime r-matrices 12
4.1 Non-skew-symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 BD triples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Skew-symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Bézout operators 20
5.1 Non-homogeneous associative classical Yang–Baxter equation . . . . . . . . . . . . . . 21
5.2 Linear quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.3 Algebraic meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.4 Rota–Baxter operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.5 ∗-multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Rime Poisson brackets 30
6.1 Rime pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.2 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.3 Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7 Orderable quadratic rime associative algebras 38
Appendix A. Equations 40
Appendix B. Blocks 41
B.1 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
B.2 GL(2) and GL(1|1) R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
B.3 Riming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Appendix C. Rimeless triple 47
References 48
1 From ice to rime
A well known class of solutions R̂ ∈End (V ⊗ V ), V is a vector space, of the Yang–Baxter equation
YB(R̂) = 0, where
YB(R̂) := (R̂⊗ 11)(11 ⊗ R̂)(R̂ ⊗ 11)− (11⊗ R̂)(R̂ ⊗ 11)(11 ⊗ R̂) , (1)
is characterized by the so called ice condition (see lectures [21] for details) which says that R̂
can be
different from zero only if the set of the upper and the set of the lower indices coincide,
kl 6= 0 ⇒ {i, j} ≡ {k, l}. (2)
We introduce the ”rime” Ansatz, relaxing the ice condition: the entry R̂
kl can be different from zero
if the set of the lower indices is a subset of the set of the upper indices,
6= 0 ⇒ {k, l} ⊂ {i, j} . (3)
Matrices for which it holds will be referred to as “rime” matrices. Figuratively, in the rime, in contrast
to the ice, situation, putting an apple and a banana in a fridge, there is a non-zero amplitude to find
next morning two apples instead (but never an apple and an orange).
The Yang–Baxter equation for a matrix R̂ is equivalent to the equality of two different reorderings
of xiyjzk, using xiyj = R̂
kxl, xizj = R̂
kxl and yizj = R̂
kyl, to the form z•y•x•. One of
advantages of the rime Ansatz is that only indices i, j and k appear in the latter expression. Another
advantage is that for fixed values of i and j, the elements x• and y• with these values of indices form
a subsystem.
A rime R-matrix has the following structure
= αijδ
+ βijδ
+ γijδ
l + γ
(no summation) . (4)
To avoid redundancy, fix βii = 0, γii = 0 = γ
ii. We denote by αi the diagonal elements R̂
ii, αi = αii.
Throughout the text we shall assume that the matrix R̂ is invertible which, in particular, implies that
αi 6= 0 for all i.
The order of growth of the number of unknowns in the Yang–Baxter system for a rime matrix is
n2, where n =dimV .
Arbitrary permutations and rescalings of coordinates preserve the rime condition.
The ice and rime matrices are made of 4× 4 elementary building blocks, respectively,
R̂ice =
α1 0 0 0
0 β12 α12 0
0 α21 β21 0
0 0 0 α2
and R̂rime =
α1 0 0 0
γ12 β12 α12 γ
γ′21 α21 β21 γ21
0 0 0 α2
. (5)
In appendix B we analyze the structure of the 4× 4 rime blocks.
We call a rime matrix strict if αijγij 6= 0 ∀ i and j, i 6= j. Note that strict rime matrices are
necessarily not ice.
Proposition 1. Let R̂ be a rime matrix (4). Then R̂ is a solution of the Yang–Baxter equation if it
is of the form
= (1− βji)δilδ
+ βijδ
+ γijδ
l − γjiδ
, (6)
where βij and γij satisfy the system
βijβji = γjiγij , (7)
βij + βji = βjk + βkj =: β , (8)
βijβjk = (βjk − βji)βik = (βij − βkj)βik , (9)
γijγjk = (βji − βjk)γik = (βkj − βij)γik . (10)
Proof. The Yang–Baxter system of equations YB(R̂)
abc = 0 for a rime matrix is given in the appendix
A. The subset (257) - (259) together with its image under the involution (256) reads
ij(γij + γ
ji) = 0 = αijγij(γji + γ
ij) , (11)
αij(βijβji + γijγ
ij) = 0 = αij(βijβji − γijγji) , (12)
ij(αij + βji − αi) = 0 = αijγij(αij + βji − αj) , (13)
ij(αji + βij − αi) = 0 = αijγij(αji + βij − αj) . (14)
These equations are implied by (and, in the strict rime situation, are equivalent to) the following
system
γ′ij = −γji , αij + βji = αi , αji + βij = αi , (15)
βijβji = γjiγij . (16)
One checks that other equations YB(R̂)
abc = 0, for which two indices among {i, j, k} are different,
follow from (15) and (16). The last two equations from (15) imply αi = αj for all i and j. As an
overall rescaling of a solution of the Yang–Baxter equation by a constant is again a solution of the
Yang–Baxter equation, we can, without loss of generality, set it to one,
αi = 1 . (17)
Eqs. (15) and (17) yield the form (6) of the matrix R̂ and eq. (7).
Using (15), we rewrite the subset (266) - (268) together with its image under the involution (256)
in the form
(βij + βji − βik − βki)γijγik = 0 , (18)
αij(βijβjk + βikβji − βikβjk) = 0 = αji(βjiβkj + βkiβij − βkiβkj) , (19)
αij(γijγjk + γik(βjk − βji)) = 0 = αji(γjiγkj + γki(βkj − βij)) . (20)
These equations are implied by (and, in the strict rime situation, are equivalent to) eqs. (8), (9) and
(10). One checks that other equations YB(R̂)
abc = 0 with three different indices {i, j, k} follow from
the system (7)–(10). The proof is finished. �
Lemma 1. The rime Yang–Baxter solution R (6) is of Hecke type,
R̂2 = βR̂+ (1− β)11⊗ 11 . (21)
Moreover, when β 6= 2, R is of GL-type: it has two eigenvalues 1 and β − 1 with multiplicities n(n+1)
n(n−1)
, respectively. When β = 2 the matrix R̂ has a nontrivial Jordanian structure.
Proof. In view of the block structure of rime matrices it is enough to check the Hecke relation (21)
for one elementary (4 × 4) block which follows from (7) and (8). When β 6= 2 the multiplicities m1
and mβ−1 are solutions of the system
m1 +mβ−1 = n
2 , m1 + (β − 1)mβ−1 = n+
n(n− 1)
β (≡ TrR̂) . (22)
When β = 2 the matrix R̂ has only one eigenvalue 1 but R̂ 6= 11⊗ 11 due to (7) and (8). �
Unitary solutions, R̂2 = 11⊗ 11, are singled out by the value of the parameter β = 0.
Lemma 2. A strict rime Yang–Baxter solution R (6) can be brought to a rime matrix
= (1 − βji)δilδ
+ βijδ
− βijδikδil + βjiδ
, (23)
that is, to a solution (6) with γij = −βij , by a change of basis.
Proof. The strict rime condition αijγij 6= 0 implies βijβji 6= 0 in view of (7). Thus for a strict rime
R-matrix all βij and γij are nonvanishing. The ratio of eqs. (9) and (10) is well-defined and it follows
from eqs. (7) and (8) that
γijγjk
βijβjk
(βji − βjk)γik
(βji − βjk)βik
= −γik
γijγji
βijβji
= 1 , (24)
ξijξjk = ξik ξijξji = 1 , (25)
where ξij = −
. Eq. (25) is solved by ξij =
with di 6= 0, i = 1, . . . , n, hence β’s and γ’s are
related by
γij = −
βij . (26)
A change of basis with a matrix D,
R̂ 7−→ (D ⊗D) R̂ (D−1 ⊗D−1) , (27)
where Dij = djδ
j , transforms R to the form (23). �
Under the strict rime condition, the Yang–Baxter system of equations (see appendix A) reduces
to eqs. (8) and (9). However, the matrix (23), where the parameters βij are subject to eqs. (8) and
(9), is a solution of the Yang–Baxter equation without a strict rime assumption.
Remark. Right and left even quantum spaces are defined by, respectively,
kxl = xixj , xjxiR
kl = xlxk ; (28)
right and left odd quantum spaces are defined by, respectively,
kξl = (β − 1)ξiξj , ξjξiRijkl = (β − 1)ξlξk . (29)
Assume that β 6= 2. The left even space is classical1 as well as the right odd space
[xi, xj ] = 0 , [ξ
i, ξj ]
= 0 , (30)
where [ , ] and [ , ]+ stand for the commutator and the anti-commutator. The relations for the right
even space are
[xi, xj ] + (βijx
i + βjix
j)(xi − xj) = 0 ; (31)
the relations for the left odd space read
(2− β)ξ2i + ξiρ+ (1− β)ρξi = 0 , (32)
[ξi, ξj ]+ − βijξiξj − βjiξjξi = 0 , i 6= j , (33)
where ρ =
j ξj.
2 Rime Yang–Baxter solutions
In this section we solve eqs. (8) and (9) thus obtaining explicitly rime Yang–Baxter solutions.
2.1 Non-unitary rime R-matrices
Proposition 2. The non-unitary strict rime Yang–Baxter solutions (23) with a parameter β = βji+
βij 6= 0 are parameterized by a point φ ∈ PCn in a projective space, φ = (φ1 : φ2 : . . . : φn), such that
φi 6= 0 for all i and φi 6= φj for all i and j, i 6= j. These solutions are given by
βij =
φi − φj
. (34)
Proof. Taking the ratio of the following pairs of equations from (9)
βijβjk = (βjk − βji)βik , βkjβji = (βji − βjk)βki (35)
we find that quantities ηij = −βij/βji verify equations
ηijηjk = ηik , ηijηji = 1 , (36)
1Let R̂ be a rime R-matrix (not necessarily strict). When β 6= 2, the following statement holds. If (i) the left even
space is classical (which implies that γ′ij = −γji, αij + βji = 1 and αi = 1 in our normalization) and (ii) the R-matrix
is Hecke (which implies that βij + βji = β) then the system of equations from the appendix A again reduces to (7), (9)
and (10) as in the strict rime situation.
whose solutions are ηij = φi/φj for some constants φi 6= 0, i = 1, . . . , n.
Substituting the relation βji = −
βij into β = βij + βji, we obtain βij −
βij = β which
establishes (34). �
Remark. There is a different parameterization, βij = −
φi − φj
, of strict rime solutions; it is related
to the parameterization (34) by φi 7−→ (φi)−1.
A direct check shows that the condition φi 6= 0 is not necessary: the formula (34) with φi 6= φj for
all i and j, i 6= j, gives a rime solution of the Yang–Baxter equation. However when one of φi is 0,
the matrix (23) is no more strict.
2.2 Unitary rime R-matrices
For a unitary strict rime Yang–Baxter solution (23), R̂2 = 11, we have β = 0, so βij = −βji.
Proposition 3. The unitary strict rime Yang–Baxter solutions (23) are parameterized by a vector
(µ1, . . . , µn) such that µi 6= µj,
βij =
µi − µj
. (37)
Proof. Since βij = −βji we can rewrite βijβjk = (βjk − βji)βik as βijβjk = (βij + βjk)βik or
. (38)
These equations are solved by
= µi − µj , (39)
which is equivalent to (37). �
Remark. The unitary R-matrices of Proposition 3 can be obtained as a limit β → 0 of the non-
unitary R-matrices of Proposition 2. Indeed, for the following expansion of the parameters φi in the
“small” parameter β,
φi = 1 + βµi + o(β) , (40)
the expression (34) has a limit (37),
βij =
β(1 + βµi + o(β))
βµi − βµj + o(β)
β→0−→ βij =
µi − µj
. (41)
2.3 Properties
1. Denote the R-matrix (23) with βij as in (34) by R̂(~φ ). Let R̂21 = PR̂12P , where P is the
permutation operator. Then the following holds:
R̂21(~φ ) = F
−1 ⊗ F−1 R12(~φ−1) F ⊗ F , (42)
where F = diag(φ1, φ2, . . . , φn) and ~φ
−1 is a vector with components φ−1i .
Denote the R-matrix (23) with βij as in (37) by R̂(~µ ). Then the following holds:
R̂21(~µ) = R̂12(−~µ) . (43)
2. The R-matrix (23) is skew invertible in the sense that there exists an operator Ψ̂R, which satisfies
(see, e.g. [21])
Tr2(R̂12(Ψ̂R)23) = P13 . (44)
The matrices of the left and right quantum traces (that is, the left and right traces of the skew inverse
Ψ̂R), (QR)1 = Tr2((Ψ̂R)12) and (Q̃R)2 = Tr1((Ψ̂R)12), are given by the formulas
j = −βjk
l: l 6=k
(1− βjl) , k 6= j , and (QR)jj =
(1− βjl) ; (45)
(Q̃R)
j = βjk
l: l 6=k
(1− βlj) , k 6= j , and (Q̃R)jj =
(1− βlj) . (46)
The matrices QR and Q̃R satisfy QRQ̃R = (1− β)n−111.
For (34), one has Spec QR = Spec Q̃R = {(1 − β)a, a = 0, . . . , n − 1}. The eigenvector wa(~φ)
of the matrix QR with the eigenvalue (1 − β)n−1−a coincides with the eigenvector of the matrix Q̃R
with the eigenvalue (1−β)a. One has (wa(~φ))j = eĵa(~φ), where eĵi (~φ) is the i-th elementary symmetric
function of (φ1, φ2, . . . , φn) with φj omitted.
For (37), the Jordanian form of the matrix QR, as well as of Q̃R, is non trivial: it is a single block.
In the basis {wi(~µ)}, i = 0, 1, 2, . . . , n− 1, where (wi(~µ))j = eĵi (~µ), one has
QR wi(~µ) =
n− 1− s
ws(~µ) . (47)
3. For an R-matrix R̂, the group of invertible matrices Y satisfying
R̂12Y1Y2 = Y1Y2R̂12 (48)
form the invariance group GR of R̂. The matrices QR and Q̃R belong to the invariance group as well
as the matrices proportional to the identity matrix. One can write down formulas for the group GR
for a rime R-matrix (23) uniformly in terms of βij as in (45) and (46) but the properties are different
in the non-unitary and unitary cases and we describe them separately.
3a. The invariance group G
R(~φ )
for the R-matrix R̂(~φ ) is 2-parametric. It consists of matrices
Y (u, v), u, v 6= 0, where
Y (u, v)
l:l 6=j
uφj − vφl
φj − φl
and Y (u, v)ij =
(u− v)φj
φj − φi
l:l 6=i,j
uφj − vφl
φj − φl
, i 6= j . (49)
One has
R(~φ )
= Y (1− β, 1) , Q̃
R(~φ )
= Y (1, 1 − β) . (50)
The composition law is the component-wise multiplication of the parameters {u, v},
Y (u1, v1)Y (u2, v2) = Y (u1u2, v1v2) . (51)
The point u = v = 1 corresponds to the identity matrix, Y (1, 1) = 11; the determinant of Y (u, v)
is (uv)n(n−1)/2; u = v corresponds to global rescalings; the connected component of unity of the
subgroup SG
R(~φ )
consisting of matrices with determinant 1 is uv = 1; the generator η of the connected
component of unity of the subgroup SG
R(~φ )
is traceless and reads
ηij =
φj − φi
, i 6= j , and ηjj = −
l:l 6=j
φj − φl
. (52)
3b. For the R-matrix R̂(~µ ), the group SGR(~µ ), consisting of matrices with determinant 1 is
1-parametric as well. It is formed by matrices Y (0)(a), where
Y (0)(a)
l:l 6=j
µj − µl
) and Y (0)(a)ij =
µj − µi
l:l 6=i,j
µj − µl
) , i 6= j . (53)
The expression (53) can be obtained by taking a limit of (49), similarly to (41) and letting additionally
u = 1 + aβ/2 + o(β) and v = 1− aβ/2 + o(β).
One has
QR(~µ ) = Y
(0)(−1) , Q̃R(~µ ) = Y (0)(1) . (54)
The composition law is Y (0)(a1)Y
(0)(a2) = Y
(0)(a1 + a2).
The point a = 0 in (53) corresponds to the identity matrix, Y (0)(0) = 11; the generator η(0) of the
invariance group SGR(~µ ) is
(η(0))ij =
µj − µi
, i 6= j , and (η(0))jj =
l:l 6=j
µj − µl
. (55)
3 Rime and Cremmer–Gervais R-matrices
The Cremmer–Gervais R-matrix arises in the exchange relations of the chiral vertex operators in
the non-linearly W -extended Virasoro algebra [6]. The Cremmer–Gervais solution [6] of the Yang–
Baxter equation in its general two-parametric form reads (see, e.g. [17]; we use a rescaled matrix with
eigenvalues 1 and −q−2)
(R̂CG,p)
kl = q
−2θijpi−jδilδ
k + (1− q
s: i≤s<j
pi−sδskδ
i+j−s
l − (1− q
s: j<s<i
pi−sδskδ
i+j−s
l , (56)
where θij is the step function (θij = 1 when i > j and θij = 0 when i ≤ j).
The parameter value p = q2/n specifies the SL(n) Cremmer–Gervais R-matrix (its diagonal twist
being the GL(n) solution (56)). The Cremmer–Gervais solution is a non-diagonal twist of the standard
Drinfeld–Jimbo solution [18, 9].
Let R̂CG := R̂CG,1, that is, the solution (56) with p = 1. The matrix
D(p)ij = δ
i−1 (57)
with arbitrary p satisfies (R̂CG)12D(p)1D(p)2 = D(p)1D(p)2(R̂CG)12. It was observed in [10] that if
R̂12D1D2 = D1D2R̂12 for some R-matrix R̂ and operator D then D1R̂12D
1 is again an R-matrix
(this operation was also used in [15] to partially change the statistics of ghosts in the super-symmetric
situation). The two-parametric matrix R̂CG,p (56) can be obtained from the Cremmer–Gervais matrix
R̂CG by this operation as well,
(R̂CG,p)12 = D(p)1(R̂CG)12D(p)
1 . (58)
Let R̂ be the non-unitary rime matrix from Proposition 2 with φi 6= φj.
Proposition 4. The matrix R̂ transforms into the Cremmer–Gervais solution R̂CG
R̂ = (X ⊗X) R̂CG (X−1 ⊗X−1) (59)
by a change of basis with the invertible matrix
Xkj = ej−1 (φ1, . . . , φ̂k, . . . , φn) =: e
j−1 (60)
whose inverse is
(X−1)ji =
(−1)j−1φn−ji
k:k 6=i
(φi − φk)
. (61)
Here the hat over φj means that this entry is omitted in the expression and ei are the elementary
symmetric polynomials ei(x1, . . . , xN ) =
s1<...<si
xs1xs2 . . . xsi . The projective parameters (φ1 : φ2 :
. . . : φn) are removed by the transformation X and the only essential parameter β in R̂ is related to
the parameter q in R̂CG by
q−2 = 1− β . (62)
Proof. Due to the Lagrange interpolation formula, the matrix, inverse to the Vandermonde matrix
||V jk ||
j,k=1 = φ
k is (V
−1)kj =
(−1)j−1ek̂j−1
l:l 6=k
(φk − φl)
. (63)
The matrix X (60) has the form X = DV −1 d, where Dmk = δ
l:l 6=k(φk − φl) and dij = (−1)j−1δij
are diagonal n× n matrices. Thus, its inverse is X−1 = d−1 V D−1, which establishes (61).
We now prove the matrix identity (59) in the form
R̂(X ⊗X) = (X ⊗X) R̂CG . (64)
The substitution of the explicit form of the rime matrix R̂ (23) with βij = βφi/(φi − φj) and R̂CG
(56) reduces (64) to a set of relations between the symmetrical polynomials eâk−1
l−1 =
eîa−1e
b−1(R̂CG)
kl . (65)
There are two subcases: i) i = j and ii) i 6= j.
i) The left hand side of eq. (65) with i = j is just eîk−1e
l−1 due to the rime condition. Eq. (65) is
satisfied because of the symmetry relation (R̂CG)
kl = δ
l + δ
k − (R̂CG)bakl .
ii) For i 6= j eq. (65), where q−2 = 1− β, reduces, after some algebraic manipulations, to
φi − φj
k−1 − φje
k−1)(e
l−1 − e
s: s≥max(1,k−l+2)
(eîl+s−2e
k−s − e
l+s−2e
k−s) , 1 ≤ i, j, k, l ≤ n .
In fact, the sum in the right hand side goes till s = min(k, n + 1 − l) since eĵr = 0 when r ≥ n − 1;
moreover we can start the summation from s = 1 because when 1 < k−l+2 the sum for 1 ≤ s ≤ k−l+1
is anti-symmetric under s←→ k − l + 2− s and thus vanishes.
To prove (66) we write er = e
r + φie
r−1; therefore e
r = e
r + φje
r−1 and e
r = e
r + φie
r−1 and eq.
(66) becomes
− (φi − φj)eîĵk−1e
l−2 = (φi − φj)
l+s−2e
k−s−1 − e
l+s−3e
k−s) . (67)
The sum in the right hand side telescopes to the value of (−eîĵl+s−3e
k−s) at s = 1, that is, to (−e
l−2).
The proof is complete. �
It should be noted that the matrix X = X(~φ ) does not depend on q. The change of the basis with
the matrix X(~φ′ )X(~φ )−1 transforms the R-matrix R̂(~φ ) to R̂(~φ′ ). We have
(X(~φ′ )X(~φ )−1)ij =
φj − φ′i
(φj − φ′k)
l:l 6=j
(φj − φl)
The structure of the matrices X andX−1 shows that when the dimension is infinite, the R-matrices
R̂CG,1 and R̂(~φ ) (as well as the R-matrices R̂(~φ ) and R̂(~φ′ ) for different φ and φ
′) are in general not
equivalent.
The right even quantum plane for the Cremmer–Gervais matrix R̂CG,1 is defined by the following
equations
yiyj = q2yjyi + (q2 − 1)(yi+1yj−1 + . . .+ yj−1yi+1), i < j . (69)
If i+ 1 < j − 1, one uses the formula (69) recursively to get the ordering relations.
The change of basis with the matrix X,
eîj−1y
j , (70)
transformes the quantum plane (69) into the rime quantum plane (31) exhibiting coordinate two-
dimensional subplanes. The change of basis (70) can be written in terms of a ”generating function”:
ej(φ1, . . . , φn) y
j . (71)
. (72)
Remark. The standard Drinfeld–Jimbo R-matrix admits, for a certain choice of multi-parameters, a
different rime form. The relations uivj = (R̂c)
kul for this choice are
uivi = viui ,
uivj = vjui + (1− q−2)viuj , i < j ,
uivj = q−2 vjui , i > j .
The left even space for this R-matrix is classical.
The change of variables with the matrix X̃ij = 1− θji,
U i := u1 + u2 + · · ·+ ui , V i := v1 + v2 + · · ·+ vi , (74)
transforms the relations (73) into
U iV i = V iU i ,
U iV j = V jU i + (1− q−2)V iU j − (1− q−2)V iU i , i < j ,
U iV j = q−2 V jU i + (1− q−2)V jU j , i > j .
The matrix X, defined by eq. (60), degenerates if φi = φj for some i and j. Interestingly,
the R-matrix X ⊗XR̂c X−1 ⊗X−1 admits limits limφσ(2)→0 limφσ(3)→0 . . . limφσ(n)→0 for an arbitrary
permutation σ ∈ Sn and the result is always rime. In particular,
X̃ ⊗ X̃R̂c X̃−1 ⊗ X̃−1 = lim
. . . lim
X ⊗XR̂c X−1 ⊗X−1 . (76)
4 Classical rime r-matrices
The classical limit of an R-matrix is a classical r-matrix, a solution of the classical Yang–Baxter (cYB)
equation
[r12, r13] + [r12, r23] + [r13, r23] = 0 . (77)
We are going to show that the classical limits of the rime R-matrices from Section 2 are equivalent
to the Cremmer-Gervais r-matrices in the non-skew-symmetric case and to the ”boundary” r-matrix
of Gerstenhaber and Giaquinto [14] (see also [4]; this r-matrix is attributed to A. G. Elashvili there)
in the skew-symmetric case. Similar equivalences appeared in the study of the gauge transformations
of the dynamical r-matrices in the Calogero-Moser model [12, 13] 2.
In the sequel we use the following conventions. An R-matrix acts in a space V ⊗ V . A basis of V
is {ei} (labeled by a lower index); an operator A in V has matrix coefficients Aji , A(ei) = A
i ej , so for
a vector ~v = viei one has (A~v)
i = Aij~v
j ; the matrix units are eij , e
j(ek) = δ
kej , so the multiplication
rule is eije
l = δ
j ; eαi are the sl(n) simple positive root elements, eαi = e
i ; P is the permutation
operator, P (ei ⊗ ej) = ej ⊗ ei, so P (eij ⊗ ekl ) = eil ⊗ ekj and (PB)klij = Blkij for an operator B in V ⊗ V
having matrix coefficients Bklij , B(ei ⊗ ej) = Bklij ek ⊗ el.
4.1 Non-skew-symmetric case
Proposition 5. The non-unitary rime R-matrix (Proposition 2) is a quantization of the non-skew-
symmetric r-matrix
i,j:i 6=j
φi − φj
(eij ⊗ e
i − e
i ⊗ e
j + e
i ∧ eij) , (78)
where x ∧ y := x ⊗ y − y ⊗ x. The change of basis with the matrix Xjk = ek−1 (φ1, . . . , φ̂j , . . . , φn)
transforms r into the parameter-free cYB solution rCG
rCG =
i,j:i<j
(ei+s−1j ⊗ e
j−s+1
i − e
i+s−1
i ⊗ e
j−s+1
j ) . (79)
Proof. The coefficients βij (34) are linear in the deformation parameter β (β = 0 is the classical
point). Hence
R = 11⊗ 11 + βr , (80)
where R = PR̂ and r is given by (78).
The matrix RCG−11⊗11, where RCG = PR̂CG, is linear with respect to the parameter β = 1−q−2
as well,
RCG = 11⊗ 11 + β rCG (81)
thus the formula (59) implies r = (X ⊗X) rCG (X−1 ⊗X−1). �
We mentioned two ways of obtaining the numerical two-parametric R-matrix (R̂CG,p) from the R-
matrix (R̂CG,1): by a diagonal twist and by the operation (58). There is one more way which consists
of changing the representation. We shall illustrate it on the example of the classical GL r-matrix (79).
A change of representation of the Lie algebra GL,
eij 7→ eij + c δij11 , (82)
2We thank László Fehér for drawing our attention to the references [12, 13].
where c is a constant, produces the following effect on the r-matrix (79):
rCG 7→ rCG + c
η ⊗ 11− 11⊗ η − (n− 1)11 ⊗ 11
, (83)
where n =dimV and
η = −
n(n+ 1)
j , tr η = 0 . (84)
The classical version of the operation (58) is as follows. Let η be an arbitrary generator of the
invariance group of an r-matrix r,
[r, η1 + η2] = 0 . (85)
Then the operator
r(c) = r + c(η1 − η2) , (86)
where c is a constant, is again a classical r-matrix (a solution of the cYBe).
The operator η in (84) is, up to a scale, the unique traceless generator of the invariance group
(see (57)) of the r-matrix (79). Thus, the representation change and the operation (86) give the same
family of r-matrices (up to an addition of a multiple of the identity operator, which does not violate
the cYBe).
4.2 BD triples.
Each block in the strict rime classical r-matrix (78) looks even more ”rimed”,
0 0 0 0
β′12 −β′12 β′21 −β′21
−β′12 β′12 −β′21 β′21
0 0 0 0
, (87)
where β′ij = βij/β = φi/(φi − φj). The multiplication from the left by P acts on each block as a
permutation of the second and third lines, so the rime r-matrix (87) enjoys the symmetry Pr = −r.
We shall now discuss this symmetry property in the context of Belavin–Drinfeld triples.
In [3] Belavin and Drinfeld gave, for a simple Lie algebra g, a description of non-unitary (non-
skew-symmetric) cYB solutions r ∈ g⊗ g, satisfying r12 + r21 = t, where t ∈ g ⊗ g is the g-invariant
element. The non-unitary solutions are put into correspondence with combinatorial objects called
Belavin–Drinfeld triples (BD-triples for short). The Belavin–Drinfeld triple (Π1,Π2, τ) for a simple
Lie algebra g consists of the following data: Π1,Π2 are subsets of the set of simple positive roots Π
of the algebra g and τ is an invertible mapping: Π1 → Π2 such that 〈τ(ρ), τ(ρ′)〉 = 〈ρ, ρ′〉 for any
ρ, ρ′ ∈ Π1 and τk(ρ) 6= ρ for any ρ ∈ Π1 and any natural k for which τk(ρ) is defined.
The r-matrix for a triple (Π1,Π2, τ) has the form
r = r0 +
e−α ⊗ eα +
α,β∈∆+:α<β
e−α ∧ eβ , (88)
where < is a partial order on the set of positive roots ∆+ defined by the rule: α < β for α, β ∈ ∆+
if there exists a natural k such that τk(α) = β. The part r0 belongs to h⊗ h, where h is the Cartan
subalgebra of g; r0 contains continuous ”multiparameters”, which satisfy
(τ(α) ⊗ id + id⊗ α)(r0) = 0 for all α ∈ Π1 . (89)
We are dealing with matrix solutions r of the cYB equation, r ∈ gl(n)⊗ gl(n), so r12 + r21 can be
a linear combination of P and 11⊗ 11.
Let Π = {α1, . . . , αn−1} be the set of the positive simple roots for the Lie algebra sl(n).
There are two Cremmer–Gervais BD triples, T+ and T−. For the Cremmer–Gervais triple BD-
triple T+, Π1 = {α1, α2, . . . , αn−2}, Π2 = {α2, α3, . . . , αn−1} and τ(αi) = αi+1. The data (Π1,Π2, τ)
is encoded in the graph
. . . •
• • • • . . . • •
The triple T− can be obtained from the triple T+ either by setting Π
1 = Π2, Π
2 = Π1 and
τ ′ = τ−1 or by applying the outer automorphism of the underlying An−1 Dynkin diagram; the graph
corresponding to the triple T− is
. . . •
• • • . . . • • •
The r-matrix (79) corresponds to the triple (90) for a certain choice of the multiparameters. Here is
the r-matrix r′ corresponding to the triple (91)
r′CG =
i,j:i<j
(eij−s+1 ⊗ e
i+s−1 − e
j−s+1 ⊗ e
i+s−1) (92)
for a certain choice of the multiparameters, for which it satisfies r′P = −r′.
For the r-matrices (79) and (92), one has r12+ r21 = P −11⊗11. The Cartan part of the r-matrices
(79) and (92) are
r0 = −
i,j:i<j
eii ⊗ e
j , r
0 = −
i,j:i<j
j ⊗ e
i . (93)
The following lemma shows that a classical r-matrix r for a triple T can have a symmetry with
respect to the multiplication by P from one side if and only if all segments (connected components)
of Π1 are mapped by τ according to either (90) or (91).
Lemma 3. A non-skew-symmetric classical r-matrix with a Belavin–Drinfeld data (Π1,Π2, τ) can
satisfy Pr = −r (respectively, rP = −r) for a certain choice of the multiparameters if and only if
τ(αi) = αi+1 (respectively, τ(αi) = αi−1) for all i ∈ Π1.
Proof. Assume that τ(αm) = αm+k for some natural k, k ≥ 1. Then r contains the term em+km+k+1∧e
with the coefficient 1. Such r-matrix cannot satisfy rP = −r for if rP = −r then r contains the term
m+k+1
∧ em+km with the coefficient (−1) but the coefficient in e−α ∧ eβ is 1 in the formula (88).
If Pr = −r then r should contain also the term em+1m+k+1 ∧ e
m . It then follows that
(i) the Lie subalgebra generated by Π1 contains e
m therefore the interval [αm, αm+1, . . . , αm+k−1]
is contained in Π1;
(ii) the Lie subalgebra generated by Π2 contains e
m+k+1
therefore the interval [αm+1, αm+2, . . . , αm+k]
is contained in Π2;
(iii) the image of the interval [αm, αm+1, . . . , αm+k−1] under τ is the interval [αm+1, αm+2, . . . , αm+k].
This implies that the interval [αm+1, αm+2, . . . , αm+k−1] is τ -invariant (since τ(αm) = αm+k) which
contradicts to the nilpotency of τ unless this interval is empty, that is, k = 1.
Similarly, rP = −r is possible only if τ(αi) = αi−1 for all i ∈ Π1.
It is left to show that when τ(αi) = αi+1 (respectively, τ(αi) = αi−1) for all i ∈ Π1 the multipa-
rameters can indeed be adjusted to fulfill Pr = −r (respectively, rP = −r). We leave it as an exercise
for the reader to check that with the assignment (93) for r (respectively, for r′) the compatibility
condition (89) is verified. The proof is finished. �
Remark. Two extreme BD triples can be rimed, the empty (Drinfeld–Jimbo) one and the “maximal”
Cremmer–Gervais one. However, not every triple can be rimed: already the triple
O • •
• • •
provides a counterexample. We outline a computer-aided proof in appendix C.
4.3 Skew-symmetric case
A skew-symmetric classical r-matrix r ∈ g ∧ g is canonically associated with a quasi-Frobenius Lie
subalgebra (f, ω) of g (see, e.g., [24]). A Lie algebra f which admits a non-degenerate 2-cocycle ω is
called quasi-Frobenius; it is Frobenius if ω is a coboundary, i.e., ω(X,Y ) = λ([X,Y ]) for some λ ∈ f∗.
We describe now the skew-symmetric r-matrix arising in the classical limit of the unitary rime
R-matrix from Proposition 3.
Proposition 6. The unitary rime R-matrix (Proposition 3) is a quantization of the skew-symmetric
r-matrix
i,j:i<j
µi − µj
(eij − e
j) ∧ (e
i − e
i) ∈ gl(n) ∧ gl(n) . (95)
This skew-symmetric classical r-matrix corresponds to a Frobenius Lie algebra (g0(n), δλn) spanned by
the generators Zij := e
j − e
j , i 6= j, with the Frobenius structure determined by the coboundary of the
1-cochain λn = −
i,j:i 6=j
j , where {zij}, i 6= j, is the basis in g∗0(n), dual to the basis {Zij} in g0(n),
zij(Z
l ) = δ
Proof. An artificial introduction of a small parameter c by a rescaling µi 7→ c−1µi in the formula for
the R-matrix R̂ in Proposition 3 gives
R = 11⊗ 11 + c r , (96)
where r is given by (95).
The n(n− 1) matrices Zij := eij − e
j , i 6= j, form an associative subalgebra of the matrix algebra,
l = (δ
− δil )(Zki − Z li) (97)
(we set Zii = 0 for all i); with respect to the commutators these matrices form a Lie subalgebra g0(n)
of the Lie algebra gl(n), g0(n) ⊂ gl(n):
[Zij , Z
i ] = Z
i − Z
j , [Z
i , Z
i ] = Z
i − Z
i , [Z
j , Z
] = Z
− Zik , i 6= j 6= k 6= i , (98)
all other brackets vanish. The skew-symmetric solution (95) of the cYB equation,
i,j:i<j
Zij ∧ Z
µi − µj
, (99)
is non-degenerate on the carrier subalgebra g0(n). The carrier subalgebra g0(n) is necessarily quasi-
Frobenius, having a 2-cocycle ω given by the inverse of the r-matrix, that is,
ω(ZA, ZB) = rAB , where r
ABrBC = δ
C , r =
rABZA ∧ ZB . (100)
We have
ω(Zij , Z
l ) = −(µi − µj)δliδ
. (101)
It is easy to check that the 2-cycle ω is a coboundary,
ω(Zij, Z
l ) = λn([Z
j , Z
l ]) , λn = −
i,j:i 6=j
j ∈ g∗0(n) , (102)
thus the subalgebra g0(n) is Frobenius. �
The ”Frobenius” r-matrix (95) (and its quantization) was considered in the work [2].
Proposition 7. The skew-symmetric rime classical r-matrix (95), r =
i<j(µi−µj)−1Zij∧Z
i , where
µ = (µ1, µ2, . . . , µn) is an arbitrary vector such that µi 6= µj, belongs to the orbit of the parameter-free
classical r-matrix
i,j:i<j
ei+ki ∧ e
j−k+1
j . (103)
More precisely,
r = AdXµ ⊗AdXµ(b) , (104)
where the element Xµ ∈ GL(n) is defined by (Xµ)jk = ek−1 (µ1, . . . , µ̂j , . . . , µn).
3This matrix is the same X as in Proposition 4 but depending on variables µi.
Proof. The equality r = AdXµ ⊗ AdXµ(b) is equivalent to a set of relations for the elementary
symmetric functions ei,
(Xµ ⊗Xµ) b = r (Xµ ⊗Xµ) ⇔
eîr−1e
s−1 b
l−1 , (105)
where
j−k+1
δi+ka −
δi−k+1a δ
and r
(δiaδ
b + δ
− δiaδ
− δjaδib)/(µi − µj) , i 6= j ,
0 , i = j .
Both operators b
ab and r
ab are symmetric in the lower indices and anti-symmetric in the upper indices,
that is,
Pb = −b , bP = b and Pr = −r , rP = r . (106)
Eqs. (105) have the following form
(eîb+s−2e
a−s−1 − e
b+s−2e
a−s−1) =
µi − µj
(eîa−1 − e
a−1)(e
b−1 − e
b−1) . (107)
Due to (66), the left hand side of (107) equals
µi − µj
a−2 − µje
a−2)(e
b−1 − e
b−1) . (108)
The right hand side of (107) equals (108) as well because eîa−1 = ea−1 − µieîa−2. �
As in the non-skew-symmetric case, in the infinite dimension the operators b and r are in general
not equivalent.
The sl(n) cYB solution. Let I =
i=1 e
i be the central element of gl(n). The generators
Z̃ij = Z
I ∈ sl(n) satisfy the same relations (98) as Zij thus they form a subalgebra g̃0(n) of the
Lie algebra sl(n) which is isomorphic to g0(n), g̃0(n) ≃ g0(n). This isomorphism gives rise to another
solution r̃ ∈ sl(n) ∧ sl(n) of the cYB equation,
i,j:i<j
Z̃ij ∧ Z̃
µi − µj
∈ sl(n) ∧ sl(n) . (109)
We have the following lemma about the carrier Lie algebra of r̃ (the Lie subalgebra of sl(n) spanned
by the generators Z̃ij).
Lemma 4. The subalgebra g̃0(n) ⊂ sl(n) of dimension dim g̃0(n) = n(n − 1) is isomorphic to the
maximal parabolic subalgebra p of sl(n) obtained by deleting the first negative root.
Proof. The vector v =
i=1 ei is an eigenvector for all elements Z̃
Z̃ij(v) =
v for all i and j , i 6= j . (110)
In a basis in which the first vector is v, the linear span of the generators Z̃ij is
∗ ∗ . . . ∗
0 ∗ ∗
0 ∗ . . . ∗
, (111)
with the traceless condition. The comparison of dimensions finishes the proof. �
Gerstenhaber and Giaquinto [14] found a classical r-matrix bCG which they called “boundary”
because it lies in the closure of the solution space of the YB equation. The cYB solution bCG corre-
sponds to a Frobenius subalgebra (p,Ω), where p is the parabolic subalgebra of sl(n) as above and the
2-cocycle Ω is a coboundary,
Ω = δλbCG , λbCG =
(eii+1)
∗ ∈ p∗ . (112)
The r-matrix bCG is a twist of b (see [8]).
Since the carriers of r̃ and bCG are isomorphic, the r-matrices are equivalent. We shall now prove
that the same matrix Xµ transforms bCG into r̃.
Proposition 8. The boundary classical r-matrix bCG ∈ sl(n) ∧ sl(n),
bCG =
) eii ∧ e
i,j:i<j
ei+ki ∧ e
j−k+1
j , (113)
transforms into the cYB solution r̃ ∈ sl(n) ∧ sl(n),
i,j:i<j
Z̃ij ∧ Z̃
µi − µj
, where Z̃ij = e
j − e
eii , (114)
by a change of basis with the matrix Xµ ∈ GL(n),
r̃ = AdXµ ⊗AdXµ(bCG) . (115)
Proof. Due to Proposition 7 we have r = AdXµ ⊗ AdXµ(b). The cYB solution bCG is the sum of
b and other terms, bCG = b +
i,j(1 −
) eii ∧ e
j . Therefore it is enough to show that r̃ − r =
AdXµ ⊗AdXµ(bCG − b). One has
r̃ − r = 1
i,j:i 6=j
µi − µj
, bCG − b = I ∧
(1− j
j . (116)
Thus we have to show that
(1− j
i,j:i 6=j
µi − µj
Xµ , (117)
which amounts to the following identities for the elementary symmetric functions:
(1− b− 1
)eîb−2 =
j:j 6=i
b−1 − e
µi − µj
. (118)
Replacing, in the right hand side, e
b−1 by e
b−1 + µie
b−2, e
b−1 by e
b−1 + µje
b−2 and noticing that
c = (n − c)ec, c = 1, 2, . . . , n (for the elementary symmetric functions in n variables) finishes the
proof. �
The passage to the sl(n) solution is another instance of the representation change. The general
representation change (82) produces the following effect on the numerical r-matrix (103):
b 7→ b− cη(0) ∧ 11 , (119)
where η(0) is the generator of the invariance group of the r-matrix (103),
η(0) =
(n− j)ej+1j . (120)
The representation change and the operation (86) produce the same 1-parametric family (119) of
skew-symmetric r-matrices. The choice c = −1/n corresponds to the r-matrix bCG.
5 Bézout operators
The Bézout operator [5] is the following endomorphism b(0) of the space P of polynomials of two
variables x and y:
b(0)f(x, y) =
f(x, y)− f(y, x)
or b(0) =
(I − P ) , (121)
where I is the identity operator and P is a permutation, Pf(x, y) = f(y, x). For any natural n, the
subspace Pn of polynomials of degree less than n in x and less than n in y is invariant with respect to
the operator b(0). The matrix of the restriction of b(0) onto Pn, written in the basis {xayb} of powers
(in the decreasing order) coincides with the operator (103).
The non-skew-symmetric matrix (79) is the matrix of the operator
(I − P ) (122)
in this basis. The rime bases are formed by the non-normalized Lagrange polynomials {li(x)lj(y)},
li(t) =
s:s 6=i
(t− φs), at points {φi}, i = 1, 2, . . . , n.
We shall call the operators b(0) and b Bézout r-matrices. The Bézout r-matrices were rediscovered
in several different contexts related to the Yang–Baxter equation (except the fact that they are the
Cremmer–Gervais r-matrices, they appear, for instance, in [7] and [19]).
The standard r-matrix r(s), for the choice of the multi-parameters for which it can be non-trivially
rimed (see the remark at the end of section 3), has the following form in terms of polynomials
r(s) : xiyj 7→ θ(i− j)xiyj − θ(j − i)xjyi . (123)
The subspaces Pn are invariant with respect to r
The properties of the Bézout r-matrices b(0) and b (and of the operator r(s)) become more trans-
parent when they are viewed as operators on polynomials. In particular,
(b(0))2 = 0 , b(0)P = −b(0) , Pb(0) = b(0) , b(0) + b(0)21 = 0 , (124)
b2 = b , bP = −b , b+ b21 = I − P , (125)
(r(s))2 = r(s) , r(s)P = −r(s) , r(s) + r(s)21 = I − P . (126)
The description of the invariance groups of the operators b(0) and b is especially transparent when
these operators are viewed as operators on the spaces of polynomials. Let ∂x and ∂y be the derivatives
in x and y. We have (∂x + ∂y)
= 0 which implies that ∂x is the generator of the invariance
group of b(0); the group is formed by translations. Similarly, (x∂x + y∂y)
= 0 which implies
that x∂x is the generator of the invariance group of b; the group is formed by dilatations. The operation
(86) implies that the operators
b(0) + c(∂x − ∂y) , b+ c(x∂x − y∂y) (127)
are solutions of the cYBe (the quantum version is easy as well) for an arbitrary constant c.
5.1 Non-homogeneous associative classical Yang–Baxter equation
The operators b(0), b and r(s) satisfy an equation stronger than the cYBe. For an endomorphism r of
V ⊗ V , define
r ◦ r := r12r13 + r13r23 − r23r12 , r ◦′ r := r13r12 + r23r13 − r12r23 . (128)
The equation r ◦ r = 0 (as well as r ◦′ r = 0) is called associative classical Yang–Baxter equation
(acYBe) [1, 20].
We introduce a non-homogeneous associative classical Yang–Baxter equation (nhacYBe):
r ◦ r = cr13 , (129)
where c is a constant.
Let F be the space of polynomials in one variable. For the space F ⊗ F of polynomials in two
variables, we denote by x (respectively, y) the generator of the first (respectively, second) copy of F .
For F ⊗ F ⊗ F , the generators are denoted by x, y and z.
Lemma 5. 1. Let M be an operator on the space F ⊗ F . Assume that
M(xf) = f + yM(f) , (130)
M(yf) = −f + xM(f) (131)
for an arbitrary f ∈ F ⊗ F . Then4
M ◦M(xF ) = zM ◦M(F ) , M ◦M(yF ) = xM ◦M(F ) , M ◦M(zF ) = yM ◦M(F ) (132)
for an arbitrary F ∈ F ⊗ F ⊗ F .
2. The operator M = b(0) verifies (130) and (131).
3. Moreover, the unique solution of eqs. (130) and (131) (for the operator M on the space F ⊗ F)
together with the ”initial” condition M(1) = 0 is M = b(0).
Proof. A direct calculation. �
Proposition 9. 1. The Bézout operator b(0) satisfies the acYBe.
2. The Bézout operator b and the operator r(s) satisfy the nhacYBe with c = 1.
Proof. A direct calculation for b(0). Another way is to notice that the relations (132) for M = b(0)
reduce the verification of b(0) ◦ b(0)(F ) = 0 for a monomial F ∈ F ⊗ F ⊗ F to the case F = 1, which
is trivial.
For the Bézout operator b ≡ xb(0) (x here is the operator of multiplication by x), we have, for an
arbitrary F ∈ F ⊗ F ⊗ F ,
b ◦ b (F ) = xb(0)12 (xb
13 (F )) + xb
13 (yb
23 (F ))− yb
23 (xb
12 (F ))
13 (F ) + yb
13 (F )
+ xyb
23 (F )− xyb
12 (F )
13 (F ) + xyb
(0) ◦ b(0)(F ) = b13(F ) .
(133)
We used eq. (130) for b(0) in the second equality.
For the operator r(s), the identity
θ(i− k)θ(i− j) + θ(i− k)θ(j − k)− θ(i− j)θ(j − k) = θ(i− k) (134)
for the step function θ is helpful. �
In each of cases (124-126), the operator r satisfies a quadratic equation r2 = u1r+u2I, the relation
r + r21 = αP + βI with some constants α and β and the nhacYBe with some constant c. Several
general comments about relations between the constants appearing in these equations are in order
here.
4Eq. M ◦M(xF ) = zM ◦M(F ) follows from (130) alone.
1. Assume that an r-matrix (a solution of the cYBe) satisfies r ◦r = cr13. Then r ◦′ r = cr13. Taking
the combinations (r ◦ r− cr13)−P23(r ◦′ r− cr13)P23 and (r ◦ r− cr13)−P12(r ◦′ r− cr13)P12, we find
r13(Sr)23 − (Sr)23r12 = c(r13 − r12) , (Sr)12r13 − r23(Sr)12 = c(r13 − r23) , (135)
where (Sr)12 := r12 + r21. If (Sr)12 = αP12 + βI with some constants α and β, as in (124-126), then
it follows from (135) that (β − c)(r13 − r12) = 0 thus
c = β . (136)
This explains the value of the constant c in lemma 9.
2. For an endomorphism r of V ⊗ V , assume that r ◦ r = βr13 and (Sr)12 = αP12 + βI. Then
P23(r ◦ r − βr13)P23 = r13r12 + r12r32 − r32r13 − βr12
= r13r12 + r12(αP23 + βI − r23)− (αP23 + βI − r23)r13 − βr12 = r ◦′ r − βr13 .
(137)
Thus, if (Sr)12 = αP12 + βI then r ◦ r = βr13 implies r ◦′ r = βr13.
3. Assume that r ◦ r = cr13 for an endomorphism r of V ⊗ V . Then for r̃ = r+ aI + bP , a and b are
constants, we have
r̃ ◦ r̃ = (c+ 2a)r̃13 + bP13(Sr)23 − a(a+ c)I − bcP13 + b2P23P12 . (138)
If, in addition, (Sr)12 = αP12 + βI, then
r̃ ◦ r̃ = (c+ 2a)r̃13 − a(c+ a)I + b(β − c)P13 + b(α+ b)P23P12 . (139)
This shows that the equation r ◦ r = c1r13 + c2I + c3P13 + c4P23P12, c1, c2, c3 and c4 are constants,
reduces to r ◦ r = c̃1r13 + c̃3P13 by a shift r 7→ r + aI + bP .
If r ◦ r = βr13 and (Sr)12 = αP12 + βI then
r̃ ◦ r̃ = (β + 2a)r̃13 − a(β + a)I + b(α+ b)P23P12 . (140)
The combination P23P12 does not appear for b = 0 or b = −α. The choice b = −α corresponds,
modulo a shift of r by a multiple of I, to r 7→ r21, so we consider only b = 0. Then, with the choice
a = −β we find that the operator r̃ = r − βI satisfies the nhacYBe (and (Sr)12 = αP12 − βI). For
the choice a = −β/2 we find that the operator r̃ = r −
I satisfies
r̃ ◦ r̃ = β
, (Sr̃)12 = αP12 . (141)
In particular, the operator
2(x− y)
P (142)
satisfies (141) with β = 1 and α = −1. Also, b̃2 = 1
4. Assume that r2 = ur + v and r12 + r21 = αP12 + βI for an endomorphism r of V ⊗ V . Squaring
the relation r12 − βI = αP12 − r21 and using the same relation again, we obtain
(u− β)(2r12 − βI − αP12) = 0 . (143)
Thus, if r is not a linear combination of I and P then
u = β . (144)
5. Assume that r ◦ r = cr13 and rP = −r for an endomorphism r of V ⊗ V . The nhacYBe has the
following equivalent form:
[r13, r23] = (r12 − cI)r13P23 . (145)
Indeed,
r13r23 − r23r13 = (−r13r23 + r23r12)P23 = (r12 − cI)r13P23 . (146)
Here in the first equality we used r23P23 = −r23 and moved P23 to the right; in the second equality
we used the nhacYBe r ◦ r = cr13.
5.2 Linear quantization
Consider an algebra with three generators r12, r13 and r23 and relations
r13r23 = r23r12 − r12r13 + βr13 ,
r13r12 = r12r23 − r23r13 + βr13 ,
r212 = βr12 + v , r
13 = βr13 + v , r
23 = βr23 + v .
(147)
Choose an order, say, r13 > r23 > r12. Consider (147) as ordering relations. The overlaps in (147)
lead to exactly one more relation:
r23r12r23 = r12r23r12 . (148)
Thus the algebra in question is 12-dimensional (it follows from (147) and (148) that a general
element of the algebra is a product AB of an element A of the Hecke algebra generated by r12 and
r23 and a polynomial B, of degree less than 2, in r13).
We conclude that the nhacYBe together with the quadratic equation for r imply the YBe. Note
that the other form of the YBe also follows:
r23r13r12 − r12r13r23 = (r12r23 − r13r12 + βr13)r12 − r12(r23r12 − r12r13 + βr13)r23
= −r13(βr12 + v) + βr13r12 + (βr12 + v)r13 − βr12r13 = 0.
(149)
Here in the first equality both nhacYBe for r were used; the quadratic relation for r was used in the
second equality.
Therefore, the quantization of such r-matrix is ”linear”5: a combination
R = I + λr , (150)
where λ is an arbitrary constant, satisfies the YBe R12R13R23 = R23R13R12.
5It was noted in [8] that the operator b(0) satisfies both forms of the YBe, squares to zero and that its quantization
has the simple form (150).
5.3 Algebraic meaning
We shall clarify the algebraic meaning of the non-homogeneous associative classical Yang–Baxter
equation in the general context of associative algebras.
Let A be an algebra. Let r ∈ A⊗ A. The operation
δ(0) : A→ A⊗A , δ(0)(u) = (u⊗ 1) r − r (1⊗ u) (151)
(the algebra A does not need to be unital, (u⊗1)(a⊗ b) stands for ua⊗ b and (a⊗ b)(u⊗1) for au⊗ b)
is coassociative if and only if [1]
(u⊗ 1⊗ 1) (r ◦′ r) = (r ◦′ r) (1⊗ 1⊗ u) ∀ u ∈ A . (152)
In particular, δ(0) is coassociative if (r ◦′ r) = 0.
Assume now that the algebra A is unital. Define the operations δ and δ̃ : A→ A⊗ A,
δ(u) := (u⊗ 1) r − r (1⊗ u)− c (u⊗ 1) , (153)
δ̃(u) := (u⊗ 1) r − r (1⊗ u) + c (1⊗ u) , (154)
where c is a constant.
Proposition 10. The coassociativity of each of the operations δ and δ̃ is equivalent to
(u⊗ 1⊗ 1) (r ◦′ r − c r13) = (r ◦′ r − c r13) (1⊗ 1⊗ u) ∀ u ∈ A . (155)
Proof. A straightforward calculation. �
In particular, the operations δ and δ̃ are coassociative if r ◦′ r = c r13.
The map (151) has the following property:
δ(0)(uv) = (u⊗ 1) δ(0)(v) + δ(0)(u) (1 ⊗ v) ; (156)
that is, δ(0) is a derivation with respect to the standard structure of A ⊗ A as a bi-module over A,
uU := (u⊗ 1)U and Uu := U(1⊗ u) for u ∈ A and U ∈ A⊗ A.
For the operations δ and δ̃, the analogue of the property (156) reads
δ(uv) = (u⊗ 1) δ(v) + δ(u) (1 ⊗ v) + c (u⊗ v) , (157)
δ̃(uv) = (u⊗ 1) δ̃(v) + δ̃(u) (1⊗ v)− c (u⊗ v) . (158)
5.4 Rota–Baxter operators
Let A be an algebra. An operator r : A → A is called Rota–Baxter operator of weight α if
r(A)r(B) + αr(AB) = r
r(A)B +Ar(B)
(159)
for arbitrary A,B ∈ A (α is a constant). We refer to [22] for further information about the Rota–
Baxter operators.
The Rota–Baxter operators of weight zero are closely related to the acYBe [23]. It turns out
that the Rota–Baxter operators of non-zero weight are related to the nhacYBe. We shall discuss this
relation and calculate the Rota–Baxter operators corresponding to the Bézout operators.
It is surprising that the Bézout operators, which rather have the sense of derivatives, become,
being interpreted as operators on matrix algebras, the Rota–Baxter operators which are designed to
axiomatize the properties of indefinite integrations and summations.
1. For an endomorphism r of V ⊗ V , define two endomorphisms, r and r′, of the matrix algebra
Mat(V ):
r(A)1 := Tr2(r12A2) , r
′(A)2 := Tr1(r12A1) , A ∈ Mat(V ) , (160)
where Tri is the trace in the copy number i of the space V .
Assume that r satisfies the nhacYBe (129). Multiplying (129) by A2B3, A,B ∈ Mat(V ), and
taking traces in the spaces 2 and 3, we find
r(A)r(B) + r
r′(A)B
Ar(B)
= cTr(A)r(B) . (161)
Assume, in addition, that r12 + r21 = αP12 + βI. Then
r(A) + r′(A) = αA+ βTr(A) 11 . (162)
If c = β then expressing r′(A) by (162) and substituting into (161), we find that the term with Tr(A)
drops out and r is the Rota–Baxter operator of weight α on the algebra of matrices. Similarly, r′ is
the Rota–Baxter operator of weight α as well.
2. We shall calculate the Rota–Baxter operators corresponding to the Bézout operators in the poly-
nomial basis.
The action of the operator b(0) on monomials xkyl reads
b(0)(xkyl) =
−(xl−1yk + xl−2yk+1 + · · ·+ xkyl−1) , k < l ,
0 , k = l ,
xk−1yl + xk−2yl+1 + · · · + xlyk−1 , k > l .
(163)
The action of the operator b on monomials xkyl reads
b(xkyl) =
−(xlyk + xl−1yk+1 + · · ·+ xk+1yl−1) , k < l ,
0 , k = l ,
xkyl + xk−1yl+1 + · · ·+ xl+1yk−1 , k > l .
(164)
Shortly,
b(0)(xkyl) = θ(k − l)
k−l−1
xl+syk−s−1 − θ(l − k)
l−k−1
xk+syl−s−1 , (165)
b(xkyl) = θ(k − l)
xl+syk−s − θ(l − k)
xk+syl−s . (166)
We list several useful matrix forms of the operators b(0) and b in the basis formed by monomials,
ea ⊗ eb := xayb; for the operator b(0):
b(0) =
i,j,a,b
θ(j − a) θ(j − b) δi+j
a+b+1
eja ∧ eib
i,j,a,b
θ(j − a) θ(j − b)− θ(i− b) θ(i− a)
a+b+1 e
a ⊗ eib
i,j:i<j
i+a−1 ∧ e
(167)
and for the operator b:
i,j,a,b
θ(j + 1− a) θ(a− i) δi+ja+b (e
a ⊗ eib − eia ⊗ e
i,j,a,b
θ(j + 1− a) θ(a− i)− θ(i+ 1− a) θ(a− j)
a+b e
a ⊗ eib
i,j:i<j
i+a ⊗ e
j−a − eii+a ⊗ e
i,j:i<j
j−i−1
i+a ∧ e
j−a + e
j ⊗ e
i − eij ⊗ e
(168)
where x ∧ y = x⊗ y − y ⊗ x.
The Rota–Baxter operator r
corresponding to b(0) reads
(A)ij = θ(j − i)
Ai−sj−s−1 − θ(i+ 1− j)
Ai+s+1j+s . (169)
In the right hand side of (169), the summations are over those s ≥ 0 for which the corresponding
matrix element in the sum makes sense; that is, the range of s in the first sum is s = 0, 1, . . . , i − 1
and, in the second sum, s = 0, 1, . . . n− i− 1;
The Rota–Baxter operator rb corresponding to b reads (with the same convention about the sum-
mation ranges)
rb(A)
j = θ(j + 1− i)
Ai−s−1j−s−1 − θ(i− j)
Ai+sj+s . (170)
Its weight is -1.
For the operator r(s), given by eq. (123), the corresponding Rota–Baxter operator r(s) is
r(s)(A)ij =
−θ(j − i)Aij , i 6= j ,
s:s<i
Ass , i = j .
(171)
Its weight is -1.
We shall give also the Rota–Baxter operator for the Bézout r-matrix b in the rime basis, that is,
for the r-matrix (78); it has weight 1 (since r12+r21 = P −I for r in (78)). The Rota–Baxter operator
has the form
r(A)ij =
φj − φi
(Aij −A
j) , i 6= j ,
s:s 6=i
φi − φs
(Ais −Ass) , i = j .
(172)
5.5 ∗-multiplication
1. Let r : A→ A be a Rota–Baxter operator of weight α (see eq.(159)) on an algebra A. It is known
that the operation
A ∗B := r(A)B +Ar(B)− αAB , A,B ∈ A , (173)
defines an associative product on the space A. This product is closely related to the coproducts (153)
and (154) by duality. We shall illustrate it in the context of the matrix algebras.
Define an operation ∗̃ by
〈δ̃(u), B ⊗A〉 = 〈u,A∗̃B〉 , (174)
where δ̃ is given by (154). We have then
〈δ̃(u), B ⊗A〉 = Tr12
u1rB1A2 − ru2B1A2 + c u2B1A2
= Tr1
u1 Tr2(rA2)B1
− Tr1
u1A1 Tr2(B2 r21)
c u1 Tr(B)A1
r(A)B −A r′(B) + cATr(B)
(175)
A∗̃B = r(A)B −A r′(B) + cATr(B) . (176)
In eq. (175), xi stands for the copy of an element x in the space number i in A ⊗ A; the operators
r and r′ are given by (160); to obtain the second and the third terms in the second line of (175) we
renumbered 1↔ 2 and then moved r cyclically under the trace in the second term.
Assume, as before, that r12+ r21 = αP12 +βI and c = β. Then, expressing r
′(A) by (162), we find
that the term with Tr(B) drops out and it follows that
A∗̃B = A ∗B . (177)
2. We shall describe the ∗-multiplication in the simplest example of the Rota–Baxter operators 169)
and (170) corresponding to the Bézout operators for the the polynomials of degree less than 2 (that
is, for 2× 2 matrices A =
a11 a
a21 a
≡ aije
For the operator b(0) = e21 ∧ e11, we have
(A) =
−a21 a11
(178)
and the ∗-multiplication reads
A ∗o Ã ≡ Ar
(Ã) + r
(A)Ã =
−a21ã11 −a21ã12 + a11(ã11 + ã22)
−a21ã21 a21ã11
. (179)
This algebra is isomorphic to the algebra of 3× 3 matrices of the form
∗ ∗ ∗
0 0 ∗
0 0 0
with the identification
e11 7→
0 1 0
0 0 1
0 0 0
 , e12 7→
−1 0 0
0 0 0
0 0 0
 , e21 7→
0 0 1
0 0 0
0 0 0
 , e22 7→
0 0 0
0 0 1
0 0 0
 . (180)
For the operator b = e22 ⊗ e11 − e12 ⊗ e21, we have
rb(A) =
−a21 a11
(181)
and the ∗-multiplication reads
A ∗ Ã ≡ Arb(Ã) + rb(A)Ã+AÃ =
a11ã
2 + a
1 + ã
a11ã
2 + a
1 + ã
. (182)
This algebra is isomorphic to the algebra of 3× 3 matrices of the form
∗ ∗ ∗
0 ∗ 0
0 0 0
with the identification
e11 7→
1 0 0
0 1 0
0 0 0
 , e12 7→
0 0 1
0 0 0
0 0 0
 , e21 7→
0 1 0
0 0 0
0 0 0
 , e22 7→
0 0 0
0 1 0
0 0 0
 . (183)
6 Rime Poisson brackets
The Poisson brackets having the form
{xi, xj} = fij(xi, xj) , i, j = 1, 2, . . . , n , (184)
with some functions fij of two variables, we shall call rime. In this section we study quadratic rime
Poisson brackets,
{xi, xj} = aij(xi)2 − aji(xj)2 + 2νijxixj , i, j = 1, 2, . . . , n . (185)
We show that there is a three-dimensional pencil of such Poisson brackets and then find the invariance
group and the normal form of each individual member of the pencil.
6.1 Rime pencil
In this subsection we establish that the quadratic rime Poisson brackets form a three-dimensional
Poisson pencil.
The left hand side of (185) contains a matrix aij with zeros on the diagonal, aii = 0, and an
anti-symmetric matrix νij, νij = −νji. The Jacobi identity constraints these matrices to satisfy
aijajk + aik(νij + νjk) = 0 , i 6= j 6= k 6= i . (186)
We shall describe a general solution of eq. (186) in the strict situation, that is, when all aij and νij
are different from zero for i 6= j.
The left hand side of νij + νjk = −aijajk/aik is anti-symmetric with respect to (i, k), that is
ΥijΥjkΥki = 1 for Υij = −aij/aji, which readily implies the existence of a vector φi such that
Υij = φ
j . Therefore,
aik = φicikφ
k , (187)
where the matrix cij is anti-symmetric, cij = −cji. Next, 2νki = −(νij +νjk)+ (νjk+νki)+ (νki+νij);
using (186) to express each bracket in the right hand side, we find
νki =
cijcki
cjkcki
cijcjk
. (188)
The right hand side of eq. (188) does not depend on j which imposes further restrictions on the matrix
cij when n > 3. Writing the sum νij + νjk + νkl + νli in two ways, as (νij + νjk) + (νkl + νli) and as
(νjk + νkl) + (νli + νij), and using (186) to express each bracket in terms of the matrix c, we obtain
cijcjk − cilclk
cjkckl − cjicil
. (189)
Replacing j by m in (188) gives the condition on the matrix c:
cijcki
cjkcki
cijcjk
cimcki
cmkcki
cimcmk
. (190)
Using eq. (189) to rewrite the combination
cijcjk
cimcmk
, we find
(cjkckm − cjicim)Ψijkm = 0 , where Ψijkm =
cjkcim
cijckm
ckicjm
. (191)
The quantity Ψijkm is totally anti-symmetric with respect to its indices. Therefore, if Ψijkm 6= 0 then
the combinations (cjkckm − cjicim) vanish for all permutations of indices. This is however impossible:
the system of three linear equations
cijcjk − cimcmk = 0 ,
cikckm − cijcjm = 0 ,
cimcmj − cikckj = 0
(192)
for unknowns {cjk, ckm, cmj} has, by definition, a non-zero solution but the determinant of the system
is different from zero. Thus the Pfaffian Ψijkm vanishes for each quadruple {i, j, k,m}; in other
words, the coefficients of the matrix 1/cij satisfy the Plücker relations; therefore the form 1/cij is
decomposable, c−1ij = sitj−sjti, for some vectors ~s and ~t. For each i, at least one of si or ti is different
from zero. Making, if necessary, a change of basis in the two dimensional plane spanned by ~s and ~t, we
can therefore always assume that all components of, say, the vector ~s are different from zero, si 6= 0
∀ i. We represent the bivector 1/cij in the form (u−1i = si and ψi = −ti/si)
= u−1i u
j (ψi − ψj) . (193)
Substituting (193) into (188) we obtain
νki +
u2k + u
ψk − ψi
u2i − u2j
ψi − ψj
u2k − u2j
ψk − ψj
. (194)
Replacing j by m in the right hand side and equating the resulting expressions, we find that the
independency of the right hand side on j implies:
Eijkm :=
(ψi − ψj)(ψi − ψk)(ψi − ψm)
(ψj − ψi)(ψj − ψk)(ψj − ψm)
(ψk − ψi)(ψk − ψj)(ψk − ψm)
(ψm − ψi)(ψm − ψj)(ψm − ψk)
= 0 .
(195)
for every quadruple {i, j, k,m}.
The quantity Eijkm is totally symmetric. Selecting three values of the index, say, 1,2 and 3, we
can form the quadruple {i, 1, 2, 3} for each i. Solving Ei123 = 0, we obtain the following expression
for u2i :
u2i = A1
(ψi −M2)(ψi −M3)
(M1 −M2)(M1 −M3)
(ψi −M1)(ψi −M3)
(M2 −M1)(M2 −M3)
(ψi −M1)(ψi −M2)
(M3 −M1)(M3 −M2)
(196)
for some constants A1, A2, A3,M1,M2 and M3. The right hand side is the value, at the point ψi, of a
quadratic polynomial which equals to Aa at the points Ma, a = 1, 2, 3. Since A1, A2, A3,M1,M2 and
M3 are arbitrary, we can simply write
u2i = aψ
i + bψi + c . (197)
With the expressions (197) for u2i , the equalities (195) are identically satisfied which shows that (197)
is the general solution.
Upon rescaling xi 7→ φiuixi with φi from (187), the Poisson brackets (185) simplify. The following
statement is established (for n = 2 or 3, (197) does not impose a restriction on the anti-symmetric
matrix cij with all off-diagonal entries different from zero).
Proposition 11. Up to a rescaling of variables, the general strict quadratic rime Poisson brackets
have the form
{xi, xj} =
̺(ψj)(x
i)2 + ̺(ψi)(x
ψi − ψj
(ψi − ψj) a−
̺(ψi) + ̺(ψj)
ψi − ψj
xixj , (198)
where ~ψ is an arbitrary vector with pairwise distinct components and ̺(t) = at2+bt+c is an arbitrary
quadratic polynomial 6.
Thus the strict quadratic rime Poisson brackets form the three-dimensional pencil (parameterized
by the polynomial ̺).
The Poisson brackets (198) can be rewritten in the following forms:
{xi, xj} =
ψi − ψj
̺(ψj)x
i − ̺(ψi)xj
(xi − xj) + a (ψi − ψj)xixj , (199)
{xi, xj} =
au2ij + buijvij + cv
ψi − ψj
ψi − ψj
, (200)
where uij = ψjx
i − ψixj and vij = xi − xj.
Remark 1. For ̺(t) = bt (respectively, ̺(t) = c) these Poisson brackets appear in the classical
limit of the commutation relations (31) in the non-unitary (respectively, unitary) case (with the
parameterization βij = −
ψi − ψj
in the non-unitary case).
Remark 2. The strict rime linear Poisson brackets
{xi, xj} = aijxi − ajixj , aij 6= 0 for all i, j = 1, 2, . . . , n : i 6= j (201)
(or strict rime Lie algebras) are less interesting. The Jacobi identity is
aikakj = aijajk for all i 6= j 6= k 6= i . (202)
6To have nonvanishing coefficients in the formula (198) one has to impose certain inequalities for the components of
the vector ~ψ and the coefficients of the polynomial ̺; however, the formula (198) defines Poisson brackets without these
inequalities.
Rescale variables x2, x3, . . . , xn to have a1i = 1, i = 2, . . . , n. Then the condition (202) with one of i, j, k
equal 1 implies aij = aji and aij = ai1/aj1, i, j = 2, . . . , n; it follows that a
i1 = a
j1, i, j = 2, . . . , n. For
n > 3, the condition (202) with i, j, k > 1 forces ai1 = aj1, i, j = 2, . . . , n. Denote by ν this common
value, ai1 = ν, i, j = 2, . . . , n. After a rescaling x
1 7→ νx1 we find a unique strict rime Lie algebra,
[xi, xj ] = xi−xj for all i and k, which is almost trivial: [xi, xk −xl] = −(xk −xl) for all i, k and l and
[xi − xj , xk − xl] = 0 for all i, j, k, l.
For n = 3, there is one more possibility: a31 = −a21. After a rescaling x1 7→ a21x1, the solution
reads
[x1, x2] = x1 − x2 , [x1, x3] = x1 + x3 , [x2, x3] = −x2 + x3 . (203)
This Lie algebra is isomorphic to sl(2); the isomorphism is given, for example, by h 7→ x1 − x3,
e 7→ x1+x3 and f 7→ x2− (x1+x3)/4 (here h, e and f are the standard generators of sl(2), [h, e] = 2e,
[h, f ] = −2f and [e, f ] = h).
6.2 Invariance
In this subsection we analyze the invariance group of each individual member of the Poisson pencil
from the proposition 11. We find that the Poisson brackets (198), with arbitrary (non-vanishing) ̺,
admit a non-trivial 1-parametric invariance group.
The transformation law of Poisson brackets {xi, xj} = f ij(x) under an infinitesimal change of
variables, x̃i = xi + ǫ ϕi(x), ǫ2 = 0, is {x̃i, x̃j} = f ij(x̃) + ǫ δxf ij, where δxf ij = {ϕi, xj}+ {xi, ϕj} +
ϕk∂kf
ij. For a linear infinitesimal transformation, ϕi(x) = Aijx
j, we have
ij = Aik{xk, xj}+A
i, xk} − xlAkl ∂k{xi, xj} . (204)
Specializing to the Poisson brackets (198), we find
ij = Uji − Uij (205)
Uij :=
+Ajs(ψija−
̺i + ̺j
s:s 6=i
(xs)2 +
(xi)2 + (ψsia−
̺s + ̺i
)xixs
(206)
where ψij = ψi − ψj and ̺s = ̺(ψs).
The Poisson brackets (198) remain rime under the infinitesimal linear transformation with the
matrix A if the coefficients in (xs)2, xsxi and xsxj, s 6= i, j, in (205) vanish which gives the following
system:
(xs)2 , s 6= i, j ⇒ Ajs
= 0 , (207)
xixs , s 6= i, j ⇒ 2Ais
ψsja−
̺i + ̺j
− ̺s + ̺i
= 0 . (208)
Eq. (207) implies that Alk = νk̺l/ψlk, l 6= k, with arbitrary constants νk. For a quadratic polynomial
̺, this solves eq. (208) as well. The coefficient in xjxs vanishes due to the anti-symmetry.
The Poisson brackets (198) are invariant under the infinitesimal linear transformation with the
matrix A if, in addition to (207) and (208), the coefficients in (xi)2, (xj)2 and xixj in (205) vanish
which gives:
xixj ⇒ ̺jAij + ̺iA
i = 0 , (209)
(xi)2 ⇒ Aii
ψija−
̺i + ̺j
s:s 6=i
= 0 . (210)
Eq. (209) implies that νk are equal, νk = ν. The matrix A is defined up to a multiplicative factor,
so we can set ν to 1. Since the Poisson brackets (198) are quadratic, a global rescaling leaves them
invariant, so we can add to A a matrix, proportional to the identity matrix and make A traceless.
The traceless condition, together with eq. (210) determines the diagonal entries, Aii = a(n − 1)ψi +
b+ ̺i
s:s 6=i
. The coefficient in (xj)2 vanishes due to the anti-symmetry. We summarize the
obtained results.
Proposition 12. (i) The infinitesimal linear transformation with the matrix A leaves the Poisson
brackets (198) rime if and only if
Alk =
, l 6= k , (211)
with arbitrary constants νk.
(ii) Up to a global rescaling of coordinates, the invariance group of the Poisson brackets (198) is
1-parametric, with a generator A,
A(̺)ij =
, i 6= j , and A(̺)ii =
̺′i + ̺iξi , ξi :=
s:s 6=i
, (212)
where ̺′i is the value of the derivative of the polynomial ̺ at the point ψi.
Since the Poisson brackets transformed with the matrix (211) are still rime, it follows from the
proposition 11 that they can be written, after an appropriate rescalings of coordinates, in the form
(198). In other words, the variation δx can be compensated by a variation of ψ’s and ̺ and a diagonal
transformation of the coordinates. We have
− δxf ij = δ(1) + δ(2) , (213)
where
δ(1) =
̺i̺j(x
i − xj)2
(νi − νj) + a
νj̺j(x
i)2 − νi̺i(xj)2
(214)
δ(2) = (Ãii − Ã
i)2 − ̺i(xj)2
, Ãii := A
i − ̺′iνi −
s:s 6=i
. (215)
Choose Aii to set Ã
i to 0; this is a diagonal transformation of the coordinates. Then δ
(2) vanishes and
the variation of f ij is reduced to δ(1).
On the other hand, under a variation of ψ′s, ψi 7→ ψi + δψi, the Poisson brackets (198) transform
in the following way:
(xi − xj)2
(̺iδψj − ̺jδψi) + a
(xj)2δψi − (xi)2δψj
(216)
and we conclude that with the choice
δψi = ǫ̺iνi (217)
the variation δ(1) is compensated by the variation δψ. The coefficients of the polynomial ̺ stay the
same. In the next subsection we will study relations between the variation of ψ’s and the polynomial
Remark. With ξi as in (212), define three operators,
(B−)ij =
, i 6= j , and (B−)ii = −ξi , (218)
(B0)ij =
, i 6= j , and (B0)ii = −(
+ ψiξi) , (219)
(B+)ij =
, i 6= j , and (B+)ii = −((n− 1)ψi + ψ2i ξi) . (220)
The operators B+, B0 and B− generate an action of the Lie algebra sl(2),
[B0, B−] = −B− , [B0, B+] = B+ , [B+, B−] = −2B0 (221)
(to obtain the usual commutation relations for the generators of sl(2), change the sign of B+).
This is the usual projective action of sl(2) on polynomials f(t) of degree less than n,
B− : f(t) 7→ f ′(t) , B0 : f(t) 7→ tf ′(t)−
f(t) , B+ : f(t) 7→ t2f ′(t)− (n− 1)t f(t) , (222)
written in the basis of the non-normalized Lagrange polynomials, li(t) =
s:s 6=i
(t − ψs), at points
{ψi}, i = 1, 2, . . . , n. Indeed, in the basis {li(t)}, a polynomial f(t), deg(f) ≤ n − 1, takes the form
f ili, where f
i = li(ψi)
−1f(ψi). We have
l′i(t) =
a:a6=i
b:b6=a,i
(t− ψb) , so l′i(ψk) =
b:b6=k,i
ψkb = lk(ψk)
, k 6= i . (223)
Also,
l′i(t) = li(t)
s:s 6=i
t− ψs
, so l′i(ψi) = −li(ψi)ξi . (224)
Therefore, l′i(t) = −ξili(t) +
k:k 6=i
lk(t), which is exactly (218). For functions on the set of points
{ψi}, the operator of multiplication by t acts as a diagonal matrix Diag(ψ1, ψ2, . . . , ψn) and (219)-(220)
follow.
Define an involution ̟ on the space of matrices7,
̟(Y )ij = Y
j , i 6= j and ̟(Y )ii = −Y ii , Y ∈ Matn . (226)
B(̺) = aB+ + bB0 + cB− , B(̺) : f 7→ ̺(t)f ′(t)− n− 1
̺′(t)f(t) . (227)
In the basis {li(t)} for B, the generator (212) of the invariance group is
A(̺) = ̟(B(̺)) . (228)
Note that the operators ̟(B−), ̟(B0) and ̟(B+) do not form a Lie algebra.
6.3 Normal form
In this subsection we derive a normal form of each individual member of the Poisson pencil from the
proposition 11. It depends only on the discriminant of the polynomial ̺. When the discriminant of ̺
is different from zero, the Poisson brackets (198) are equivalent to the Poisson brackets defined by the
r-matrix (79). When the polynomial ̺ is different from zero but its discriminant is zero, the Poisson
brackets (198) are equivalent to the Poisson brackets defined by the r-matrix (103).
Under a variation of the polynomial ̺, ̺(t) 7→ (a + δa)t2 + (b + δb)t + (c + δc), we have for the
Poisson brackets (200):
u2ijδa+ uijvijδb+ v
ψi − ψj
. (229)
The variation of ̺ can be compensated by a variation (216) of ψ’s if the coefficients in (xj)2, xixj
and (xi)2 in the combination (δψ + δ̺)f
ij vanish, which gives the following system:
(xj)2 ⇒
̺iδψj − ̺jδψi
+ aδψi +
ψ2i δa+ ψiδb+ δc
= 0 , (230)
xixj ⇒ −
2(̺iδψj − ̺jδψi)
2ψiψjδa+ (ψi + ψj)δb + 2δc
= 0 . (231)
A combination 2×(230)+(231) gives
2aδψi + 2ψiδa+ δb = 0 . (232)
7The involution ̟ is the difference of two complementary projectors. The involution ̟ satisfies
̟(Y1)̟(Y2) +̟(Y1Y2) =
̟(̟(Y1)̟(Y2)) + Y1Y2 ,
̟(̟(Y1)Y2) + Y1̟(Y2) ,
̟(Y1̟(Y2)) +̟(Y1)Y2
(225)
for arbitrary Y1, Y2 ∈ Matn. All other linear dependencies between Y1Y2, ̟(Y1)Y2, Y1̟(Y2), ̟(Y1)̟(Y2), ̟(Y1Y2),
̟(̟(Y1)Y2), ̟(Y1̟(Y2)) and ̟(̟(Y1)̟(Y2)) are consequences of the three identities (225).
Substituting the expression (232) for δψ’s into (230) gives
δD(̺) = 0 , where D(̺) = b2 − 4ac . (233)
The coefficient in (xi)2 in (δψ + δ̺)f
ij vanishes due to the anti-symmetry.
Therefore, a necessary condition for a variation of ̺ to be compensated by a variation of ψ’s is that
the discriminant D(̺) does not vary. We shall now see that the discriminant is the unique invariant.
Explicitly, under a shift ψj 7→ ψj + ζ, we have uij 7→ uij + ζvij and vij 7→ vij (in the notation
(200)), which produces the following effect on the coefficients of the polynomial ̺:
a 7→ a , b 7→ b+ 2ζa , c 7→ c+ ζb+ ζ2a . (234)
A dilatation ψj 7→ λψj produces the following effect on the coefficients of ̺:
a 7→ λa , b 7→ b , c 7→ λ−1c . (235)
The inversion ψj 7→ ψ−1j accompanied by a change of variables x̃i = ψ
i produces the following
effect on the coefficients of ̺:
a 7→ −c , b 7→ −b , c 7→ −a . (236)
The set of operators (234) and (235) generates the action of the affine group on the space of the
polynomials ̺. The affine group, together with the inversion (236) generates an action8 of so(3) (the
spin 1 representation of sl(2)) on the space of the polynomials ̺ and the classification reduces to that
of orbits. The orbits (in the complex situation) of non-zero polynomials are of two types: ”massive”,
D(̺) 6= 0, or ”light-like”, D(̺) = 0. Particular representatives of both types appear in the Poisson
brackets, corresponding to the rime r-matrices (see the remark 1 after the proposition 11) and thus
to the r-matrices studied in subsections 4.1 and 4.3. We obtain the following statement.
Proposition 13. Let ̺(t) be a non-zero quadratic polynomial.
If the discriminant of ̺ is different from zero, D(̺) 6= 0, then there exists a change of the param-
eters ψi in the Poisson brackets (198) which sets ̺(t) to bt, ̺(t) 7→ bt; these are the Poisson brackets
corresponding to the r-matrix rCG (subsection 4.1).
If the discriminant of ̺ is zero, D(̺) = 0, then there exists a change of the parameters ψi in the
Poisson brackets (198) which sets ̺(t) to c, ̺(t) 7→ c; these are the Poisson brackets corresponding to
the r-matrix bCG (subsection 4.3).
The generator A(̺) of the invariance group can be easily described in both cases, D(̺) 6= 0 and
D(̺) = 0, in the parameter-free basis (that is, for the r-matrices rCG and bCG; in the rime basis the
generators are given by (52) and (55), respectively). For D(̺) 6= 0 (respectively, D(̺) = 0), it coincides
with the matrix of the operator B0 (respectively, B−), as in the remark in subsection 6.2, in the basis
{ti} of powers of the variable t. This implies somewhat unexpectedly that for an arbitrary polynomial
̺(t) the matrices A(̺) and ̟(A(̺)) are related by a similarity transformation. Note that in the basis
{ti} of powers, the operators aB++bB0+cB− and ̟(aB++bB0+cB−) are also related by a similarity
transformation for arbitrary a, b and c but here it is obvious: ̟(aB++bB0+cB−) = aB+−bB0+cB−,
so the operator ̟(aB+ + bB0 + cB−) belongs to sl(2) and moreover lies on the same (complex) orbit
as aB+ + bB0 + cB− with respect to the adjoint action.
8Let e+ be the generator of the 1-parametric group (234) and h the generator of the 1-parametric group (235). Denote
by I the inversion (236). The remaining generator e− is Ie+I.
7 Orderable quadratic rime associative algebras
Consider an associative algebra A defined by quadratic relations giving a lexicographical order. This
means that xjxk for j < k is a linear combination of terms xaxb with a ≥ b and either a > j or a = j
and b > k.
We shall say that such algebra A is rime if {a, b} ⊂ {j, k}. In other words, the relations in the
algebra are
xjxk = fjkx
kxj + gjkx
kxk , j < k . (237)
We shall classify the strict rime algebras A (that is, the algebras for which all coefficients fij and
gij are different from zero for i < j).
The only possible overlaps for the set of relations (237) are of the form (xjxk)xl = xj(xkxl),
j < k < l. The ordered form of the expression (xjxk)xl is
(xjxk)xl = fjkfjlfkl x
lxkxj + fjkfjlgkl x
lxlxj + f2klgjk x
lxkxk
+ (fklgjkgkl + f
kl(fjkgjl + gjkgkl))x
lxlxk
+ (fjkgjlgkl + gjkg
kl + fklgkl(fjkgjl + gjkgkl))x
lxlxl .
(238)
The ordered form of the expression xj(xkxl) is
xj(xkxl) = fjkfjlfkl x
lxkxj + f2jlgkl x
lxlxj + fklfjlgjk x
lxkxk
+ fklgjl x
lxlxk + (gklgjl + fjlgklgjl)x
lxlxl .
(239)
Equating coefficients, we find
xlxlxj : fjkfjlgkl = f
jlgkl , (240)
xlxkxk : f2klgjk = fklfjlgjk , (241)
xlxlxk : fklgjkgkl + f
kl(fjkgjl + gjkgkl) = fklgjl , (242)
xlxlxl : fjkgjlgkl + gjkg
kl + fklgkl(fjkgjl + gjkgkl) = gklgjl + fjlgklgjl . (243)
In the strict situation, eqs. (240) and (241) simplify, respectively, to
fjk = fjl , for j < k and j < l , (244)
fkl = fjl , for j < l and k < l . (245)
Eqs. (244) and (245) imply that fjk’s are all equal,
fjk =: f . (246)
The substitution of (246) into (242) gives (in the strict situation)
(f + 1)
gjkgkl + gjl(f − 1)
= 0 for j < k < l . (247)
Eq. (243) follows from (246) and (247).
We have thus two cases:
(i) f = −1 and no extra conditions on gjk’s;
(ii) f 6= −1 and
gjkgkl = (1− f) gjl for j < k < l ; (248)
1− f 6= 0 since gjk 6= 0 and gkl 6= 0.
In the case (ii), make an appropriate rescaling of generators, xi 7→ dixi to achieve
gi,i+1 = 1− f for all i = 1, . . . , n− 1 . (249)
It then follows from eq. (248) that
gij = 1− f for all i < j . (250)
We summarize the obtained results.
Proposition 14. Up to a rescaling of variables, the general orderable quadratic strict rime algebra
has relations
(i) either of the form
xjxk = −xkxj + gjkxkxk , j < k , (251)
with no conditions on the coefficients gjk;
(ii) or of the form
xjxk = fxkxj + (1− f)xkxk , j < k , (252)
with arbitrary f (it is strict when f 6= 0, 1).
By construction, the algebras of types (i) and (ii) possess a basis formed by ordered monomials
and thus have the Poincaré series of the algebra of commuting variables.
The algebra with defining relations (252) is the quantum space for the R-matrix (75). The relations
(252) can be written in the form
(xj − xk)xk = fxk(xj − xk) , j < k ; (253)
this is a quantization of the Poisson brackets
{xj , xk} = xk(xj − xk) , j < k . (254)
It would be interesting to know if the algebra with the defining relations (251) admits an R-matrix
description.
Acknowledgements
It is our pleasure to thank László Fehér, Alexei Isaev and Milen Yakimov for enlightening discussions.
The work was partially supported by the ANR project GIMP No.ANR-05-BLAN-0029-01. The second
author (T. Popov) was also partially supported by the Program “Bourses d’échanges scientifiques pour
les pays de l’Est européen” and by the Bulgarian National Council for Scientific Research project PH-
1406.
Appendix A. Equations
Here we give the list of the equations YB(R̂)
abc = 0 for the rime matrix
kl = αijδ
k + βijδ
l + γijδ
l + γ
l , (255)
with a convention αi = αii and βii = γii = γ
ii = 0.
The rime Ansatz implies that YB(R̂)
abc can be different from zero only if the set of lower indices is
contained in the set of upper indices. Therefore, the equations split into two lists: the first one with
two different indices among {i, j, k} and the second one with three different indices.
The full set of equations YB(R̂)
abc = 0 is invariant under the involution ι,
ι : αi ↔ αi , αij ↔ αji , βij ↔ βji , γij ↔ γ′ji , (256)
for if R̂ is a solution of the YBe then R̂21 = PR̂P is a solution of the YBe as well. We shall write
only the necessary part of the equations, the rest can be obtained by the involution ι.
The equations YB(R̂)
abc = 0 with two different indices are:
αijγij(γji + γ
ij) = 0 , (257)
αij(βijβji + γijγ
ij) = 0 = αij(βijβji − γijγji) , (258)
αijγij(αij + βji − αj) = 0 = αijγij(αji + βij − αj) , (259)
βij(α
i − αijαji − αiβij) + (αi − βij)γijγ′ij = 0 , (260)
(αi − αj)γ2ij + αijγij(γij + γ′ji) = 0 , (261)
αijβijγ
ji + (αiβij + γ
ijγij)γij = 0 , (262)
(αij − αji − βij + βji)γijγ′ji = 0 = (αij − αji − βij + βji)βijβji , (263)
ji(αj − αij) + βjiγij(αi − βji) + γij(βijβji + γjiγ′ji) = 0 , (264)
(α2i − αi(αji + βji) + βijβji − γijγji)γij = (α2i − αi(αij + βij) + βijβji − γ′ijγ′ji)γ′ji . (265)
The equations with three different indices {i, j, k} are:
(αij − αki − βij + βki)γijγ′ki = 0 , (266)
αij(βijβjk + βikβji − βikβjk) = 0 , (267)
αij(γijγjk + γik(βjk − βji)) = αij(γijγ′kj + γik(βkj − βij)) = 0 , (268)
(αijαji − αjkαkj)βik + βijβjk(βij − βjk) = 0 , (269)
(αi + βik − βji)βjiγik + γikγjiγ′ji + αik(γjkγ′ji + βjkγ′ki) = 0 , (270)
(αi + αij − αkj − βkj)γijγik − γ2ikγkj + γij(αikγ′ki − γijγ′kj) = 0 , (271)
(αi − βkj)βijγik + (βikβkj + γijγ′ij)γik + αikβijγ′ki − (βij − βik)γijγ′kj = 0 , (272)
αij(γijγjk + γik(αjk − αji))=αji(γijγjk + γik(αjk − αji))=αij(γijγ′kj + γik(αkj − αij))=0. (273)
Appendix B. Blocks
We analyze here the structure of 4×4 blocks of an invertible and skew-invertible rime R-matrix cor-
responding to two-dimensional coordinate planes.
We denote the matrix elements as in (5).
The skew-invertibility of a rime R-matrix imposes restrictions on its entries: in the line R̂i∗j∗ only
two entries can be non-zero, R̂
ji and R̂
jj; in the line R̂
∗i only two entries can be non-zero, R̂
ji and
ii . Therefore,
αij = 0 ⇒ γijγ′ij 6= 0 and γijγ′ij = 0 ⇒ αij 6= 0 . (274)
Dealing with a single block, this becomes especially clear: to skew invert a 4×4 block is the same
as to invert the matrix
α1 0 γ12 β12
0 0 α12 γ
γ′21 α21 0 0
β21 γ21 0 α2
, (275)
whose determinant is
(α12β12 − γ12γ′12)(α21β21 − γ21γ′21)− α1α2α12α21 . (276)
B.1 Solutions
Here we classify solutions which are neither ice nor strict rime. For an ice R-matrix, α12 6= 0 and
α21 6= 0. For a rime R-matrix, αij might vanish and we consider the cases according to the number of
αij ’s which can be zero.
1. Both α12 and α21 do not vanish, α12α21 6= 0.
If γ12γ21 6= 0 then by (257), γ′12γ′21 6= 0. This is strict rime.
If both γ12 = 0 and γ21 = 0 then eq. (259) implies (αji + βij − αj)γ′ji = 0; eq. (261) implies
(αi − αj + αji)γ′ji = 0 and eq. (262) implies βijγ′ji = 0. Combining these, we find γ′ij = 0, this is ice.
It is left to analyze the situation when only one of γ’s is different from zero, say γ12 6= 0 and
γ21 = 0. We have the following chain of implications:
(257) ⇒ γ′12 = 0 , (277)
(259) ⇒
β12 = α2 − α21 ,
β21 = α2 − α12 ,
(278)
(258) ⇒ (α2 − α12)(α2 − α21) = 0 , (279)
(260) ⇒
(α1 − α2)(α2 − α21)(α1 + α21) = 0 ,
(α1 − α2)(α2 − α12)(α1 + α12) = 0 ,
(280)
(261)&(265) ⇒
(α1 − α2 + α12)γ12 + α12γ′21 = 0 ,
(α1 − α2 + α21)γ′21 + α21γ12 = 0 .
(281)
Eqs. (262), (263) and (264) are satisfied. By the second line in (281), γ′21 6= 0.
Now the system of inequalities and equations is invariant under R̂ ↔ R̂21, so up to this trans-
formation we can solve eq. (279) by setting α21 = α2. Then, by (281), γ
21 = −γ12α2/α1, β’s are
expressed in terms of α’s by (278) and the remaining system for α’s reduces to a single equation
(α1 − α2)(α1 + α12) = 0. We obtain two solutions:
1a. α2 = α1; α1, α12 and γ12 are arbitrary non-zero numbers; we rescale the R-matrix to set
α1α12 = 1 and denote q = α1, γ = γ12:
R̂(q;γ) =
α1 0 0 0
γ12 0 α12 0
−γ12 α1 α1 − α12 0
0 0 0 α1
q 0 0 0
γ 0 q−1 0
−γ q q − q−1 0
0 0 0 q
. (282)
The R-matrix (282) is semi-simple if (and only if) q+q−1 6= 0 and it is then an R-matrix of GL(2)-type,
Spec(R̂) = {q, q, q,−q−1}. This solution is a specialization of (15)-(16).
1b. α12 = −α1; α1, α2 and γ12 are arbitrary non-zero numbers; we rescale the R-matrix to set
α1α2 = −1 and denote q = α1, γ = γ12/q:
R̂(q;γ) =
α1 0 0 0
γ12 0 −α1 0
−γ12α2/α1 α2 α1 + α2 0
0 0 0 α2
q 0 0 0
qγ 0 −q 0
q−1γ −q−1 q − q−1 0
0 0 0 −q−1
. (283)
The R-matrix (283) is semi-simple if (and only if) q+ q−1 6= 0 and it is then an R-matrix of GL(1|1)-
type, Spec(R̂) = {q, q,−q−1,−q−1}.
2. Assume that α12 = 0.
By the invertibility, β12β21 6= 0; by the skew-invertibility, γ12γ′12 6= 0; now eqs. (257) and (258)
imply β12β21 = γ12γ21, γ
12 = −γ21 and γ′21 = −γ12. Eq. (259) implies α2 = α1, β12 = α1 − α21 and
β21 = α1.
The rest is satisfied and we obtain a solution, in which α1, β12 and γ12 are arbitrary non-zero
numbers; we rescale the R-matrix to set α1β12 = −1 and denote q = α1, γ = γ12:
R̂(q;γ) =
α1 0 0 0
γ12 β12 0 −α1β12/γ12
−γ12 α1 − β12 α1 α1β12/γ12
0 0 0 α1
q 0 0 0
γ −q−1 0 1/γ
−γ q + q−1 q −1/γ
0 0 0 q
. (284)
The R-matrix (284) is semi-simple if (and only if) q+q−1 6= 0 and it is then an R-matrix of GL(2)-type,
Spec(R̂) = {q, q, q,−q−1}. This solution is a specialization of (15)-(16).
3. Finally, assume that α12 = α21 = 0.
By the invertibility, β12β21 6= 0; by the skew-invertibility, γ12γ′12γ21γ′21 6= 0; now eq. (261) implies
α2 = α1, eq. (263) implies β21 = β12; eq. (262) implies γ12γ
12 = γ21γ
21 = −α1β12; eq. (265) implies
that γ12γ21 can take three values: α
12 or (−α1β12).
The rest is satisfied and we obtain a solution, in which α1, β12 and γ12 are arbitrary non-zero
numbers; we rescale the R-matrix to set α1β12 = −1 and denote q = α1, γ = γ12:
R̂(q,ω;γ) =
α1 0 0 0
γ12 β12 0 −α1β12/γ12
−α1β12γ12/ω 0 β12 ω/γ12
0 0 0 α1
q 0 0 0
γ −q−1 0 1/γ
γ/ω 0 −q−1 ω/γ
0 0 0 q
, (285)
where ω = q2, 1, q−2. The R-matrix (285) is semi-simple if (and only if) q + q−1 6= 0 and it is then an
R-matrix of GL(1|1)-type, Spec(R̂) = {q, q,−q−1,−q−1}.
It follows from the analysis above that if γij 6= 0 in an invertible and skew-invertible rime R-matrix
then γ′ji 6= 0.
In each of the cases (282)-(285), the parameter γ 6= 0 can be set to an arbitrary (non-zero) value
by a diagonal change of basis. The R-matrices (282)-(285) are skew-invertible.
B.2 GL(2) and GL(1|1) R-matrices
1. In dimension 2, except the standard R-matrices of GL-type,
GL(2)
(q,p)
q 0 0 0
0 0 p 0
0 p−1 q − q−1 0
0 0 0 q
GL(1|1)
(q,p)
q 0 0 0
0 0 p 0
0 p−1 q − q−1 0
0 0 0 −q−1
, (286)
there are two non-standard one-parametric families of non-unitary R-matrices of the type GL(1|1):
the eight-vertex one,
R̂I(q) =
q − q−1 + 2 0 0 q − q−1
0 q − q−1 q + q−1 0
0 q + q−1 q − q−1 0
q − q−1 0 0 q − q−1 − 2
, (287)
and the matrix R̂(II) for which the matrix R = PR̂ can be given an upper-triangular form,
R̂II(q,ε) =
q 0 0 q + q−1
0 0 εq−1 0
0 εq q − q−1 0
0 0 0 −q−1
, (288)
where ε = ±1.
The R-matrices (286), (287) and (288) are semi-simple if (and only if) q + q−1 6= 0.
Up to the transformations R̂↔ R̂21 and R̂↔ R̂t (the transposition), basis changes and rescalings
R̂ 7→ c R̂ (where c is a constant), the complete list of semi-simple invertible and skew-invertible
R-matrices includes (see [16] for a description of all solutions of the Yang–Baxter equation in two
dimensions and [11] for the classification of GL(2)-type R-matrices), in addition to (286)-(288), the
one-parametric family of Jordanian solutions R̂
(h1:h2)
(h1:h2)
1 h1 −h1 h1h2
0 0 1 −h2
0 1 0 h2
0 0 0 1
(289)
(the Jordanian R-matrix is of GL(2)-type; it is unitary; the essential parameter is the projective vector
(h1 : h2)), as well as the permutation-like solution R̂
(a,b,c)
and one more solution R̂
(a,b,c)
1 0 0 0
0 0 a 0
0 b 0 0
0 0 0 c
0 0 0 a
0 1 0 0
0 0 1 0
a 0 0 0
. (290)
The R-matrix R̂
(a,b,c)
is Hecke when ab = 1 and c = ±1 and it is then standard (and unitary). The
R-matrix R̂
is Hecke when a2 = 1; it is then unitary and related to the standard R-matrix by a
change of basis with the matrix
Without the demand of semi-simplicity, the full list of invertible and skew-invertible R-matrices
contains two more solutions,
(h1:h2:
1 h1 h2 h3
0 0 1 h1
0 1 0 h2
0 0 0 1
, R̂(
′′′) =
1 0 0 1
0 0 −1 0
0 −1 0 0
0 0 0 1
. (291)
The essential parameter for the R-matrix R̂
(h1:h2:
is the projective vector (h1 : h2 :
h3). The
R-matrix R̂
(h1:h2:
is semi-simple if and only if h2 = −h1 and h3 = −h21; it then belongs to the
family (289) of Jordanian R-matrices.
2. For the R-matrices from the list above, the transformations R̂ ↔ R̂21, R̂ ↔ R̂t and R̂ ↔ R̂−1
partly overlap or reduce to parameter or basis changes. We shall write formulas for the Hecke R-
matrices only.
For the standard R-matrix R̂(q,p) := R̂
GL(2)
(q,p)
R̂t(q,p) = R̂(q,p−1) , (R̂(q,p))21 = (π ⊗ π)R̂(q,p)(π
−1 ⊗ π−1) , R̂−1
(q,p)
= (R̂(q−1,p−1))21 , (292)
where π =
For the standard R-matrix R̂(q,p) := R̂
GL(1|1)
(q,p)
R̂t(q,p) = R̂(q,p−1) , (R̂(q,p))21 = (π ⊗ π)R̂(−q−1,p)(π
−1 ⊗ π−1) , R̂−1
(q,p)
= (R̂(q−1,p−1))21 . (293)
For the non-standard GL(1|1) R-matrix R̂(q) := R̂I(q),
R̂t(q) = R̂(q) , (R̂(q))21 = R̂(q) , R̂
= (D ⊗D)R̂(q−1)(D ⊗D)−1 , (294)
where D =
For the non-standard GL(1|1) R-matrix R̂(q,ε) := R̂II(q,ε),
R̂t(q,ε) = (π̃ ⊗ π̃)(R̂(−q−1,−ε))21(π̃
−1 ⊗ π̃−1) , R̂−1
(q,ε)
= (R̂(q−1,ε))21 , (295)
where π̃ =
For the Jordanian R-matrix R̂(h1:h2) := R̂
(h1:h2)
R̂t(h1:h2) = (π ⊗ π)R̂(h2:h1)(π
−1 ⊗ π−1) , (R̂(h1:h2))21 = R̂(−h1:−h2) , R̂
(h1:h2)
= R̂(h1:h2) . (296)
B.3 Riming
We shall now identify the rime R-matrices (282)-(285).
1. GL(2)
The R-matrices (282) and (284) are related by a change of basis (the number in brackets refers to the
corresponding equation),
(282)
(q;γ)
T ⊗ T = T ⊗ T R̂(284)
(q;γ)
, T =
q −1/γ
. (297)
In turn, the R-matrix (282) is related to the standard R-matrix R̂
GL(2)
(q,q−1)
by a change of basis,
(282)
(q;γ)
T ⊗ T = T ⊗ T R̂GL(2)
(q,q−1)
, T =
q − q−1 0
. (298)
In the unitary situation (that is, q − q−1 = 0), the R-matrix R̂(282)
(q;γ)
belongs to the family of
Jordanian R-matrices.
Note that for the R-matrices (282) and (284), the left even quantum spaces are classical.
2. GL(1|1)
The R-matrix (283) is related to the R-matrix (285) with the choice ω = β212,
(283)
(q;γ)
T ⊗ T = T ⊗ T R̂(285)
(−q−1,q2;1) , T =
. (299)
We have
(285)
(q,1;γ)
T ⊗ T = T ⊗ T R̂I
, T =
γ −γτ
, where τ2 =
q − 1
q + 1
, (300)
(285)
(q,q2;γ)
T ⊗ T = T ⊗ T R̂II
(q,1)
, T =
γq−1 −γq−1
, (301)
(285)
(q,q−2;γ)
T ⊗ T = T ⊗ T (R̂II
(q,1)
)21 , T =
γq −γq
. (302)
In the unitary situation (that is, for q = ±1) only eq. (300) changes; but now different choices for
ω coincide.
3. Since the standard R-matrices are rime as well, we conclude that in dimension 2, all non-unitary
Hecke R-matrices fit into the rime Ansatz. When h1 = 0, the Jordanian R-matrix R̂
(0:h2)
is rime as
well. However, when h1 6= 0, the Jordanian R-matrix R̂
(h1:h2)
cannot be rimed. Indeed, assume that
h1 6= 0 and let A = (T ⊗ T )R̂
(h1:h2)
(T ⊗ T )−1 with some invertible matrix T . Then
(Det(T ))2A1112 = h1 (T
2 (Det(T )− h2 T 11 T 21 ) ,
(Det(T ))2A1121 = −h1 (T 11 )2 (Det(T ) + h2 T 11 T 21 ) ,
(Det(T ))2A2212 = h1 (T
2 (Det(T )− h2 T 11 T 21 ) ,
(Det(T ))2A1121 = −h1 (T 21 )2 (Det(T ) + h2 T 11 T 21 ) .
(303)
For an invertible T , the non-rime entries (303) of A cannot vanish simultaneously.
4. We remark also that all non-standard R-matrices of GL(1|1)-type are uniformly described by the
formula (285). The right quantum spaces for the R-matrix R̂
(285)
(q,ω;γ)
, with γ = 1, read
(R̂− q11⊗ 11)ijkl x
kxl = 0 :
(q + q−1)xy = x2 + y2 ,
(q + q−1)xy = ω−1x2 + ωy2 ;
(304)
(R̂ + q−111⊗ 11)ij
xkxl = 0 :
x2 = 0 ,
y2 = 0 .
(305)
Using the diamond lemma, it is straightforward to verify that the Poincaré series of the quantum
space (304) is of GL(1|1)-type if and only if ω = q−2, 1 or q2.
Appendix C. Rimeless triple
We sketch here a proof that the triple (94) cannot be rimed. Relations xiyj = R̂
kxl, where R̂ is
the R-matrix for the triple (94) with arbitrary multiparameters, read
xiyi = yixi , i = 1, 2, 3, 4 (306)
x1y2 =
y2x1 ,
x1y3 =
y3x1 ,
x1y4 =
y4x1 − rs
y3x2 ,
x2y1 =
y1x2 + (1− q−2)y2x1 ,
x2y3 =
y3x2 ,
x2y4 =
y4x2 ,
(307)
x3y1 =
y1x3 + (1− q−2)y3x1 ,
x3y2 =
y2x3 + (1− q−2)y3x2 ,
x3y4 =
y4x3 ,
x4y1 =
y1x4 + (1− q−2)y4x1 + 1
y2x3 ,
x4y2 =
y2x4 + (1− q−2)y4x2 ,
x4y3 =
y3x4 + (1− q−2)y4x3 .
(308)
The parameter q enters the characteristic equation for R̂, R̂2 = (1− q−2)R̂+ q−211⊗ 11; p, r and s are
the multiparameters. The only needed restriction is q2 6= 1.
Denote by 〈l(1), l(2)〉 a two-dimensional plane spanned by l(1) and l(2). We say that two linear forms
l(1) and l(2) (in four variables) form a rime pair if, for the ordering relations (306) and (307)-(308),
each product l(α)(x)l(β)(y), α = 1, 2, β = 1, 2, is a linear combination of l(1)(y)l(1)(x), l(1)(y)l(2)(x),
l(2)(y)l(1)(x) and l(2)(y)l(2)(x). If, in addition, l(α)(x)l(α)(y) is proportional to l(α)(y)l(α)(x) for α = 1
and 2, we say that l(1) and l(2) form a rime basis in the plane 〈l(1), l(2)〉. We call a plane rime if it
admits a rime basis.
Fork Lemma. Assume that l(1)(x) = x1 + a2x
2 + a3x
3 and l(4)(x) = b2x
2 + b3x
3 + x4 form a rime
pair for some a2, a3, b2 and b3. Then either a3b2 6= 0 and a2 = b3 = 0 or a2b3 6= 0 and a3 = b2 = 0.
If a3b2 6= 0 then
r = s = 1 , l(1)(x) = x1 + wx3 and l(4)(x) = x4 +
q − q−1
x2 , w 6= 0 is arbitrary . (309)
If a2b3 6= 0 then
, r = s , l(1)(x) = x1 + wx2 and l(4)(x) = x4 +
q − q−1
x3 , w 6= 0 is arbitrary . (310)
Moreover, if r = s = 1 and p 6= q−1 then the rime plane 〈l(1), l(4)〉 admits a unique, up to
rescalings, rime basis {l(1), l(4)}; if p = q−1 and r = s 6= 1 then the rime plane 〈l(1), l(4)〉 admits a
unique, up to rescalings, rime basis {l(1), l(4)}; if p = q−1 and r = s = 1 then any two independent
linear combinations of l(1) and l(4) form a rime basis in the plane 〈l(1), l(4)〉.
Proof. A straightforward calculation. �
Assume that a rime basis {x̃i} for the triple (94) exists, x̃i = Aijxj , the matrix Aij is invertible.
Rename the rime variables x̃i in such a way that the minor
A11 A
A41 A
is non-zero and A11A
4 6= 0;
normalize the variables x̃1 and x̃4 to have A11 = A
4 = 1. The plane 〈x̃1, x̃4〉 is, by definition, rime,
with a rime basis {x̃1, x̃4}.
Suppose that r = s = 1 and p 6= q−1 or p = q−1 and r = s 6= 1. Then, by Fork Lemma, the
rime basis in the plane 〈x̃1, x̃4〉 is, up to proportionality, unique, so we know the variables x̃1 and
x̃4. The variables x̃1 and x̃2 form a rime plane. Therefore, if the variable x̃2 contains x4 with a
non-zero coefficient then, by Fork Lemma, x̃2 must be proportional to x̃4, contradicting to the linear
independence of the variables x̃2 and x̃4. Similarly, the variable x̃2 cannot contain x1 with a non-zero
coefficient (the plane 〈x̃2, x̃4〉 is rime). Thus, x̃2 is a linear combination of x2 and x3. Same for x̃3:
it is a linear combination of x2 and x3. One of the variables, x̃2 or x̃3, say, x̃2, contains x2 with a
non-zero coefficient. Writing rime equations for the plane 〈x̃1, x̃2〉 in the case r = s = 1 and p 6= q−1
(for the plane 〈x̃2, x̃4〉 in the case p = q−1 and r = s 6= 1) quickly leads to a contradiction.
Therefore, if the relations (306) and (307)-(307) can be rimed then p = q−1 and r = s = 1. It
follows from Fork Lemma that
x̃4 = (q − q−1)c2c3x1 + c2x2 + c3x3 + x4
for some c2 and c3. The planes 〈x̃a, x̃4〉, a = 1, 2, 3, are rime. Subtracting from the variables x̃a the
variable x̃4 with appropriate coefficients, we find three linearly independent combinations
l(x) = d1x
1 + d2x
2 + d3x
3 , (311)
each forming a rime pair with x̃4. We must have: l(x)l(y) is a linear combination of l(y)l(x), l(y)x̃4,
ỹ4l(x) and ỹ4x̃4. It follows, after a straightforward calculation, that d2d3 = 0. Moreover, d2 = d3 = 0
is excluded by Fork Lemma. In the case d2 6= 0 and d3 = 0 (respectively, d3 6= 0 and d2 = 0), the rime
condition implies that d1 = (q − q−1)c2d3 (respectively, d1 = (q − q−1)c3d2). Thus, only two linearly
independent combinations (311) can form a rime pair with x̃4, the final contradiction.
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	From ice to rime
	Rime Yang–Baxter solutions.25cm
	Non-unitary rime R-matrices
	Unitary rime R-matrices
	Properties
	Rime and Cremmer–Gervais R-matrices
	Classical rime r-matrices.25cm
	Non-skew-symmetric case
	BD triples.
	Skew-symmetric case
	Bézout operators.25cm
	Non-homogeneous associative classical Yang–Baxter equation
	Linear quantization
	Algebraic meaning
	Rota–Baxter operators
	*-multiplication
	Rime Poisson brackets.25cm
	Rime pencil
	Invariance
	Normal form
	Orderable quadratic rime associative algebras.1cm
	Appendix A.  Equations
	Appendix B.  Blocks.25cm
	B.1  Solutions
	B.2  GL(2) and GL(1|1) R-matrices
	B.3  Riming
	Appendix C.  Rimeless triple
	References

  We replace the ice Ansatz on matrix solutions of the Yang-Baxter equation by
a weaker condition which we call "rime". Rime solutions include the standard
Drinfeld-Jimbo R-matrix. Solutions of the Yang--Baxter equation within the rime
Ansatz which are maximally different from the standard one we call "strict
rime". A strict rime non-unitary solution is parameterized by a projective
vector. We show that this solution transforms to the Cremmer-Gervais R-matrix
by a change of basis with a matrix containing symmetric functions in the
components of the parameterizing vector. A strict unitary solution (the rime
Ansatz is well adapted for taking a unitary limit) is shown to be equivalent to
a quantization of a classical "boundary" r-matrix of Gerstenhaber and
Giaquinto. We analyze the structure of the elementary rime blocks and find, as
a by-product, that all non-standard R-matrices of GL(1|1)-type can be uniformly
described in a rime form.
  We discuss then connections of the classical rime solutions with the Bezout
operators. The Bezout operators satisfy the (non-)homogeneous associative
classical Yang--Baxter equation which is related to the Rota-Baxter operators.
  We classify the rime Poisson brackets: they form a 3-dimensional pencil. A
normal form of each individual member of the pencil depends on the discriminant
of a certain quadratic polynomial. We also classify orderable quadratic rime
associative algebras.
  For the standard Drinfeld-Jimbo solution, there is a choice of the
multiparameters, for which it can be non-trivially rimed. However, not every
Belavin-Drinfeld triple admits a choice of the multiparameters for which it can
be rimed. We give a minimal example.

CPT-P49-2006
R-MATRICES IN RIME
Oleg Ogievetsky∗
Centre de Physique Théorique†, Luminy, 13288 Marseille, France
Todor Popov
Institute for Nuclear Research and Nuclear Energy,
Bulgarian Academy of Sciences, Sofia, BG-1784, Bulgaria
Abstract
We replace the ice Ansatz on matrix solutions of the Yang–Baxter equation by a weaker condition
which we call rime. Rime solutions include the standard Drinfeld–Jimbo R-matrix. Solutions of
the Yang–Baxter equation within the rime Ansatz which are maximally different from the standard
one we call strict rime. A strict rime non-unitary solution is parameterized by a projective vector
~φ. We show that in the finite dimension this solution transforms to the Cremmer–Gervais R-matrix
by a change of basis with a matrix containing symmetric functions in the components of ~φ. A strict
unitary solution (the rime Ansatz is well adapted for taking a unitary limit) in the finite dimension
is shown to be equivalent to a quantization of a classical ”boundary” r-matrix of Gerstenhaber and
Giaquinto. We analyze the structure of the elementary rime blocks and find, as a by-product, that
all non-standard R-matrices of GL(1|1)-type can be uniformly described in a rime form.
We discuss then connections of the classical rime solutions with the Bézout operators. The Bézout
operators satisfy the (non-)homogeneous associative classical Yang–Baxter equation which is related
to the Rota–Baxter operators. We calculate the Rota–Baxter operators corresponding to the Bézout
operators.
We classify the rime Poisson brackets: they form a 3-dimensional pencil. A normal form of each
individual member of the pencil depends on the discriminant of a certain quadratic polynomial.
We also classify orderable quadratic rime associative algebras
For the standard Drinfeld–Jimbo solution, there is a choice of the multiparameters, for which it
can be non-trivially rimed. However, not every Belavin–Drinfeld triple admits a choice of the
multiparameters for which it can be rimed. We give a minimal example.
∗On leave of absence from P.N. Lebedev Physical Institute, Theoretical Department, Leninsky prospekt 53, 119991
Moscow, Russia
†Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix–Marseille I, Aix–Marseille II et du Sud
Toulon – Var; laboratoire affilié à la FRUMAM (FR 2291)
http://arxiv.org/abs/0704.1947v3
Contents
1 From ice to rime 3
2 Rime Yang–Baxter solutions 6
2.1 Non-unitary rime R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Unitary rime R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3 Rime and Cremmer–Gervais R-matrices 9
4 Classical rime r-matrices 12
4.1 Non-skew-symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2 BD triples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Skew-symmetric case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Bézout operators 20
5.1 Non-homogeneous associative classical Yang–Baxter equation . . . . . . . . . . . . . . 21
5.2 Linear quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.3 Algebraic meaning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.4 Rota–Baxter operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5.5 ∗-multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6 Rime Poisson brackets 30
6.1 Rime pencil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
6.2 Invariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
6.3 Normal form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7 Orderable quadratic rime associative algebras 38
Appendix A. Equations 40
Appendix B. Blocks 41
B.1 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
B.2 GL(2) and GL(1|1) R-matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
B.3 Riming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Appendix C. Rimeless triple 47
References 48
1 From ice to rime
A well known class of solutions R̂ ∈End (V ⊗ V ), V is a vector space, of the Yang–Baxter equation
YB(R̂) = 0, where
YB(R̂) := (R̂⊗ 11)(11 ⊗ R̂)(R̂ ⊗ 11)− (11⊗ R̂)(R̂ ⊗ 11)(11 ⊗ R̂) , (1)
is characterized by the so called ice condition (see lectures [21] for details) which says that R̂
can be
different from zero only if the set of the upper and the set of the lower indices coincide,
kl 6= 0 ⇒ {i, j} ≡ {k, l}. (2)
We introduce the ”rime” Ansatz, relaxing the ice condition: the entry R̂
kl can be different from zero
if the set of the lower indices is a subset of the set of the upper indices,
6= 0 ⇒ {k, l} ⊂ {i, j} . (3)
Matrices for which it holds will be referred to as “rime” matrices. Figuratively, in the rime, in contrast
to the ice, situation, putting an apple and a banana in a fridge, there is a non-zero amplitude to find
next morning two apples instead (but never an apple and an orange).
The Yang–Baxter equation for a matrix R̂ is equivalent to the equality of two different reorderings
of xiyjzk, using xiyj = R̂
kxl, xizj = R̂
kxl and yizj = R̂
kyl, to the form z•y•x•. One of
advantages of the rime Ansatz is that only indices i, j and k appear in the latter expression. Another
advantage is that for fixed values of i and j, the elements x• and y• with these values of indices form
a subsystem.
A rime R-matrix has the following structure
= αijδ
+ βijδ
+ γijδ
l + γ
(no summation) . (4)
To avoid redundancy, fix βii = 0, γii = 0 = γ
ii. We denote by αi the diagonal elements R̂
ii, αi = αii.
Throughout the text we shall assume that the matrix R̂ is invertible which, in particular, implies that
αi 6= 0 for all i.
The order of growth of the number of unknowns in the Yang–Baxter system for a rime matrix is
n2, where n =dimV .
Arbitrary permutations and rescalings of coordinates preserve the rime condition.
The ice and rime matrices are made of 4× 4 elementary building blocks, respectively,
R̂ice =
α1 0 0 0
0 β12 α12 0
0 α21 β21 0
0 0 0 α2
and R̂rime =
α1 0 0 0
γ12 β12 α12 γ
γ′21 α21 β21 γ21
0 0 0 α2
. (5)
In appendix B we analyze the structure of the 4× 4 rime blocks.
We call a rime matrix strict if αijγij 6= 0 ∀ i and j, i 6= j. Note that strict rime matrices are
necessarily not ice.
Proposition 1. Let R̂ be a rime matrix (4). Then R̂ is a solution of the Yang–Baxter equation if it
is of the form
= (1− βji)δilδ
+ βijδ
+ γijδ
l − γjiδ
, (6)
where βij and γij satisfy the system
βijβji = γjiγij , (7)
βij + βji = βjk + βkj =: β , (8)
βijβjk = (βjk − βji)βik = (βij − βkj)βik , (9)
γijγjk = (βji − βjk)γik = (βkj − βij)γik . (10)
Proof. The Yang–Baxter system of equations YB(R̂)
abc = 0 for a rime matrix is given in the appendix
A. The subset (257) - (259) together with its image under the involution (256) reads
ij(γij + γ
ji) = 0 = αijγij(γji + γ
ij) , (11)
αij(βijβji + γijγ
ij) = 0 = αij(βijβji − γijγji) , (12)
ij(αij + βji − αi) = 0 = αijγij(αij + βji − αj) , (13)
ij(αji + βij − αi) = 0 = αijγij(αji + βij − αj) . (14)
These equations are implied by (and, in the strict rime situation, are equivalent to) the following
system
γ′ij = −γji , αij + βji = αi , αji + βij = αi , (15)
βijβji = γjiγij . (16)
One checks that other equations YB(R̂)
abc = 0, for which two indices among {i, j, k} are different,
follow from (15) and (16). The last two equations from (15) imply αi = αj for all i and j. As an
overall rescaling of a solution of the Yang–Baxter equation by a constant is again a solution of the
Yang–Baxter equation, we can, without loss of generality, set it to one,
αi = 1 . (17)
Eqs. (15) and (17) yield the form (6) of the matrix R̂ and eq. (7).
Using (15), we rewrite the subset (266) - (268) together with its image under the involution (256)
in the form
(βij + βji − βik − βki)γijγik = 0 , (18)
αij(βijβjk + βikβji − βikβjk) = 0 = αji(βjiβkj + βkiβij − βkiβkj) , (19)
αij(γijγjk + γik(βjk − βji)) = 0 = αji(γjiγkj + γki(βkj − βij)) . (20)
These equations are implied by (and, in the strict rime situation, are equivalent to) eqs. (8), (9) and
(10). One checks that other equations YB(R̂)
abc = 0 with three different indices {i, j, k} follow from
the system (7)–(10). The proof is finished. �
Lemma 1. The rime Yang–Baxter solution R (6) is of Hecke type,
R̂2 = βR̂+ (1− β)11⊗ 11 . (21)
Moreover, when β 6= 2, R is of GL-type: it has two eigenvalues 1 and β − 1 with multiplicities n(n+1)
n(n−1)
, respectively. When β = 2 the matrix R̂ has a nontrivial Jordanian structure.
Proof. In view of the block structure of rime matrices it is enough to check the Hecke relation (21)
for one elementary (4 × 4) block which follows from (7) and (8). When β 6= 2 the multiplicities m1
and mβ−1 are solutions of the system
m1 +mβ−1 = n
2 , m1 + (β − 1)mβ−1 = n+
n(n− 1)
β (≡ TrR̂) . (22)
When β = 2 the matrix R̂ has only one eigenvalue 1 but R̂ 6= 11⊗ 11 due to (7) and (8). �
Unitary solutions, R̂2 = 11⊗ 11, are singled out by the value of the parameter β = 0.
Lemma 2. A strict rime Yang–Baxter solution R (6) can be brought to a rime matrix
= (1 − βji)δilδ
+ βijδ
− βijδikδil + βjiδ
, (23)
that is, to a solution (6) with γij = −βij , by a change of basis.
Proof. The strict rime condition αijγij 6= 0 implies βijβji 6= 0 in view of (7). Thus for a strict rime
R-matrix all βij and γij are nonvanishing. The ratio of eqs. (9) and (10) is well-defined and it follows
from eqs. (7) and (8) that
γijγjk
βijβjk
(βji − βjk)γik
(βji − βjk)βik
= −γik
γijγji
βijβji
= 1 , (24)
ξijξjk = ξik ξijξji = 1 , (25)
where ξij = −
. Eq. (25) is solved by ξij =
with di 6= 0, i = 1, . . . , n, hence β’s and γ’s are
related by
γij = −
βij . (26)
A change of basis with a matrix D,
R̂ 7−→ (D ⊗D) R̂ (D−1 ⊗D−1) , (27)
where Dij = djδ
j , transforms R to the form (23). �
Under the strict rime condition, the Yang–Baxter system of equations (see appendix A) reduces
to eqs. (8) and (9). However, the matrix (23), where the parameters βij are subject to eqs. (8) and
(9), is a solution of the Yang–Baxter equation without a strict rime assumption.
Remark. Right and left even quantum spaces are defined by, respectively,
kxl = xixj , xjxiR
kl = xlxk ; (28)
right and left odd quantum spaces are defined by, respectively,
kξl = (β − 1)ξiξj , ξjξiRijkl = (β − 1)ξlξk . (29)
Assume that β 6= 2. The left even space is classical1 as well as the right odd space
[xi, xj ] = 0 , [ξ
i, ξj ]
= 0 , (30)
where [ , ] and [ , ]+ stand for the commutator and the anti-commutator. The relations for the right
even space are
[xi, xj ] + (βijx
i + βjix
j)(xi − xj) = 0 ; (31)
the relations for the left odd space read
(2− β)ξ2i + ξiρ+ (1− β)ρξi = 0 , (32)
[ξi, ξj ]+ − βijξiξj − βjiξjξi = 0 , i 6= j , (33)
where ρ =
j ξj.
2 Rime Yang–Baxter solutions
In this section we solve eqs. (8) and (9) thus obtaining explicitly rime Yang–Baxter solutions.
2.1 Non-unitary rime R-matrices
Proposition 2. The non-unitary strict rime Yang–Baxter solutions (23) with a parameter β = βji+
βij 6= 0 are parameterized by a point φ ∈ PCn in a projective space, φ = (φ1 : φ2 : . . . : φn), such that
φi 6= 0 for all i and φi 6= φj for all i and j, i 6= j. These solutions are given by
βij =
φi − φj
. (34)
Proof. Taking the ratio of the following pairs of equations from (9)
βijβjk = (βjk − βji)βik , βkjβji = (βji − βjk)βki (35)
we find that quantities ηij = −βij/βji verify equations
ηijηjk = ηik , ηijηji = 1 , (36)
1Let R̂ be a rime R-matrix (not necessarily strict). When β 6= 2, the following statement holds. If (i) the left even
space is classical (which implies that γ′ij = −γji, αij + βji = 1 and αi = 1 in our normalization) and (ii) the R-matrix
is Hecke (which implies that βij + βji = β) then the system of equations from the appendix A again reduces to (7), (9)
and (10) as in the strict rime situation.
whose solutions are ηij = φi/φj for some constants φi 6= 0, i = 1, . . . , n.
Substituting the relation βji = −
βij into β = βij + βji, we obtain βij −
βij = β which
establishes (34). �
Remark. There is a different parameterization, βij = −
φi − φj
, of strict rime solutions; it is related
to the parameterization (34) by φi 7−→ (φi)−1.
A direct check shows that the condition φi 6= 0 is not necessary: the formula (34) with φi 6= φj for
all i and j, i 6= j, gives a rime solution of the Yang–Baxter equation. However when one of φi is 0,
the matrix (23) is no more strict.
2.2 Unitary rime R-matrices
For a unitary strict rime Yang–Baxter solution (23), R̂2 = 11, we have β = 0, so βij = −βji.
Proposition 3. The unitary strict rime Yang–Baxter solutions (23) are parameterized by a vector
(µ1, . . . , µn) such that µi 6= µj,
βij =
µi − µj
. (37)
Proof. Since βij = −βji we can rewrite βijβjk = (βjk − βji)βik as βijβjk = (βij + βjk)βik or
. (38)
These equations are solved by
= µi − µj , (39)
which is equivalent to (37). �
Remark. The unitary R-matrices of Proposition 3 can be obtained as a limit β → 0 of the non-
unitary R-matrices of Proposition 2. Indeed, for the following expansion of the parameters φi in the
“small” parameter β,
φi = 1 + βµi + o(β) , (40)
the expression (34) has a limit (37),
βij =
β(1 + βµi + o(β))
βµi − βµj + o(β)
β→0−→ βij =
µi − µj
. (41)
2.3 Properties
1. Denote the R-matrix (23) with βij as in (34) by R̂(~φ ). Let R̂21 = PR̂12P , where P is the
permutation operator. Then the following holds:
R̂21(~φ ) = F
−1 ⊗ F−1 R12(~φ−1) F ⊗ F , (42)
where F = diag(φ1, φ2, . . . , φn) and ~φ
−1 is a vector with components φ−1i .
Denote the R-matrix (23) with βij as in (37) by R̂(~µ ). Then the following holds:
R̂21(~µ) = R̂12(−~µ) . (43)
2. The R-matrix (23) is skew invertible in the sense that there exists an operator Ψ̂R, which satisfies
(see, e.g. [21])
Tr2(R̂12(Ψ̂R)23) = P13 . (44)
The matrices of the left and right quantum traces (that is, the left and right traces of the skew inverse
Ψ̂R), (QR)1 = Tr2((Ψ̂R)12) and (Q̃R)2 = Tr1((Ψ̂R)12), are given by the formulas
j = −βjk
l: l 6=k
(1− βjl) , k 6= j , and (QR)jj =
(1− βjl) ; (45)
(Q̃R)
j = βjk
l: l 6=k
(1− βlj) , k 6= j , and (Q̃R)jj =
(1− βlj) . (46)
The matrices QR and Q̃R satisfy QRQ̃R = (1− β)n−111.
For (34), one has Spec QR = Spec Q̃R = {(1 − β)a, a = 0, . . . , n − 1}. The eigenvector wa(~φ)
of the matrix QR with the eigenvalue (1 − β)n−1−a coincides with the eigenvector of the matrix Q̃R
with the eigenvalue (1−β)a. One has (wa(~φ))j = eĵa(~φ), where eĵi (~φ) is the i-th elementary symmetric
function of (φ1, φ2, . . . , φn) with φj omitted.
For (37), the Jordanian form of the matrix QR, as well as of Q̃R, is non trivial: it is a single block.
In the basis {wi(~µ)}, i = 0, 1, 2, . . . , n− 1, where (wi(~µ))j = eĵi (~µ), one has
QR wi(~µ) =
n− 1− s
ws(~µ) . (47)
3. For an R-matrix R̂, the group of invertible matrices Y satisfying
R̂12Y1Y2 = Y1Y2R̂12 (48)
form the invariance group GR of R̂. The matrices QR and Q̃R belong to the invariance group as well
as the matrices proportional to the identity matrix. One can write down formulas for the group GR
for a rime R-matrix (23) uniformly in terms of βij as in (45) and (46) but the properties are different
in the non-unitary and unitary cases and we describe them separately.
3a. The invariance group G
R(~φ )
for the R-matrix R̂(~φ ) is 2-parametric. It consists of matrices
Y (u, v), u, v 6= 0, where
Y (u, v)
l:l 6=j
uφj − vφl
φj − φl
and Y (u, v)ij =
(u− v)φj
φj − φi
l:l 6=i,j
uφj − vφl
φj − φl
, i 6= j . (49)
One has
R(~φ )
= Y (1− β, 1) , Q̃
R(~φ )
= Y (1, 1 − β) . (50)
The composition law is the component-wise multiplication of the parameters {u, v},
Y (u1, v1)Y (u2, v2) = Y (u1u2, v1v2) . (51)
The point u = v = 1 corresponds to the identity matrix, Y (1, 1) = 11; the determinant of Y (u, v)
is (uv)n(n−1)/2; u = v corresponds to global rescalings; the connected component of unity of the
subgroup SG
R(~φ )
consisting of matrices with determinant 1 is uv = 1; the generator η of the connected
component of unity of the subgroup SG
R(~φ )
is traceless and reads
ηij =
φj − φi
, i 6= j , and ηjj = −
l:l 6=j
φj − φl
. (52)
3b. For the R-matrix R̂(~µ ), the group SGR(~µ ), consisting of matrices with determinant 1 is
1-parametric as well. It is formed by matrices Y (0)(a), where
Y (0)(a)
l:l 6=j
µj − µl
) and Y (0)(a)ij =
µj − µi
l:l 6=i,j
µj − µl
) , i 6= j . (53)
The expression (53) can be obtained by taking a limit of (49), similarly to (41) and letting additionally
u = 1 + aβ/2 + o(β) and v = 1− aβ/2 + o(β).
One has
QR(~µ ) = Y
(0)(−1) , Q̃R(~µ ) = Y (0)(1) . (54)
The composition law is Y (0)(a1)Y
(0)(a2) = Y
(0)(a1 + a2).
The point a = 0 in (53) corresponds to the identity matrix, Y (0)(0) = 11; the generator η(0) of the
invariance group SGR(~µ ) is
(η(0))ij =
µj − µi
, i 6= j , and (η(0))jj =
l:l 6=j
µj − µl
. (55)
3 Rime and Cremmer–Gervais R-matrices
The Cremmer–Gervais R-matrix arises in the exchange relations of the chiral vertex operators in
the non-linearly W -extended Virasoro algebra [6]. The Cremmer–Gervais solution [6] of the Yang–
Baxter equation in its general two-parametric form reads (see, e.g. [17]; we use a rescaled matrix with
eigenvalues 1 and −q−2)
(R̂CG,p)
kl = q
−2θijpi−jδilδ
k + (1− q
s: i≤s<j
pi−sδskδ
i+j−s
l − (1− q
s: j<s<i
pi−sδskδ
i+j−s
l , (56)
where θij is the step function (θij = 1 when i > j and θij = 0 when i ≤ j).
The parameter value p = q2/n specifies the SL(n) Cremmer–Gervais R-matrix (its diagonal twist
being the GL(n) solution (56)). The Cremmer–Gervais solution is a non-diagonal twist of the standard
Drinfeld–Jimbo solution [18, 9].
Let R̂CG := R̂CG,1, that is, the solution (56) with p = 1. The matrix
D(p)ij = δ
i−1 (57)
with arbitrary p satisfies (R̂CG)12D(p)1D(p)2 = D(p)1D(p)2(R̂CG)12. It was observed in [10] that if
R̂12D1D2 = D1D2R̂12 for some R-matrix R̂ and operator D then D1R̂12D
1 is again an R-matrix
(this operation was also used in [15] to partially change the statistics of ghosts in the super-symmetric
situation). The two-parametric matrix R̂CG,p (56) can be obtained from the Cremmer–Gervais matrix
R̂CG by this operation as well,
(R̂CG,p)12 = D(p)1(R̂CG)12D(p)
1 . (58)
Let R̂ be the non-unitary rime matrix from Proposition 2 with φi 6= φj.
Proposition 4. The matrix R̂ transforms into the Cremmer–Gervais solution R̂CG
R̂ = (X ⊗X) R̂CG (X−1 ⊗X−1) (59)
by a change of basis with the invertible matrix
Xkj = ej−1 (φ1, . . . , φ̂k, . . . , φn) =: e
j−1 (60)
whose inverse is
(X−1)ji =
(−1)j−1φn−ji
k:k 6=i
(φi − φk)
. (61)
Here the hat over φj means that this entry is omitted in the expression and ei are the elementary
symmetric polynomials ei(x1, . . . , xN ) =
s1<...<si
xs1xs2 . . . xsi . The projective parameters (φ1 : φ2 :
. . . : φn) are removed by the transformation X and the only essential parameter β in R̂ is related to
the parameter q in R̂CG by
q−2 = 1− β . (62)
Proof. Due to the Lagrange interpolation formula, the matrix, inverse to the Vandermonde matrix
||V jk ||
j,k=1 = φ
k is (V
−1)kj =
(−1)j−1ek̂j−1
l:l 6=k
(φk − φl)
. (63)
The matrix X (60) has the form X = DV −1 d, where Dmk = δ
l:l 6=k(φk − φl) and dij = (−1)j−1δij
are diagonal n× n matrices. Thus, its inverse is X−1 = d−1 V D−1, which establishes (61).
We now prove the matrix identity (59) in the form
R̂(X ⊗X) = (X ⊗X) R̂CG . (64)
The substitution of the explicit form of the rime matrix R̂ (23) with βij = βφi/(φi − φj) and R̂CG
(56) reduces (64) to a set of relations between the symmetrical polynomials eâk−1
l−1 =
eîa−1e
b−1(R̂CG)
kl . (65)
There are two subcases: i) i = j and ii) i 6= j.
i) The left hand side of eq. (65) with i = j is just eîk−1e
l−1 due to the rime condition. Eq. (65) is
satisfied because of the symmetry relation (R̂CG)
kl = δ
l + δ
k − (R̂CG)bakl .
ii) For i 6= j eq. (65), where q−2 = 1− β, reduces, after some algebraic manipulations, to
φi − φj
k−1 − φje
k−1)(e
l−1 − e
s: s≥max(1,k−l+2)
(eîl+s−2e
k−s − e
l+s−2e
k−s) , 1 ≤ i, j, k, l ≤ n .
In fact, the sum in the right hand side goes till s = min(k, n + 1 − l) since eĵr = 0 when r ≥ n − 1;
moreover we can start the summation from s = 1 because when 1 < k−l+2 the sum for 1 ≤ s ≤ k−l+1
is anti-symmetric under s←→ k − l + 2− s and thus vanishes.
To prove (66) we write er = e
r + φie
r−1; therefore e
r = e
r + φje
r−1 and e
r = e
r + φie
r−1 and eq.
(66) becomes
− (φi − φj)eîĵk−1e
l−2 = (φi − φj)
l+s−2e
k−s−1 − e
l+s−3e
k−s) . (67)
The sum in the right hand side telescopes to the value of (−eîĵl+s−3e
k−s) at s = 1, that is, to (−e
l−2).
The proof is complete. �
It should be noted that the matrix X = X(~φ ) does not depend on q. The change of the basis with
the matrix X(~φ′ )X(~φ )−1 transforms the R-matrix R̂(~φ ) to R̂(~φ′ ). We have
(X(~φ′ )X(~φ )−1)ij =
φj − φ′i
(φj − φ′k)
l:l 6=j
(φj − φl)
The structure of the matrices X andX−1 shows that when the dimension is infinite, the R-matrices
R̂CG,1 and R̂(~φ ) (as well as the R-matrices R̂(~φ ) and R̂(~φ′ ) for different φ and φ
′) are in general not
equivalent.
The right even quantum plane for the Cremmer–Gervais matrix R̂CG,1 is defined by the following
equations
yiyj = q2yjyi + (q2 − 1)(yi+1yj−1 + . . .+ yj−1yi+1), i < j . (69)
If i+ 1 < j − 1, one uses the formula (69) recursively to get the ordering relations.
The change of basis with the matrix X,
eîj−1y
j , (70)
transformes the quantum plane (69) into the rime quantum plane (31) exhibiting coordinate two-
dimensional subplanes. The change of basis (70) can be written in terms of a ”generating function”:
ej(φ1, . . . , φn) y
j . (71)
. (72)
Remark. The standard Drinfeld–Jimbo R-matrix admits, for a certain choice of multi-parameters, a
different rime form. The relations uivj = (R̂c)
kul for this choice are
uivi = viui ,
uivj = vjui + (1− q−2)viuj , i < j ,
uivj = q−2 vjui , i > j .
The left even space for this R-matrix is classical.
The change of variables with the matrix X̃ij = 1− θji,
U i := u1 + u2 + · · ·+ ui , V i := v1 + v2 + · · ·+ vi , (74)
transforms the relations (73) into
U iV i = V iU i ,
U iV j = V jU i + (1− q−2)V iU j − (1− q−2)V iU i , i < j ,
U iV j = q−2 V jU i + (1− q−2)V jU j , i > j .
The matrix X, defined by eq. (60), degenerates if φi = φj for some i and j. Interestingly,
the R-matrix X ⊗XR̂c X−1 ⊗X−1 admits limits limφσ(2)→0 limφσ(3)→0 . . . limφσ(n)→0 for an arbitrary
permutation σ ∈ Sn and the result is always rime. In particular,
X̃ ⊗ X̃R̂c X̃−1 ⊗ X̃−1 = lim
. . . lim
X ⊗XR̂c X−1 ⊗X−1 . (76)
4 Classical rime r-matrices
The classical limit of an R-matrix is a classical r-matrix, a solution of the classical Yang–Baxter (cYB)
equation
[r12, r13] + [r12, r23] + [r13, r23] = 0 . (77)
We are going to show that the classical limits of the rime R-matrices from Section 2 are equivalent
to the Cremmer-Gervais r-matrices in the non-skew-symmetric case and to the ”boundary” r-matrix
of Gerstenhaber and Giaquinto [14] (see also [4]; this r-matrix is attributed to A. G. Elashvili there)
in the skew-symmetric case. Similar equivalences appeared in the study of the gauge transformations
of the dynamical r-matrices in the Calogero-Moser model [12, 13] 2.
In the sequel we use the following conventions. An R-matrix acts in a space V ⊗ V . A basis of V
is {ei} (labeled by a lower index); an operator A in V has matrix coefficients Aji , A(ei) = A
i ej , so for
a vector ~v = viei one has (A~v)
i = Aij~v
j ; the matrix units are eij , e
j(ek) = δ
kej , so the multiplication
rule is eije
l = δ
j ; eαi are the sl(n) simple positive root elements, eαi = e
i ; P is the permutation
operator, P (ei ⊗ ej) = ej ⊗ ei, so P (eij ⊗ ekl ) = eil ⊗ ekj and (PB)klij = Blkij for an operator B in V ⊗ V
having matrix coefficients Bklij , B(ei ⊗ ej) = Bklij ek ⊗ el.
4.1 Non-skew-symmetric case
Proposition 5. The non-unitary rime R-matrix (Proposition 2) is a quantization of the non-skew-
symmetric r-matrix
i,j:i 6=j
φi − φj
(eij ⊗ e
i − e
i ⊗ e
j + e
i ∧ eij) , (78)
where x ∧ y := x ⊗ y − y ⊗ x. The change of basis with the matrix Xjk = ek−1 (φ1, . . . , φ̂j , . . . , φn)
transforms r into the parameter-free cYB solution rCG
rCG =
i,j:i<j
(ei+s−1j ⊗ e
j−s+1
i − e
i+s−1
i ⊗ e
j−s+1
j ) . (79)
Proof. The coefficients βij (34) are linear in the deformation parameter β (β = 0 is the classical
point). Hence
R = 11⊗ 11 + βr , (80)
where R = PR̂ and r is given by (78).
The matrix RCG−11⊗11, where RCG = PR̂CG, is linear with respect to the parameter β = 1−q−2
as well,
RCG = 11⊗ 11 + β rCG (81)
thus the formula (59) implies r = (X ⊗X) rCG (X−1 ⊗X−1). �
We mentioned two ways of obtaining the numerical two-parametric R-matrix (R̂CG,p) from the R-
matrix (R̂CG,1): by a diagonal twist and by the operation (58). There is one more way which consists
of changing the representation. We shall illustrate it on the example of the classical GL r-matrix (79).
A change of representation of the Lie algebra GL,
eij 7→ eij + c δij11 , (82)
2We thank László Fehér for drawing our attention to the references [12, 13].
where c is a constant, produces the following effect on the r-matrix (79):
rCG 7→ rCG + c
η ⊗ 11− 11⊗ η − (n− 1)11 ⊗ 11
, (83)
where n =dimV and
η = −
n(n+ 1)
j , tr η = 0 . (84)
The classical version of the operation (58) is as follows. Let η be an arbitrary generator of the
invariance group of an r-matrix r,
[r, η1 + η2] = 0 . (85)
Then the operator
r(c) = r + c(η1 − η2) , (86)
where c is a constant, is again a classical r-matrix (a solution of the cYBe).
The operator η in (84) is, up to a scale, the unique traceless generator of the invariance group
(see (57)) of the r-matrix (79). Thus, the representation change and the operation (86) give the same
family of r-matrices (up to an addition of a multiple of the identity operator, which does not violate
the cYBe).
4.2 BD triples.
Each block in the strict rime classical r-matrix (78) looks even more ”rimed”,
0 0 0 0
β′12 −β′12 β′21 −β′21
−β′12 β′12 −β′21 β′21
0 0 0 0
, (87)
where β′ij = βij/β = φi/(φi − φj). The multiplication from the left by P acts on each block as a
permutation of the second and third lines, so the rime r-matrix (87) enjoys the symmetry Pr = −r.
We shall now discuss this symmetry property in the context of Belavin–Drinfeld triples.
In [3] Belavin and Drinfeld gave, for a simple Lie algebra g, a description of non-unitary (non-
skew-symmetric) cYB solutions r ∈ g⊗ g, satisfying r12 + r21 = t, where t ∈ g ⊗ g is the g-invariant
element. The non-unitary solutions are put into correspondence with combinatorial objects called
Belavin–Drinfeld triples (BD-triples for short). The Belavin–Drinfeld triple (Π1,Π2, τ) for a simple
Lie algebra g consists of the following data: Π1,Π2 are subsets of the set of simple positive roots Π
of the algebra g and τ is an invertible mapping: Π1 → Π2 such that 〈τ(ρ), τ(ρ′)〉 = 〈ρ, ρ′〉 for any
ρ, ρ′ ∈ Π1 and τk(ρ) 6= ρ for any ρ ∈ Π1 and any natural k for which τk(ρ) is defined.
The r-matrix for a triple (Π1,Π2, τ) has the form
r = r0 +
e−α ⊗ eα +
α,β∈∆+:α<β
e−α ∧ eβ , (88)
where < is a partial order on the set of positive roots ∆+ defined by the rule: α < β for α, β ∈ ∆+
if there exists a natural k such that τk(α) = β. The part r0 belongs to h⊗ h, where h is the Cartan
subalgebra of g; r0 contains continuous ”multiparameters”, which satisfy
(τ(α) ⊗ id + id⊗ α)(r0) = 0 for all α ∈ Π1 . (89)
We are dealing with matrix solutions r of the cYB equation, r ∈ gl(n)⊗ gl(n), so r12 + r21 can be
a linear combination of P and 11⊗ 11.
Let Π = {α1, . . . , αn−1} be the set of the positive simple roots for the Lie algebra sl(n).
There are two Cremmer–Gervais BD triples, T+ and T−. For the Cremmer–Gervais triple BD-
triple T+, Π1 = {α1, α2, . . . , αn−2}, Π2 = {α2, α3, . . . , αn−1} and τ(αi) = αi+1. The data (Π1,Π2, τ)
is encoded in the graph
. . . •
• • • • . . . • •
The triple T− can be obtained from the triple T+ either by setting Π
1 = Π2, Π
2 = Π1 and
τ ′ = τ−1 or by applying the outer automorphism of the underlying An−1 Dynkin diagram; the graph
corresponding to the triple T− is
. . . •
• • • . . . • • •
The r-matrix (79) corresponds to the triple (90) for a certain choice of the multiparameters. Here is
the r-matrix r′ corresponding to the triple (91)
r′CG =
i,j:i<j
(eij−s+1 ⊗ e
i+s−1 − e
j−s+1 ⊗ e
i+s−1) (92)
for a certain choice of the multiparameters, for which it satisfies r′P = −r′.
For the r-matrices (79) and (92), one has r12+ r21 = P −11⊗11. The Cartan part of the r-matrices
(79) and (92) are
r0 = −
i,j:i<j
eii ⊗ e
j , r
0 = −
i,j:i<j
j ⊗ e
i . (93)
The following lemma shows that a classical r-matrix r for a triple T can have a symmetry with
respect to the multiplication by P from one side if and only if all segments (connected components)
of Π1 are mapped by τ according to either (90) or (91).
Lemma 3. A non-skew-symmetric classical r-matrix with a Belavin–Drinfeld data (Π1,Π2, τ) can
satisfy Pr = −r (respectively, rP = −r) for a certain choice of the multiparameters if and only if
τ(αi) = αi+1 (respectively, τ(αi) = αi−1) for all i ∈ Π1.
Proof. Assume that τ(αm) = αm+k for some natural k, k ≥ 1. Then r contains the term em+km+k+1∧e
with the coefficient 1. Such r-matrix cannot satisfy rP = −r for if rP = −r then r contains the term
m+k+1
∧ em+km with the coefficient (−1) but the coefficient in e−α ∧ eβ is 1 in the formula (88).
If Pr = −r then r should contain also the term em+1m+k+1 ∧ e
m . It then follows that
(i) the Lie subalgebra generated by Π1 contains e
m therefore the interval [αm, αm+1, . . . , αm+k−1]
is contained in Π1;
(ii) the Lie subalgebra generated by Π2 contains e
m+k+1
therefore the interval [αm+1, αm+2, . . . , αm+k]
is contained in Π2;
(iii) the image of the interval [αm, αm+1, . . . , αm+k−1] under τ is the interval [αm+1, αm+2, . . . , αm+k].
This implies that the interval [αm+1, αm+2, . . . , αm+k−1] is τ -invariant (since τ(αm) = αm+k) which
contradicts to the nilpotency of τ unless this interval is empty, that is, k = 1.
Similarly, rP = −r is possible only if τ(αi) = αi−1 for all i ∈ Π1.
It is left to show that when τ(αi) = αi+1 (respectively, τ(αi) = αi−1) for all i ∈ Π1 the multipa-
rameters can indeed be adjusted to fulfill Pr = −r (respectively, rP = −r). We leave it as an exercise
for the reader to check that with the assignment (93) for r (respectively, for r′) the compatibility
condition (89) is verified. The proof is finished. �
Remark. Two extreme BD triples can be rimed, the empty (Drinfeld–Jimbo) one and the “maximal”
Cremmer–Gervais one. However, not every triple can be rimed: already the triple
O • •
• • •
provides a counterexample. We outline a computer-aided proof in appendix C.
4.3 Skew-symmetric case
A skew-symmetric classical r-matrix r ∈ g ∧ g is canonically associated with a quasi-Frobenius Lie
subalgebra (f, ω) of g (see, e.g., [24]). A Lie algebra f which admits a non-degenerate 2-cocycle ω is
called quasi-Frobenius; it is Frobenius if ω is a coboundary, i.e., ω(X,Y ) = λ([X,Y ]) for some λ ∈ f∗.
We describe now the skew-symmetric r-matrix arising in the classical limit of the unitary rime
R-matrix from Proposition 3.
Proposition 6. The unitary rime R-matrix (Proposition 3) is a quantization of the skew-symmetric
r-matrix
i,j:i<j
µi − µj
(eij − e
j) ∧ (e
i − e
i) ∈ gl(n) ∧ gl(n) . (95)
This skew-symmetric classical r-matrix corresponds to a Frobenius Lie algebra (g0(n), δλn) spanned by
the generators Zij := e
j − e
j , i 6= j, with the Frobenius structure determined by the coboundary of the
1-cochain λn = −
i,j:i 6=j
j , where {zij}, i 6= j, is the basis in g∗0(n), dual to the basis {Zij} in g0(n),
zij(Z
l ) = δ
Proof. An artificial introduction of a small parameter c by a rescaling µi 7→ c−1µi in the formula for
the R-matrix R̂ in Proposition 3 gives
R = 11⊗ 11 + c r , (96)
where r is given by (95).
The n(n− 1) matrices Zij := eij − e
j , i 6= j, form an associative subalgebra of the matrix algebra,
l = (δ
− δil )(Zki − Z li) (97)
(we set Zii = 0 for all i); with respect to the commutators these matrices form a Lie subalgebra g0(n)
of the Lie algebra gl(n), g0(n) ⊂ gl(n):
[Zij , Z
i ] = Z
i − Z
j , [Z
i , Z
i ] = Z
i − Z
i , [Z
j , Z
] = Z
− Zik , i 6= j 6= k 6= i , (98)
all other brackets vanish. The skew-symmetric solution (95) of the cYB equation,
i,j:i<j
Zij ∧ Z
µi − µj
, (99)
is non-degenerate on the carrier subalgebra g0(n). The carrier subalgebra g0(n) is necessarily quasi-
Frobenius, having a 2-cocycle ω given by the inverse of the r-matrix, that is,
ω(ZA, ZB) = rAB , where r
ABrBC = δ
C , r =
rABZA ∧ ZB . (100)
We have
ω(Zij , Z
l ) = −(µi − µj)δliδ
. (101)
It is easy to check that the 2-cycle ω is a coboundary,
ω(Zij, Z
l ) = λn([Z
j , Z
l ]) , λn = −
i,j:i 6=j
j ∈ g∗0(n) , (102)
thus the subalgebra g0(n) is Frobenius. �
The ”Frobenius” r-matrix (95) (and its quantization) was considered in the work [2].
Proposition 7. The skew-symmetric rime classical r-matrix (95), r =
i<j(µi−µj)−1Zij∧Z
i , where
µ = (µ1, µ2, . . . , µn) is an arbitrary vector such that µi 6= µj, belongs to the orbit of the parameter-free
classical r-matrix
i,j:i<j
ei+ki ∧ e
j−k+1
j . (103)
More precisely,
r = AdXµ ⊗AdXµ(b) , (104)
where the element Xµ ∈ GL(n) is defined by (Xµ)jk = ek−1 (µ1, . . . , µ̂j , . . . , µn).
3This matrix is the same X as in Proposition 4 but depending on variables µi.
Proof. The equality r = AdXµ ⊗ AdXµ(b) is equivalent to a set of relations for the elementary
symmetric functions ei,
(Xµ ⊗Xµ) b = r (Xµ ⊗Xµ) ⇔
eîr−1e
s−1 b
l−1 , (105)
where
j−k+1
δi+ka −
δi−k+1a δ
and r
(δiaδ
b + δ
− δiaδ
− δjaδib)/(µi − µj) , i 6= j ,
0 , i = j .
Both operators b
ab and r
ab are symmetric in the lower indices and anti-symmetric in the upper indices,
that is,
Pb = −b , bP = b and Pr = −r , rP = r . (106)
Eqs. (105) have the following form
(eîb+s−2e
a−s−1 − e
b+s−2e
a−s−1) =
µi − µj
(eîa−1 − e
a−1)(e
b−1 − e
b−1) . (107)
Due to (66), the left hand side of (107) equals
µi − µj
a−2 − µje
a−2)(e
b−1 − e
b−1) . (108)
The right hand side of (107) equals (108) as well because eîa−1 = ea−1 − µieîa−2. �
As in the non-skew-symmetric case, in the infinite dimension the operators b and r are in general
not equivalent.
The sl(n) cYB solution. Let I =
i=1 e
i be the central element of gl(n). The generators
Z̃ij = Z
I ∈ sl(n) satisfy the same relations (98) as Zij thus they form a subalgebra g̃0(n) of the
Lie algebra sl(n) which is isomorphic to g0(n), g̃0(n) ≃ g0(n). This isomorphism gives rise to another
solution r̃ ∈ sl(n) ∧ sl(n) of the cYB equation,
i,j:i<j
Z̃ij ∧ Z̃
µi − µj
∈ sl(n) ∧ sl(n) . (109)
We have the following lemma about the carrier Lie algebra of r̃ (the Lie subalgebra of sl(n) spanned
by the generators Z̃ij).
Lemma 4. The subalgebra g̃0(n) ⊂ sl(n) of dimension dim g̃0(n) = n(n − 1) is isomorphic to the
maximal parabolic subalgebra p of sl(n) obtained by deleting the first negative root.
Proof. The vector v =
i=1 ei is an eigenvector for all elements Z̃
Z̃ij(v) =
v for all i and j , i 6= j . (110)
In a basis in which the first vector is v, the linear span of the generators Z̃ij is
∗ ∗ . . . ∗
0 ∗ ∗
0 ∗ . . . ∗
, (111)
with the traceless condition. The comparison of dimensions finishes the proof. �
Gerstenhaber and Giaquinto [14] found a classical r-matrix bCG which they called “boundary”
because it lies in the closure of the solution space of the YB equation. The cYB solution bCG corre-
sponds to a Frobenius subalgebra (p,Ω), where p is the parabolic subalgebra of sl(n) as above and the
2-cocycle Ω is a coboundary,
Ω = δλbCG , λbCG =
(eii+1)
∗ ∈ p∗ . (112)
The r-matrix bCG is a twist of b (see [8]).
Since the carriers of r̃ and bCG are isomorphic, the r-matrices are equivalent. We shall now prove
that the same matrix Xµ transforms bCG into r̃.
Proposition 8. The boundary classical r-matrix bCG ∈ sl(n) ∧ sl(n),
bCG =
) eii ∧ e
i,j:i<j
ei+ki ∧ e
j−k+1
j , (113)
transforms into the cYB solution r̃ ∈ sl(n) ∧ sl(n),
i,j:i<j
Z̃ij ∧ Z̃
µi − µj
, where Z̃ij = e
j − e
eii , (114)
by a change of basis with the matrix Xµ ∈ GL(n),
r̃ = AdXµ ⊗AdXµ(bCG) . (115)
Proof. Due to Proposition 7 we have r = AdXµ ⊗ AdXµ(b). The cYB solution bCG is the sum of
b and other terms, bCG = b +
i,j(1 −
) eii ∧ e
j . Therefore it is enough to show that r̃ − r =
AdXµ ⊗AdXµ(bCG − b). One has
r̃ − r = 1
i,j:i 6=j
µi − µj
, bCG − b = I ∧
(1− j
j . (116)
Thus we have to show that
(1− j
i,j:i 6=j
µi − µj
Xµ , (117)
which amounts to the following identities for the elementary symmetric functions:
(1− b− 1
)eîb−2 =
j:j 6=i
b−1 − e
µi − µj
. (118)
Replacing, in the right hand side, e
b−1 by e
b−1 + µie
b−2, e
b−1 by e
b−1 + µje
b−2 and noticing that
c = (n − c)ec, c = 1, 2, . . . , n (for the elementary symmetric functions in n variables) finishes the
proof. �
The passage to the sl(n) solution is another instance of the representation change. The general
representation change (82) produces the following effect on the numerical r-matrix (103):
b 7→ b− cη(0) ∧ 11 , (119)
where η(0) is the generator of the invariance group of the r-matrix (103),
η(0) =
(n− j)ej+1j . (120)
The representation change and the operation (86) produce the same 1-parametric family (119) of
skew-symmetric r-matrices. The choice c = −1/n corresponds to the r-matrix bCG.
5 Bézout operators
The Bézout operator [5] is the following endomorphism b(0) of the space P of polynomials of two
variables x and y:
b(0)f(x, y) =
f(x, y)− f(y, x)
or b(0) =
(I − P ) , (121)
where I is the identity operator and P is a permutation, Pf(x, y) = f(y, x). For any natural n, the
subspace Pn of polynomials of degree less than n in x and less than n in y is invariant with respect to
the operator b(0). The matrix of the restriction of b(0) onto Pn, written in the basis {xayb} of powers
(in the decreasing order) coincides with the operator (103).
The non-skew-symmetric matrix (79) is the matrix of the operator
(I − P ) (122)
in this basis. The rime bases are formed by the non-normalized Lagrange polynomials {li(x)lj(y)},
li(t) =
s:s 6=i
(t− φs), at points {φi}, i = 1, 2, . . . , n.
We shall call the operators b(0) and b Bézout r-matrices. The Bézout r-matrices were rediscovered
in several different contexts related to the Yang–Baxter equation (except the fact that they are the
Cremmer–Gervais r-matrices, they appear, for instance, in [7] and [19]).
The standard r-matrix r(s), for the choice of the multi-parameters for which it can be non-trivially
rimed (see the remark at the end of section 3), has the following form in terms of polynomials
r(s) : xiyj 7→ θ(i− j)xiyj − θ(j − i)xjyi . (123)
The subspaces Pn are invariant with respect to r
The properties of the Bézout r-matrices b(0) and b (and of the operator r(s)) become more trans-
parent when they are viewed as operators on polynomials. In particular,
(b(0))2 = 0 , b(0)P = −b(0) , Pb(0) = b(0) , b(0) + b(0)21 = 0 , (124)
b2 = b , bP = −b , b+ b21 = I − P , (125)
(r(s))2 = r(s) , r(s)P = −r(s) , r(s) + r(s)21 = I − P . (126)
The description of the invariance groups of the operators b(0) and b is especially transparent when
these operators are viewed as operators on the spaces of polynomials. Let ∂x and ∂y be the derivatives
in x and y. We have (∂x + ∂y)
= 0 which implies that ∂x is the generator of the invariance
group of b(0); the group is formed by translations. Similarly, (x∂x + y∂y)
= 0 which implies
that x∂x is the generator of the invariance group of b; the group is formed by dilatations. The operation
(86) implies that the operators
b(0) + c(∂x − ∂y) , b+ c(x∂x − y∂y) (127)
are solutions of the cYBe (the quantum version is easy as well) for an arbitrary constant c.
5.1 Non-homogeneous associative classical Yang–Baxter equation
The operators b(0), b and r(s) satisfy an equation stronger than the cYBe. For an endomorphism r of
V ⊗ V , define
r ◦ r := r12r13 + r13r23 − r23r12 , r ◦′ r := r13r12 + r23r13 − r12r23 . (128)
The equation r ◦ r = 0 (as well as r ◦′ r = 0) is called associative classical Yang–Baxter equation
(acYBe) [1, 20].
We introduce a non-homogeneous associative classical Yang–Baxter equation (nhacYBe):
r ◦ r = cr13 , (129)
where c is a constant.
Let F be the space of polynomials in one variable. For the space F ⊗ F of polynomials in two
variables, we denote by x (respectively, y) the generator of the first (respectively, second) copy of F .
For F ⊗ F ⊗ F , the generators are denoted by x, y and z.
Lemma 5. 1. Let M be an operator on the space F ⊗ F . Assume that
M(xf) = f + yM(f) , (130)
M(yf) = −f + xM(f) (131)
for an arbitrary f ∈ F ⊗ F . Then4
M ◦M(xF ) = zM ◦M(F ) , M ◦M(yF ) = xM ◦M(F ) , M ◦M(zF ) = yM ◦M(F ) (132)
for an arbitrary F ∈ F ⊗ F ⊗ F .
2. The operator M = b(0) verifies (130) and (131).
3. Moreover, the unique solution of eqs. (130) and (131) (for the operator M on the space F ⊗ F)
together with the ”initial” condition M(1) = 0 is M = b(0).
Proof. A direct calculation. �
Proposition 9. 1. The Bézout operator b(0) satisfies the acYBe.
2. The Bézout operator b and the operator r(s) satisfy the nhacYBe with c = 1.
Proof. A direct calculation for b(0). Another way is to notice that the relations (132) for M = b(0)
reduce the verification of b(0) ◦ b(0)(F ) = 0 for a monomial F ∈ F ⊗ F ⊗ F to the case F = 1, which
is trivial.
For the Bézout operator b ≡ xb(0) (x here is the operator of multiplication by x), we have, for an
arbitrary F ∈ F ⊗ F ⊗ F ,
b ◦ b (F ) = xb(0)12 (xb
13 (F )) + xb
13 (yb
23 (F ))− yb
23 (xb
12 (F ))
13 (F ) + yb
13 (F )
+ xyb
23 (F )− xyb
12 (F )
13 (F ) + xyb
(0) ◦ b(0)(F ) = b13(F ) .
(133)
We used eq. (130) for b(0) in the second equality.
For the operator r(s), the identity
θ(i− k)θ(i− j) + θ(i− k)θ(j − k)− θ(i− j)θ(j − k) = θ(i− k) (134)
for the step function θ is helpful. �
In each of cases (124-126), the operator r satisfies a quadratic equation r2 = u1r+u2I, the relation
r + r21 = αP + βI with some constants α and β and the nhacYBe with some constant c. Several
general comments about relations between the constants appearing in these equations are in order
here.
4Eq. M ◦M(xF ) = zM ◦M(F ) follows from (130) alone.
1. Assume that an r-matrix (a solution of the cYBe) satisfies r ◦r = cr13. Then r ◦′ r = cr13. Taking
the combinations (r ◦ r− cr13)−P23(r ◦′ r− cr13)P23 and (r ◦ r− cr13)−P12(r ◦′ r− cr13)P12, we find
r13(Sr)23 − (Sr)23r12 = c(r13 − r12) , (Sr)12r13 − r23(Sr)12 = c(r13 − r23) , (135)
where (Sr)12 := r12 + r21. If (Sr)12 = αP12 + βI with some constants α and β, as in (124-126), then
it follows from (135) that (β − c)(r13 − r12) = 0 thus
c = β . (136)
This explains the value of the constant c in lemma 9.
2. For an endomorphism r of V ⊗ V , assume that r ◦ r = βr13 and (Sr)12 = αP12 + βI. Then
P23(r ◦ r − βr13)P23 = r13r12 + r12r32 − r32r13 − βr12
= r13r12 + r12(αP23 + βI − r23)− (αP23 + βI − r23)r13 − βr12 = r ◦′ r − βr13 .
(137)
Thus, if (Sr)12 = αP12 + βI then r ◦ r = βr13 implies r ◦′ r = βr13.
3. Assume that r ◦ r = cr13 for an endomorphism r of V ⊗ V . Then for r̃ = r+ aI + bP , a and b are
constants, we have
r̃ ◦ r̃ = (c+ 2a)r̃13 + bP13(Sr)23 − a(a+ c)I − bcP13 + b2P23P12 . (138)
If, in addition, (Sr)12 = αP12 + βI, then
r̃ ◦ r̃ = (c+ 2a)r̃13 − a(c+ a)I + b(β − c)P13 + b(α+ b)P23P12 . (139)
This shows that the equation r ◦ r = c1r13 + c2I + c3P13 + c4P23P12, c1, c2, c3 and c4 are constants,
reduces to r ◦ r = c̃1r13 + c̃3P13 by a shift r 7→ r + aI + bP .
If r ◦ r = βr13 and (Sr)12 = αP12 + βI then
r̃ ◦ r̃ = (β + 2a)r̃13 − a(β + a)I + b(α+ b)P23P12 . (140)
The combination P23P12 does not appear for b = 0 or b = −α. The choice b = −α corresponds,
modulo a shift of r by a multiple of I, to r 7→ r21, so we consider only b = 0. Then, with the choice
a = −β we find that the operator r̃ = r − βI satisfies the nhacYBe (and (Sr)12 = αP12 − βI). For
the choice a = −β/2 we find that the operator r̃ = r −
I satisfies
r̃ ◦ r̃ = β
, (Sr̃)12 = αP12 . (141)
In particular, the operator
2(x− y)
P (142)
satisfies (141) with β = 1 and α = −1. Also, b̃2 = 1
4. Assume that r2 = ur + v and r12 + r21 = αP12 + βI for an endomorphism r of V ⊗ V . Squaring
the relation r12 − βI = αP12 − r21 and using the same relation again, we obtain
(u− β)(2r12 − βI − αP12) = 0 . (143)
Thus, if r is not a linear combination of I and P then
u = β . (144)
5. Assume that r ◦ r = cr13 and rP = −r for an endomorphism r of V ⊗ V . The nhacYBe has the
following equivalent form:
[r13, r23] = (r12 − cI)r13P23 . (145)
Indeed,
r13r23 − r23r13 = (−r13r23 + r23r12)P23 = (r12 − cI)r13P23 . (146)
Here in the first equality we used r23P23 = −r23 and moved P23 to the right; in the second equality
we used the nhacYBe r ◦ r = cr13.
5.2 Linear quantization
Consider an algebra with three generators r12, r13 and r23 and relations
r13r23 = r23r12 − r12r13 + βr13 ,
r13r12 = r12r23 − r23r13 + βr13 ,
r212 = βr12 + v , r
13 = βr13 + v , r
23 = βr23 + v .
(147)
Choose an order, say, r13 > r23 > r12. Consider (147) as ordering relations. The overlaps in (147)
lead to exactly one more relation:
r23r12r23 = r12r23r12 . (148)
Thus the algebra in question is 12-dimensional (it follows from (147) and (148) that a general
element of the algebra is a product AB of an element A of the Hecke algebra generated by r12 and
r23 and a polynomial B, of degree less than 2, in r13).
We conclude that the nhacYBe together with the quadratic equation for r imply the YBe. Note
that the other form of the YBe also follows:
r23r13r12 − r12r13r23 = (r12r23 − r13r12 + βr13)r12 − r12(r23r12 − r12r13 + βr13)r23
= −r13(βr12 + v) + βr13r12 + (βr12 + v)r13 − βr12r13 = 0.
(149)
Here in the first equality both nhacYBe for r were used; the quadratic relation for r was used in the
second equality.
Therefore, the quantization of such r-matrix is ”linear”5: a combination
R = I + λr , (150)
where λ is an arbitrary constant, satisfies the YBe R12R13R23 = R23R13R12.
5It was noted in [8] that the operator b(0) satisfies both forms of the YBe, squares to zero and that its quantization
has the simple form (150).
5.3 Algebraic meaning
We shall clarify the algebraic meaning of the non-homogeneous associative classical Yang–Baxter
equation in the general context of associative algebras.
Let A be an algebra. Let r ∈ A⊗ A. The operation
δ(0) : A→ A⊗A , δ(0)(u) = (u⊗ 1) r − r (1⊗ u) (151)
(the algebra A does not need to be unital, (u⊗1)(a⊗ b) stands for ua⊗ b and (a⊗ b)(u⊗1) for au⊗ b)
is coassociative if and only if [1]
(u⊗ 1⊗ 1) (r ◦′ r) = (r ◦′ r) (1⊗ 1⊗ u) ∀ u ∈ A . (152)
In particular, δ(0) is coassociative if (r ◦′ r) = 0.
Assume now that the algebra A is unital. Define the operations δ and δ̃ : A→ A⊗ A,
δ(u) := (u⊗ 1) r − r (1⊗ u)− c (u⊗ 1) , (153)
δ̃(u) := (u⊗ 1) r − r (1⊗ u) + c (1⊗ u) , (154)
where c is a constant.
Proposition 10. The coassociativity of each of the operations δ and δ̃ is equivalent to
(u⊗ 1⊗ 1) (r ◦′ r − c r13) = (r ◦′ r − c r13) (1⊗ 1⊗ u) ∀ u ∈ A . (155)
Proof. A straightforward calculation. �
In particular, the operations δ and δ̃ are coassociative if r ◦′ r = c r13.
The map (151) has the following property:
δ(0)(uv) = (u⊗ 1) δ(0)(v) + δ(0)(u) (1 ⊗ v) ; (156)
that is, δ(0) is a derivation with respect to the standard structure of A ⊗ A as a bi-module over A,
uU := (u⊗ 1)U and Uu := U(1⊗ u) for u ∈ A and U ∈ A⊗ A.
For the operations δ and δ̃, the analogue of the property (156) reads
δ(uv) = (u⊗ 1) δ(v) + δ(u) (1 ⊗ v) + c (u⊗ v) , (157)
δ̃(uv) = (u⊗ 1) δ̃(v) + δ̃(u) (1⊗ v)− c (u⊗ v) . (158)
5.4 Rota–Baxter operators
Let A be an algebra. An operator r : A → A is called Rota–Baxter operator of weight α if
r(A)r(B) + αr(AB) = r
r(A)B +Ar(B)
(159)
for arbitrary A,B ∈ A (α is a constant). We refer to [22] for further information about the Rota–
Baxter operators.
The Rota–Baxter operators of weight zero are closely related to the acYBe [23]. It turns out
that the Rota–Baxter operators of non-zero weight are related to the nhacYBe. We shall discuss this
relation and calculate the Rota–Baxter operators corresponding to the Bézout operators.
It is surprising that the Bézout operators, which rather have the sense of derivatives, become,
being interpreted as operators on matrix algebras, the Rota–Baxter operators which are designed to
axiomatize the properties of indefinite integrations and summations.
1. For an endomorphism r of V ⊗ V , define two endomorphisms, r and r′, of the matrix algebra
Mat(V ):
r(A)1 := Tr2(r12A2) , r
′(A)2 := Tr1(r12A1) , A ∈ Mat(V ) , (160)
where Tri is the trace in the copy number i of the space V .
Assume that r satisfies the nhacYBe (129). Multiplying (129) by A2B3, A,B ∈ Mat(V ), and
taking traces in the spaces 2 and 3, we find
r(A)r(B) + r
r′(A)B
Ar(B)
= cTr(A)r(B) . (161)
Assume, in addition, that r12 + r21 = αP12 + βI. Then
r(A) + r′(A) = αA+ βTr(A) 11 . (162)
If c = β then expressing r′(A) by (162) and substituting into (161), we find that the term with Tr(A)
drops out and r is the Rota–Baxter operator of weight α on the algebra of matrices. Similarly, r′ is
the Rota–Baxter operator of weight α as well.
2. We shall calculate the Rota–Baxter operators corresponding to the Bézout operators in the poly-
nomial basis.
The action of the operator b(0) on monomials xkyl reads
b(0)(xkyl) =
−(xl−1yk + xl−2yk+1 + · · ·+ xkyl−1) , k < l ,
0 , k = l ,
xk−1yl + xk−2yl+1 + · · · + xlyk−1 , k > l .
(163)
The action of the operator b on monomials xkyl reads
b(xkyl) =
−(xlyk + xl−1yk+1 + · · ·+ xk+1yl−1) , k < l ,
0 , k = l ,
xkyl + xk−1yl+1 + · · ·+ xl+1yk−1 , k > l .
(164)
Shortly,
b(0)(xkyl) = θ(k − l)
k−l−1
xl+syk−s−1 − θ(l − k)
l−k−1
xk+syl−s−1 , (165)
b(xkyl) = θ(k − l)
xl+syk−s − θ(l − k)
xk+syl−s . (166)
We list several useful matrix forms of the operators b(0) and b in the basis formed by monomials,
ea ⊗ eb := xayb; for the operator b(0):
b(0) =
i,j,a,b
θ(j − a) θ(j − b) δi+j
a+b+1
eja ∧ eib
i,j,a,b
θ(j − a) θ(j − b)− θ(i− b) θ(i− a)
a+b+1 e
a ⊗ eib
i,j:i<j
i+a−1 ∧ e
(167)
and for the operator b:
i,j,a,b
θ(j + 1− a) θ(a− i) δi+ja+b (e
a ⊗ eib − eia ⊗ e
i,j,a,b
θ(j + 1− a) θ(a− i)− θ(i+ 1− a) θ(a− j)
a+b e
a ⊗ eib
i,j:i<j
i+a ⊗ e
j−a − eii+a ⊗ e
i,j:i<j
j−i−1
i+a ∧ e
j−a + e
j ⊗ e
i − eij ⊗ e
(168)
where x ∧ y = x⊗ y − y ⊗ x.
The Rota–Baxter operator r
corresponding to b(0) reads
(A)ij = θ(j − i)
Ai−sj−s−1 − θ(i+ 1− j)
Ai+s+1j+s . (169)
In the right hand side of (169), the summations are over those s ≥ 0 for which the corresponding
matrix element in the sum makes sense; that is, the range of s in the first sum is s = 0, 1, . . . , i − 1
and, in the second sum, s = 0, 1, . . . n− i− 1;
The Rota–Baxter operator rb corresponding to b reads (with the same convention about the sum-
mation ranges)
rb(A)
j = θ(j + 1− i)
Ai−s−1j−s−1 − θ(i− j)
Ai+sj+s . (170)
Its weight is -1.
For the operator r(s), given by eq. (123), the corresponding Rota–Baxter operator r(s) is
r(s)(A)ij =
−θ(j − i)Aij , i 6= j ,
s:s<i
Ass , i = j .
(171)
Its weight is -1.
We shall give also the Rota–Baxter operator for the Bézout r-matrix b in the rime basis, that is,
for the r-matrix (78); it has weight 1 (since r12+r21 = P −I for r in (78)). The Rota–Baxter operator
has the form
r(A)ij =
φj − φi
(Aij −A
j) , i 6= j ,
s:s 6=i
φi − φs
(Ais −Ass) , i = j .
(172)
5.5 ∗-multiplication
1. Let r : A→ A be a Rota–Baxter operator of weight α (see eq.(159)) on an algebra A. It is known
that the operation
A ∗B := r(A)B +Ar(B)− αAB , A,B ∈ A , (173)
defines an associative product on the space A. This product is closely related to the coproducts (153)
and (154) by duality. We shall illustrate it in the context of the matrix algebras.
Define an operation ∗̃ by
〈δ̃(u), B ⊗A〉 = 〈u,A∗̃B〉 , (174)
where δ̃ is given by (154). We have then
〈δ̃(u), B ⊗A〉 = Tr12
u1rB1A2 − ru2B1A2 + c u2B1A2
= Tr1
u1 Tr2(rA2)B1
− Tr1
u1A1 Tr2(B2 r21)
c u1 Tr(B)A1
r(A)B −A r′(B) + cATr(B)
(175)
A∗̃B = r(A)B −A r′(B) + cATr(B) . (176)
In eq. (175), xi stands for the copy of an element x in the space number i in A ⊗ A; the operators
r and r′ are given by (160); to obtain the second and the third terms in the second line of (175) we
renumbered 1↔ 2 and then moved r cyclically under the trace in the second term.
Assume, as before, that r12+ r21 = αP12 +βI and c = β. Then, expressing r
′(A) by (162), we find
that the term with Tr(B) drops out and it follows that
A∗̃B = A ∗B . (177)
2. We shall describe the ∗-multiplication in the simplest example of the Rota–Baxter operators 169)
and (170) corresponding to the Bézout operators for the the polynomials of degree less than 2 (that
is, for 2× 2 matrices A =
a11 a
a21 a
≡ aije
For the operator b(0) = e21 ∧ e11, we have
(A) =
−a21 a11
(178)
and the ∗-multiplication reads
A ∗o Ã ≡ Ar
(Ã) + r
(A)Ã =
−a21ã11 −a21ã12 + a11(ã11 + ã22)
−a21ã21 a21ã11
. (179)
This algebra is isomorphic to the algebra of 3× 3 matrices of the form
∗ ∗ ∗
0 0 ∗
0 0 0
with the identification
e11 7→
0 1 0
0 0 1
0 0 0
 , e12 7→
−1 0 0
0 0 0
0 0 0
 , e21 7→
0 0 1
0 0 0
0 0 0
 , e22 7→
0 0 0
0 0 1
0 0 0
 . (180)
For the operator b = e22 ⊗ e11 − e12 ⊗ e21, we have
rb(A) =
−a21 a11
(181)
and the ∗-multiplication reads
A ∗ Ã ≡ Arb(Ã) + rb(A)Ã+AÃ =
a11ã
2 + a
1 + ã
a11ã
2 + a
1 + ã
. (182)
This algebra is isomorphic to the algebra of 3× 3 matrices of the form
∗ ∗ ∗
0 ∗ 0
0 0 0
with the identification
e11 7→
1 0 0
0 1 0
0 0 0
 , e12 7→
0 0 1
0 0 0
0 0 0
 , e21 7→
0 1 0
0 0 0
0 0 0
 , e22 7→
0 0 0
0 1 0
0 0 0
 . (183)
6 Rime Poisson brackets
The Poisson brackets having the form
{xi, xj} = fij(xi, xj) , i, j = 1, 2, . . . , n , (184)
with some functions fij of two variables, we shall call rime. In this section we study quadratic rime
Poisson brackets,
{xi, xj} = aij(xi)2 − aji(xj)2 + 2νijxixj , i, j = 1, 2, . . . , n . (185)
We show that there is a three-dimensional pencil of such Poisson brackets and then find the invariance
group and the normal form of each individual member of the pencil.
6.1 Rime pencil
In this subsection we establish that the quadratic rime Poisson brackets form a three-dimensional
Poisson pencil.
The left hand side of (185) contains a matrix aij with zeros on the diagonal, aii = 0, and an
anti-symmetric matrix νij, νij = −νji. The Jacobi identity constraints these matrices to satisfy
aijajk + aik(νij + νjk) = 0 , i 6= j 6= k 6= i . (186)
We shall describe a general solution of eq. (186) in the strict situation, that is, when all aij and νij
are different from zero for i 6= j.
The left hand side of νij + νjk = −aijajk/aik is anti-symmetric with respect to (i, k), that is
ΥijΥjkΥki = 1 for Υij = −aij/aji, which readily implies the existence of a vector φi such that
Υij = φ
j . Therefore,
aik = φicikφ
k , (187)
where the matrix cij is anti-symmetric, cij = −cji. Next, 2νki = −(νij +νjk)+ (νjk+νki)+ (νki+νij);
using (186) to express each bracket in the right hand side, we find
νki =
cijcki
cjkcki
cijcjk
. (188)
The right hand side of eq. (188) does not depend on j which imposes further restrictions on the matrix
cij when n > 3. Writing the sum νij + νjk + νkl + νli in two ways, as (νij + νjk) + (νkl + νli) and as
(νjk + νkl) + (νli + νij), and using (186) to express each bracket in terms of the matrix c, we obtain
cijcjk − cilclk
cjkckl − cjicil
. (189)
Replacing j by m in (188) gives the condition on the matrix c:
cijcki
cjkcki
cijcjk
cimcki
cmkcki
cimcmk
. (190)
Using eq. (189) to rewrite the combination
cijcjk
cimcmk
, we find
(cjkckm − cjicim)Ψijkm = 0 , where Ψijkm =
cjkcim
cijckm
ckicjm
. (191)
The quantity Ψijkm is totally anti-symmetric with respect to its indices. Therefore, if Ψijkm 6= 0 then
the combinations (cjkckm − cjicim) vanish for all permutations of indices. This is however impossible:
the system of three linear equations
cijcjk − cimcmk = 0 ,
cikckm − cijcjm = 0 ,
cimcmj − cikckj = 0
(192)
for unknowns {cjk, ckm, cmj} has, by definition, a non-zero solution but the determinant of the system
is different from zero. Thus the Pfaffian Ψijkm vanishes for each quadruple {i, j, k,m}; in other
words, the coefficients of the matrix 1/cij satisfy the Plücker relations; therefore the form 1/cij is
decomposable, c−1ij = sitj−sjti, for some vectors ~s and ~t. For each i, at least one of si or ti is different
from zero. Making, if necessary, a change of basis in the two dimensional plane spanned by ~s and ~t, we
can therefore always assume that all components of, say, the vector ~s are different from zero, si 6= 0
∀ i. We represent the bivector 1/cij in the form (u−1i = si and ψi = −ti/si)
= u−1i u
j (ψi − ψj) . (193)
Substituting (193) into (188) we obtain
νki +
u2k + u
ψk − ψi
u2i − u2j
ψi − ψj
u2k − u2j
ψk − ψj
. (194)
Replacing j by m in the right hand side and equating the resulting expressions, we find that the
independency of the right hand side on j implies:
Eijkm :=
(ψi − ψj)(ψi − ψk)(ψi − ψm)
(ψj − ψi)(ψj − ψk)(ψj − ψm)
(ψk − ψi)(ψk − ψj)(ψk − ψm)
(ψm − ψi)(ψm − ψj)(ψm − ψk)
= 0 .
(195)
for every quadruple {i, j, k,m}.
The quantity Eijkm is totally symmetric. Selecting three values of the index, say, 1,2 and 3, we
can form the quadruple {i, 1, 2, 3} for each i. Solving Ei123 = 0, we obtain the following expression
for u2i :
u2i = A1
(ψi −M2)(ψi −M3)
(M1 −M2)(M1 −M3)
(ψi −M1)(ψi −M3)
(M2 −M1)(M2 −M3)
(ψi −M1)(ψi −M2)
(M3 −M1)(M3 −M2)
(196)
for some constants A1, A2, A3,M1,M2 and M3. The right hand side is the value, at the point ψi, of a
quadratic polynomial which equals to Aa at the points Ma, a = 1, 2, 3. Since A1, A2, A3,M1,M2 and
M3 are arbitrary, we can simply write
u2i = aψ
i + bψi + c . (197)
With the expressions (197) for u2i , the equalities (195) are identically satisfied which shows that (197)
is the general solution.
Upon rescaling xi 7→ φiuixi with φi from (187), the Poisson brackets (185) simplify. The following
statement is established (for n = 2 or 3, (197) does not impose a restriction on the anti-symmetric
matrix cij with all off-diagonal entries different from zero).
Proposition 11. Up to a rescaling of variables, the general strict quadratic rime Poisson brackets
have the form
{xi, xj} =
̺(ψj)(x
i)2 + ̺(ψi)(x
ψi − ψj
(ψi − ψj) a−
̺(ψi) + ̺(ψj)
ψi − ψj
xixj , (198)
where ~ψ is an arbitrary vector with pairwise distinct components and ̺(t) = at2+bt+c is an arbitrary
quadratic polynomial 6.
Thus the strict quadratic rime Poisson brackets form the three-dimensional pencil (parameterized
by the polynomial ̺).
The Poisson brackets (198) can be rewritten in the following forms:
{xi, xj} =
ψi − ψj
̺(ψj)x
i − ̺(ψi)xj
(xi − xj) + a (ψi − ψj)xixj , (199)
{xi, xj} =
au2ij + buijvij + cv
ψi − ψj
ψi − ψj
, (200)
where uij = ψjx
i − ψixj and vij = xi − xj.
Remark 1. For ̺(t) = bt (respectively, ̺(t) = c) these Poisson brackets appear in the classical
limit of the commutation relations (31) in the non-unitary (respectively, unitary) case (with the
parameterization βij = −
ψi − ψj
in the non-unitary case).
Remark 2. The strict rime linear Poisson brackets
{xi, xj} = aijxi − ajixj , aij 6= 0 for all i, j = 1, 2, . . . , n : i 6= j (201)
(or strict rime Lie algebras) are less interesting. The Jacobi identity is
aikakj = aijajk for all i 6= j 6= k 6= i . (202)
6To have nonvanishing coefficients in the formula (198) one has to impose certain inequalities for the components of
the vector ~ψ and the coefficients of the polynomial ̺; however, the formula (198) defines Poisson brackets without these
inequalities.
Rescale variables x2, x3, . . . , xn to have a1i = 1, i = 2, . . . , n. Then the condition (202) with one of i, j, k
equal 1 implies aij = aji and aij = ai1/aj1, i, j = 2, . . . , n; it follows that a
i1 = a
j1, i, j = 2, . . . , n. For
n > 3, the condition (202) with i, j, k > 1 forces ai1 = aj1, i, j = 2, . . . , n. Denote by ν this common
value, ai1 = ν, i, j = 2, . . . , n. After a rescaling x
1 7→ νx1 we find a unique strict rime Lie algebra,
[xi, xj ] = xi−xj for all i and k, which is almost trivial: [xi, xk −xl] = −(xk −xl) for all i, k and l and
[xi − xj , xk − xl] = 0 for all i, j, k, l.
For n = 3, there is one more possibility: a31 = −a21. After a rescaling x1 7→ a21x1, the solution
reads
[x1, x2] = x1 − x2 , [x1, x3] = x1 + x3 , [x2, x3] = −x2 + x3 . (203)
This Lie algebra is isomorphic to sl(2); the isomorphism is given, for example, by h 7→ x1 − x3,
e 7→ x1+x3 and f 7→ x2− (x1+x3)/4 (here h, e and f are the standard generators of sl(2), [h, e] = 2e,
[h, f ] = −2f and [e, f ] = h).
6.2 Invariance
In this subsection we analyze the invariance group of each individual member of the Poisson pencil
from the proposition 11. We find that the Poisson brackets (198), with arbitrary (non-vanishing) ̺,
admit a non-trivial 1-parametric invariance group.
The transformation law of Poisson brackets {xi, xj} = f ij(x) under an infinitesimal change of
variables, x̃i = xi + ǫ ϕi(x), ǫ2 = 0, is {x̃i, x̃j} = f ij(x̃) + ǫ δxf ij, where δxf ij = {ϕi, xj}+ {xi, ϕj} +
ϕk∂kf
ij. For a linear infinitesimal transformation, ϕi(x) = Aijx
j, we have
ij = Aik{xk, xj}+A
i, xk} − xlAkl ∂k{xi, xj} . (204)
Specializing to the Poisson brackets (198), we find
ij = Uji − Uij (205)
Uij :=
+Ajs(ψija−
̺i + ̺j
s:s 6=i
(xs)2 +
(xi)2 + (ψsia−
̺s + ̺i
)xixs
(206)
where ψij = ψi − ψj and ̺s = ̺(ψs).
The Poisson brackets (198) remain rime under the infinitesimal linear transformation with the
matrix A if the coefficients in (xs)2, xsxi and xsxj, s 6= i, j, in (205) vanish which gives the following
system:
(xs)2 , s 6= i, j ⇒ Ajs
= 0 , (207)
xixs , s 6= i, j ⇒ 2Ais
ψsja−
̺i + ̺j
− ̺s + ̺i
= 0 . (208)
Eq. (207) implies that Alk = νk̺l/ψlk, l 6= k, with arbitrary constants νk. For a quadratic polynomial
̺, this solves eq. (208) as well. The coefficient in xjxs vanishes due to the anti-symmetry.
The Poisson brackets (198) are invariant under the infinitesimal linear transformation with the
matrix A if, in addition to (207) and (208), the coefficients in (xi)2, (xj)2 and xixj in (205) vanish
which gives:
xixj ⇒ ̺jAij + ̺iA
i = 0 , (209)
(xi)2 ⇒ Aii
ψija−
̺i + ̺j
s:s 6=i
= 0 . (210)
Eq. (209) implies that νk are equal, νk = ν. The matrix A is defined up to a multiplicative factor,
so we can set ν to 1. Since the Poisson brackets (198) are quadratic, a global rescaling leaves them
invariant, so we can add to A a matrix, proportional to the identity matrix and make A traceless.
The traceless condition, together with eq. (210) determines the diagonal entries, Aii = a(n − 1)ψi +
b+ ̺i
s:s 6=i
. The coefficient in (xj)2 vanishes due to the anti-symmetry. We summarize the
obtained results.
Proposition 12. (i) The infinitesimal linear transformation with the matrix A leaves the Poisson
brackets (198) rime if and only if
Alk =
, l 6= k , (211)
with arbitrary constants νk.
(ii) Up to a global rescaling of coordinates, the invariance group of the Poisson brackets (198) is
1-parametric, with a generator A,
A(̺)ij =
, i 6= j , and A(̺)ii =
̺′i + ̺iξi , ξi :=
s:s 6=i
, (212)
where ̺′i is the value of the derivative of the polynomial ̺ at the point ψi.
Since the Poisson brackets transformed with the matrix (211) are still rime, it follows from the
proposition 11 that they can be written, after an appropriate rescalings of coordinates, in the form
(198). In other words, the variation δx can be compensated by a variation of ψ’s and ̺ and a diagonal
transformation of the coordinates. We have
− δxf ij = δ(1) + δ(2) , (213)
where
δ(1) =
̺i̺j(x
i − xj)2
(νi − νj) + a
νj̺j(x
i)2 − νi̺i(xj)2
(214)
δ(2) = (Ãii − Ã
i)2 − ̺i(xj)2
, Ãii := A
i − ̺′iνi −
s:s 6=i
. (215)
Choose Aii to set Ã
i to 0; this is a diagonal transformation of the coordinates. Then δ
(2) vanishes and
the variation of f ij is reduced to δ(1).
On the other hand, under a variation of ψ′s, ψi 7→ ψi + δψi, the Poisson brackets (198) transform
in the following way:
(xi − xj)2
(̺iδψj − ̺jδψi) + a
(xj)2δψi − (xi)2δψj
(216)
and we conclude that with the choice
δψi = ǫ̺iνi (217)
the variation δ(1) is compensated by the variation δψ. The coefficients of the polynomial ̺ stay the
same. In the next subsection we will study relations between the variation of ψ’s and the polynomial
Remark. With ξi as in (212), define three operators,
(B−)ij =
, i 6= j , and (B−)ii = −ξi , (218)
(B0)ij =
, i 6= j , and (B0)ii = −(
+ ψiξi) , (219)
(B+)ij =
, i 6= j , and (B+)ii = −((n− 1)ψi + ψ2i ξi) . (220)
The operators B+, B0 and B− generate an action of the Lie algebra sl(2),
[B0, B−] = −B− , [B0, B+] = B+ , [B+, B−] = −2B0 (221)
(to obtain the usual commutation relations for the generators of sl(2), change the sign of B+).
This is the usual projective action of sl(2) on polynomials f(t) of degree less than n,
B− : f(t) 7→ f ′(t) , B0 : f(t) 7→ tf ′(t)−
f(t) , B+ : f(t) 7→ t2f ′(t)− (n− 1)t f(t) , (222)
written in the basis of the non-normalized Lagrange polynomials, li(t) =
s:s 6=i
(t − ψs), at points
{ψi}, i = 1, 2, . . . , n. Indeed, in the basis {li(t)}, a polynomial f(t), deg(f) ≤ n − 1, takes the form
f ili, where f
i = li(ψi)
−1f(ψi). We have
l′i(t) =
a:a6=i
b:b6=a,i
(t− ψb) , so l′i(ψk) =
b:b6=k,i
ψkb = lk(ψk)
, k 6= i . (223)
Also,
l′i(t) = li(t)
s:s 6=i
t− ψs
, so l′i(ψi) = −li(ψi)ξi . (224)
Therefore, l′i(t) = −ξili(t) +
k:k 6=i
lk(t), which is exactly (218). For functions on the set of points
{ψi}, the operator of multiplication by t acts as a diagonal matrix Diag(ψ1, ψ2, . . . , ψn) and (219)-(220)
follow.
Define an involution ̟ on the space of matrices7,
̟(Y )ij = Y
j , i 6= j and ̟(Y )ii = −Y ii , Y ∈ Matn . (226)
B(̺) = aB+ + bB0 + cB− , B(̺) : f 7→ ̺(t)f ′(t)− n− 1
̺′(t)f(t) . (227)
In the basis {li(t)} for B, the generator (212) of the invariance group is
A(̺) = ̟(B(̺)) . (228)
Note that the operators ̟(B−), ̟(B0) and ̟(B+) do not form a Lie algebra.
6.3 Normal form
In this subsection we derive a normal form of each individual member of the Poisson pencil from the
proposition 11. It depends only on the discriminant of the polynomial ̺. When the discriminant of ̺
is different from zero, the Poisson brackets (198) are equivalent to the Poisson brackets defined by the
r-matrix (79). When the polynomial ̺ is different from zero but its discriminant is zero, the Poisson
brackets (198) are equivalent to the Poisson brackets defined by the r-matrix (103).
Under a variation of the polynomial ̺, ̺(t) 7→ (a + δa)t2 + (b + δb)t + (c + δc), we have for the
Poisson brackets (200):
u2ijδa+ uijvijδb+ v
ψi − ψj
. (229)
The variation of ̺ can be compensated by a variation (216) of ψ’s if the coefficients in (xj)2, xixj
and (xi)2 in the combination (δψ + δ̺)f
ij vanish, which gives the following system:
(xj)2 ⇒
̺iδψj − ̺jδψi
+ aδψi +
ψ2i δa+ ψiδb+ δc
= 0 , (230)
xixj ⇒ −
2(̺iδψj − ̺jδψi)
2ψiψjδa+ (ψi + ψj)δb + 2δc
= 0 . (231)
A combination 2×(230)+(231) gives
2aδψi + 2ψiδa+ δb = 0 . (232)
7The involution ̟ is the difference of two complementary projectors. The involution ̟ satisfies
̟(Y1)̟(Y2) +̟(Y1Y2) =
̟(̟(Y1)̟(Y2)) + Y1Y2 ,
̟(̟(Y1)Y2) + Y1̟(Y2) ,
̟(Y1̟(Y2)) +̟(Y1)Y2
(225)
for arbitrary Y1, Y2 ∈ Matn. All other linear dependencies between Y1Y2, ̟(Y1)Y2, Y1̟(Y2), ̟(Y1)̟(Y2), ̟(Y1Y2),
̟(̟(Y1)Y2), ̟(Y1̟(Y2)) and ̟(̟(Y1)̟(Y2)) are consequences of the three identities (225).
Substituting the expression (232) for δψ’s into (230) gives
δD(̺) = 0 , where D(̺) = b2 − 4ac . (233)
The coefficient in (xi)2 in (δψ + δ̺)f
ij vanishes due to the anti-symmetry.
Therefore, a necessary condition for a variation of ̺ to be compensated by a variation of ψ’s is that
the discriminant D(̺) does not vary. We shall now see that the discriminant is the unique invariant.
Explicitly, under a shift ψj 7→ ψj + ζ, we have uij 7→ uij + ζvij and vij 7→ vij (in the notation
(200)), which produces the following effect on the coefficients of the polynomial ̺:
a 7→ a , b 7→ b+ 2ζa , c 7→ c+ ζb+ ζ2a . (234)
A dilatation ψj 7→ λψj produces the following effect on the coefficients of ̺:
a 7→ λa , b 7→ b , c 7→ λ−1c . (235)
The inversion ψj 7→ ψ−1j accompanied by a change of variables x̃i = ψ
i produces the following
effect on the coefficients of ̺:
a 7→ −c , b 7→ −b , c 7→ −a . (236)
The set of operators (234) and (235) generates the action of the affine group on the space of the
polynomials ̺. The affine group, together with the inversion (236) generates an action8 of so(3) (the
spin 1 representation of sl(2)) on the space of the polynomials ̺ and the classification reduces to that
of orbits. The orbits (in the complex situation) of non-zero polynomials are of two types: ”massive”,
D(̺) 6= 0, or ”light-like”, D(̺) = 0. Particular representatives of both types appear in the Poisson
brackets, corresponding to the rime r-matrices (see the remark 1 after the proposition 11) and thus
to the r-matrices studied in subsections 4.1 and 4.3. We obtain the following statement.
Proposition 13. Let ̺(t) be a non-zero quadratic polynomial.
If the discriminant of ̺ is different from zero, D(̺) 6= 0, then there exists a change of the param-
eters ψi in the Poisson brackets (198) which sets ̺(t) to bt, ̺(t) 7→ bt; these are the Poisson brackets
corresponding to the r-matrix rCG (subsection 4.1).
If the discriminant of ̺ is zero, D(̺) = 0, then there exists a change of the parameters ψi in the
Poisson brackets (198) which sets ̺(t) to c, ̺(t) 7→ c; these are the Poisson brackets corresponding to
the r-matrix bCG (subsection 4.3).
The generator A(̺) of the invariance group can be easily described in both cases, D(̺) 6= 0 and
D(̺) = 0, in the parameter-free basis (that is, for the r-matrices rCG and bCG; in the rime basis the
generators are given by (52) and (55), respectively). For D(̺) 6= 0 (respectively, D(̺) = 0), it coincides
with the matrix of the operator B0 (respectively, B−), as in the remark in subsection 6.2, in the basis
{ti} of powers of the variable t. This implies somewhat unexpectedly that for an arbitrary polynomial
̺(t) the matrices A(̺) and ̟(A(̺)) are related by a similarity transformation. Note that in the basis
{ti} of powers, the operators aB++bB0+cB− and ̟(aB++bB0+cB−) are also related by a similarity
transformation for arbitrary a, b and c but here it is obvious: ̟(aB++bB0+cB−) = aB+−bB0+cB−,
so the operator ̟(aB+ + bB0 + cB−) belongs to sl(2) and moreover lies on the same (complex) orbit
as aB+ + bB0 + cB− with respect to the adjoint action.
8Let e+ be the generator of the 1-parametric group (234) and h the generator of the 1-parametric group (235). Denote
by I the inversion (236). The remaining generator e− is Ie+I.
7 Orderable quadratic rime associative algebras
Consider an associative algebra A defined by quadratic relations giving a lexicographical order. This
means that xjxk for j < k is a linear combination of terms xaxb with a ≥ b and either a > j or a = j
and b > k.
We shall say that such algebra A is rime if {a, b} ⊂ {j, k}. In other words, the relations in the
algebra are
xjxk = fjkx
kxj + gjkx
kxk , j < k . (237)
We shall classify the strict rime algebras A (that is, the algebras for which all coefficients fij and
gij are different from zero for i < j).
The only possible overlaps for the set of relations (237) are of the form (xjxk)xl = xj(xkxl),
j < k < l. The ordered form of the expression (xjxk)xl is
(xjxk)xl = fjkfjlfkl x
lxkxj + fjkfjlgkl x
lxlxj + f2klgjk x
lxkxk
+ (fklgjkgkl + f
kl(fjkgjl + gjkgkl))x
lxlxk
+ (fjkgjlgkl + gjkg
kl + fklgkl(fjkgjl + gjkgkl))x
lxlxl .
(238)
The ordered form of the expression xj(xkxl) is
xj(xkxl) = fjkfjlfkl x
lxkxj + f2jlgkl x
lxlxj + fklfjlgjk x
lxkxk
+ fklgjl x
lxlxk + (gklgjl + fjlgklgjl)x
lxlxl .
(239)
Equating coefficients, we find
xlxlxj : fjkfjlgkl = f
jlgkl , (240)
xlxkxk : f2klgjk = fklfjlgjk , (241)
xlxlxk : fklgjkgkl + f
kl(fjkgjl + gjkgkl) = fklgjl , (242)
xlxlxl : fjkgjlgkl + gjkg
kl + fklgkl(fjkgjl + gjkgkl) = gklgjl + fjlgklgjl . (243)
In the strict situation, eqs. (240) and (241) simplify, respectively, to
fjk = fjl , for j < k and j < l , (244)
fkl = fjl , for j < l and k < l . (245)
Eqs. (244) and (245) imply that fjk’s are all equal,
fjk =: f . (246)
The substitution of (246) into (242) gives (in the strict situation)
(f + 1)
gjkgkl + gjl(f − 1)
= 0 for j < k < l . (247)
Eq. (243) follows from (246) and (247).
We have thus two cases:
(i) f = −1 and no extra conditions on gjk’s;
(ii) f 6= −1 and
gjkgkl = (1− f) gjl for j < k < l ; (248)
1− f 6= 0 since gjk 6= 0 and gkl 6= 0.
In the case (ii), make an appropriate rescaling of generators, xi 7→ dixi to achieve
gi,i+1 = 1− f for all i = 1, . . . , n− 1 . (249)
It then follows from eq. (248) that
gij = 1− f for all i < j . (250)
We summarize the obtained results.
Proposition 14. Up to a rescaling of variables, the general orderable quadratic strict rime algebra
has relations
(i) either of the form
xjxk = −xkxj + gjkxkxk , j < k , (251)
with no conditions on the coefficients gjk;
(ii) or of the form
xjxk = fxkxj + (1− f)xkxk , j < k , (252)
with arbitrary f (it is strict when f 6= 0, 1).
By construction, the algebras of types (i) and (ii) possess a basis formed by ordered monomials
and thus have the Poincaré series of the algebra of commuting variables.
The algebra with defining relations (252) is the quantum space for the R-matrix (75). The relations
(252) can be written in the form
(xj − xk)xk = fxk(xj − xk) , j < k ; (253)
this is a quantization of the Poisson brackets
{xj , xk} = xk(xj − xk) , j < k . (254)
It would be interesting to know if the algebra with the defining relations (251) admits an R-matrix
description.
Acknowledgements
It is our pleasure to thank László Fehér, Alexei Isaev and Milen Yakimov for enlightening discussions.
The work was partially supported by the ANR project GIMP No.ANR-05-BLAN-0029-01. The second
author (T. Popov) was also partially supported by the Program “Bourses d’échanges scientifiques pour
les pays de l’Est européen” and by the Bulgarian National Council for Scientific Research project PH-
1406.
Appendix A. Equations
Here we give the list of the equations YB(R̂)
abc = 0 for the rime matrix
kl = αijδ
k + βijδ
l + γijδ
l + γ
l , (255)
with a convention αi = αii and βii = γii = γ
ii = 0.
The rime Ansatz implies that YB(R̂)
abc can be different from zero only if the set of lower indices is
contained in the set of upper indices. Therefore, the equations split into two lists: the first one with
two different indices among {i, j, k} and the second one with three different indices.
The full set of equations YB(R̂)
abc = 0 is invariant under the involution ι,
ι : αi ↔ αi , αij ↔ αji , βij ↔ βji , γij ↔ γ′ji , (256)
for if R̂ is a solution of the YBe then R̂21 = PR̂P is a solution of the YBe as well. We shall write
only the necessary part of the equations, the rest can be obtained by the involution ι.
The equations YB(R̂)
abc = 0 with two different indices are:
αijγij(γji + γ
ij) = 0 , (257)
αij(βijβji + γijγ
ij) = 0 = αij(βijβji − γijγji) , (258)
αijγij(αij + βji − αj) = 0 = αijγij(αji + βij − αj) , (259)
βij(α
i − αijαji − αiβij) + (αi − βij)γijγ′ij = 0 , (260)
(αi − αj)γ2ij + αijγij(γij + γ′ji) = 0 , (261)
αijβijγ
ji + (αiβij + γ
ijγij)γij = 0 , (262)
(αij − αji − βij + βji)γijγ′ji = 0 = (αij − αji − βij + βji)βijβji , (263)
ji(αj − αij) + βjiγij(αi − βji) + γij(βijβji + γjiγ′ji) = 0 , (264)
(α2i − αi(αji + βji) + βijβji − γijγji)γij = (α2i − αi(αij + βij) + βijβji − γ′ijγ′ji)γ′ji . (265)
The equations with three different indices {i, j, k} are:
(αij − αki − βij + βki)γijγ′ki = 0 , (266)
αij(βijβjk + βikβji − βikβjk) = 0 , (267)
αij(γijγjk + γik(βjk − βji)) = αij(γijγ′kj + γik(βkj − βij)) = 0 , (268)
(αijαji − αjkαkj)βik + βijβjk(βij − βjk) = 0 , (269)
(αi + βik − βji)βjiγik + γikγjiγ′ji + αik(γjkγ′ji + βjkγ′ki) = 0 , (270)
(αi + αij − αkj − βkj)γijγik − γ2ikγkj + γij(αikγ′ki − γijγ′kj) = 0 , (271)
(αi − βkj)βijγik + (βikβkj + γijγ′ij)γik + αikβijγ′ki − (βij − βik)γijγ′kj = 0 , (272)
αij(γijγjk + γik(αjk − αji))=αji(γijγjk + γik(αjk − αji))=αij(γijγ′kj + γik(αkj − αij))=0. (273)
Appendix B. Blocks
We analyze here the structure of 4×4 blocks of an invertible and skew-invertible rime R-matrix cor-
responding to two-dimensional coordinate planes.
We denote the matrix elements as in (5).
The skew-invertibility of a rime R-matrix imposes restrictions on its entries: in the line R̂i∗j∗ only
two entries can be non-zero, R̂
ji and R̂
jj; in the line R̂
∗i only two entries can be non-zero, R̂
ji and
ii . Therefore,
αij = 0 ⇒ γijγ′ij 6= 0 and γijγ′ij = 0 ⇒ αij 6= 0 . (274)
Dealing with a single block, this becomes especially clear: to skew invert a 4×4 block is the same
as to invert the matrix
α1 0 γ12 β12
0 0 α12 γ
γ′21 α21 0 0
β21 γ21 0 α2
, (275)
whose determinant is
(α12β12 − γ12γ′12)(α21β21 − γ21γ′21)− α1α2α12α21 . (276)
B.1 Solutions
Here we classify solutions which are neither ice nor strict rime. For an ice R-matrix, α12 6= 0 and
α21 6= 0. For a rime R-matrix, αij might vanish and we consider the cases according to the number of
αij ’s which can be zero.
1. Both α12 and α21 do not vanish, α12α21 6= 0.
If γ12γ21 6= 0 then by (257), γ′12γ′21 6= 0. This is strict rime.
If both γ12 = 0 and γ21 = 0 then eq. (259) implies (αji + βij − αj)γ′ji = 0; eq. (261) implies
(αi − αj + αji)γ′ji = 0 and eq. (262) implies βijγ′ji = 0. Combining these, we find γ′ij = 0, this is ice.
It is left to analyze the situation when only one of γ’s is different from zero, say γ12 6= 0 and
γ21 = 0. We have the following chain of implications:
(257) ⇒ γ′12 = 0 , (277)
(259) ⇒
β12 = α2 − α21 ,
β21 = α2 − α12 ,
(278)
(258) ⇒ (α2 − α12)(α2 − α21) = 0 , (279)
(260) ⇒
(α1 − α2)(α2 − α21)(α1 + α21) = 0 ,
(α1 − α2)(α2 − α12)(α1 + α12) = 0 ,
(280)
(261)&(265) ⇒
(α1 − α2 + α12)γ12 + α12γ′21 = 0 ,
(α1 − α2 + α21)γ′21 + α21γ12 = 0 .
(281)
Eqs. (262), (263) and (264) are satisfied. By the second line in (281), γ′21 6= 0.
Now the system of inequalities and equations is invariant under R̂ ↔ R̂21, so up to this trans-
formation we can solve eq. (279) by setting α21 = α2. Then, by (281), γ
21 = −γ12α2/α1, β’s are
expressed in terms of α’s by (278) and the remaining system for α’s reduces to a single equation
(α1 − α2)(α1 + α12) = 0. We obtain two solutions:
1a. α2 = α1; α1, α12 and γ12 are arbitrary non-zero numbers; we rescale the R-matrix to set
α1α12 = 1 and denote q = α1, γ = γ12:
R̂(q;γ) =
α1 0 0 0
γ12 0 α12 0
−γ12 α1 α1 − α12 0
0 0 0 α1
q 0 0 0
γ 0 q−1 0
−γ q q − q−1 0
0 0 0 q
. (282)
The R-matrix (282) is semi-simple if (and only if) q+q−1 6= 0 and it is then an R-matrix of GL(2)-type,
Spec(R̂) = {q, q, q,−q−1}. This solution is a specialization of (15)-(16).
1b. α12 = −α1; α1, α2 and γ12 are arbitrary non-zero numbers; we rescale the R-matrix to set
α1α2 = −1 and denote q = α1, γ = γ12/q:
R̂(q;γ) =
α1 0 0 0
γ12 0 −α1 0
−γ12α2/α1 α2 α1 + α2 0
0 0 0 α2
q 0 0 0
qγ 0 −q 0
q−1γ −q−1 q − q−1 0
0 0 0 −q−1
. (283)
The R-matrix (283) is semi-simple if (and only if) q+ q−1 6= 0 and it is then an R-matrix of GL(1|1)-
type, Spec(R̂) = {q, q,−q−1,−q−1}.
2. Assume that α12 = 0.
By the invertibility, β12β21 6= 0; by the skew-invertibility, γ12γ′12 6= 0; now eqs. (257) and (258)
imply β12β21 = γ12γ21, γ
12 = −γ21 and γ′21 = −γ12. Eq. (259) implies α2 = α1, β12 = α1 − α21 and
β21 = α1.
The rest is satisfied and we obtain a solution, in which α1, β12 and γ12 are arbitrary non-zero
numbers; we rescale the R-matrix to set α1β12 = −1 and denote q = α1, γ = γ12:
R̂(q;γ) =
α1 0 0 0
γ12 β12 0 −α1β12/γ12
−γ12 α1 − β12 α1 α1β12/γ12
0 0 0 α1
q 0 0 0
γ −q−1 0 1/γ
−γ q + q−1 q −1/γ
0 0 0 q
. (284)
The R-matrix (284) is semi-simple if (and only if) q+q−1 6= 0 and it is then an R-matrix of GL(2)-type,
Spec(R̂) = {q, q, q,−q−1}. This solution is a specialization of (15)-(16).
3. Finally, assume that α12 = α21 = 0.
By the invertibility, β12β21 6= 0; by the skew-invertibility, γ12γ′12γ21γ′21 6= 0; now eq. (261) implies
α2 = α1, eq. (263) implies β21 = β12; eq. (262) implies γ12γ
12 = γ21γ
21 = −α1β12; eq. (265) implies
that γ12γ21 can take three values: α
12 or (−α1β12).
The rest is satisfied and we obtain a solution, in which α1, β12 and γ12 are arbitrary non-zero
numbers; we rescale the R-matrix to set α1β12 = −1 and denote q = α1, γ = γ12:
R̂(q,ω;γ) =
α1 0 0 0
γ12 β12 0 −α1β12/γ12
−α1β12γ12/ω 0 β12 ω/γ12
0 0 0 α1
q 0 0 0
γ −q−1 0 1/γ
γ/ω 0 −q−1 ω/γ
0 0 0 q
, (285)
where ω = q2, 1, q−2. The R-matrix (285) is semi-simple if (and only if) q + q−1 6= 0 and it is then an
R-matrix of GL(1|1)-type, Spec(R̂) = {q, q,−q−1,−q−1}.
It follows from the analysis above that if γij 6= 0 in an invertible and skew-invertible rime R-matrix
then γ′ji 6= 0.
In each of the cases (282)-(285), the parameter γ 6= 0 can be set to an arbitrary (non-zero) value
by a diagonal change of basis. The R-matrices (282)-(285) are skew-invertible.
B.2 GL(2) and GL(1|1) R-matrices
1. In dimension 2, except the standard R-matrices of GL-type,
GL(2)
(q,p)
q 0 0 0
0 0 p 0
0 p−1 q − q−1 0
0 0 0 q
GL(1|1)
(q,p)
q 0 0 0
0 0 p 0
0 p−1 q − q−1 0
0 0 0 −q−1
, (286)
there are two non-standard one-parametric families of non-unitary R-matrices of the type GL(1|1):
the eight-vertex one,
R̂I(q) =
q − q−1 + 2 0 0 q − q−1
0 q − q−1 q + q−1 0
0 q + q−1 q − q−1 0
q − q−1 0 0 q − q−1 − 2
, (287)
and the matrix R̂(II) for which the matrix R = PR̂ can be given an upper-triangular form,
R̂II(q,ε) =
q 0 0 q + q−1
0 0 εq−1 0
0 εq q − q−1 0
0 0 0 −q−1
, (288)
where ε = ±1.
The R-matrices (286), (287) and (288) are semi-simple if (and only if) q + q−1 6= 0.
Up to the transformations R̂↔ R̂21 and R̂↔ R̂t (the transposition), basis changes and rescalings
R̂ 7→ c R̂ (where c is a constant), the complete list of semi-simple invertible and skew-invertible
R-matrices includes (see [16] for a description of all solutions of the Yang–Baxter equation in two
dimensions and [11] for the classification of GL(2)-type R-matrices), in addition to (286)-(288), the
one-parametric family of Jordanian solutions R̂
(h1:h2)
(h1:h2)
1 h1 −h1 h1h2
0 0 1 −h2
0 1 0 h2
0 0 0 1
(289)
(the Jordanian R-matrix is of GL(2)-type; it is unitary; the essential parameter is the projective vector
(h1 : h2)), as well as the permutation-like solution R̂
(a,b,c)
and one more solution R̂
(a,b,c)
1 0 0 0
0 0 a 0
0 b 0 0
0 0 0 c
0 0 0 a
0 1 0 0
0 0 1 0
a 0 0 0
. (290)
The R-matrix R̂
(a,b,c)
is Hecke when ab = 1 and c = ±1 and it is then standard (and unitary). The
R-matrix R̂
is Hecke when a2 = 1; it is then unitary and related to the standard R-matrix by a
change of basis with the matrix
Without the demand of semi-simplicity, the full list of invertible and skew-invertible R-matrices
contains two more solutions,
(h1:h2:
1 h1 h2 h3
0 0 1 h1
0 1 0 h2
0 0 0 1
, R̂(
′′′) =
1 0 0 1
0 0 −1 0
0 −1 0 0
0 0 0 1
. (291)
The essential parameter for the R-matrix R̂
(h1:h2:
is the projective vector (h1 : h2 :
h3). The
R-matrix R̂
(h1:h2:
is semi-simple if and only if h2 = −h1 and h3 = −h21; it then belongs to the
family (289) of Jordanian R-matrices.
2. For the R-matrices from the list above, the transformations R̂ ↔ R̂21, R̂ ↔ R̂t and R̂ ↔ R̂−1
partly overlap or reduce to parameter or basis changes. We shall write formulas for the Hecke R-
matrices only.
For the standard R-matrix R̂(q,p) := R̂
GL(2)
(q,p)
R̂t(q,p) = R̂(q,p−1) , (R̂(q,p))21 = (π ⊗ π)R̂(q,p)(π
−1 ⊗ π−1) , R̂−1
(q,p)
= (R̂(q−1,p−1))21 , (292)
where π =
For the standard R-matrix R̂(q,p) := R̂
GL(1|1)
(q,p)
R̂t(q,p) = R̂(q,p−1) , (R̂(q,p))21 = (π ⊗ π)R̂(−q−1,p)(π
−1 ⊗ π−1) , R̂−1
(q,p)
= (R̂(q−1,p−1))21 . (293)
For the non-standard GL(1|1) R-matrix R̂(q) := R̂I(q),
R̂t(q) = R̂(q) , (R̂(q))21 = R̂(q) , R̂
= (D ⊗D)R̂(q−1)(D ⊗D)−1 , (294)
where D =
For the non-standard GL(1|1) R-matrix R̂(q,ε) := R̂II(q,ε),
R̂t(q,ε) = (π̃ ⊗ π̃)(R̂(−q−1,−ε))21(π̃
−1 ⊗ π̃−1) , R̂−1
(q,ε)
= (R̂(q−1,ε))21 , (295)
where π̃ =
For the Jordanian R-matrix R̂(h1:h2) := R̂
(h1:h2)
R̂t(h1:h2) = (π ⊗ π)R̂(h2:h1)(π
−1 ⊗ π−1) , (R̂(h1:h2))21 = R̂(−h1:−h2) , R̂
(h1:h2)
= R̂(h1:h2) . (296)
B.3 Riming
We shall now identify the rime R-matrices (282)-(285).
1. GL(2)
The R-matrices (282) and (284) are related by a change of basis (the number in brackets refers to the
corresponding equation),
(282)
(q;γ)
T ⊗ T = T ⊗ T R̂(284)
(q;γ)
, T =
q −1/γ
. (297)
In turn, the R-matrix (282) is related to the standard R-matrix R̂
GL(2)
(q,q−1)
by a change of basis,
(282)
(q;γ)
T ⊗ T = T ⊗ T R̂GL(2)
(q,q−1)
, T =
q − q−1 0
. (298)
In the unitary situation (that is, q − q−1 = 0), the R-matrix R̂(282)
(q;γ)
belongs to the family of
Jordanian R-matrices.
Note that for the R-matrices (282) and (284), the left even quantum spaces are classical.
2. GL(1|1)
The R-matrix (283) is related to the R-matrix (285) with the choice ω = β212,
(283)
(q;γ)
T ⊗ T = T ⊗ T R̂(285)
(−q−1,q2;1) , T =
. (299)
We have
(285)
(q,1;γ)
T ⊗ T = T ⊗ T R̂I
, T =
γ −γτ
, where τ2 =
q − 1
q + 1
, (300)
(285)
(q,q2;γ)
T ⊗ T = T ⊗ T R̂II
(q,1)
, T =
γq−1 −γq−1
, (301)
(285)
(q,q−2;γ)
T ⊗ T = T ⊗ T (R̂II
(q,1)
)21 , T =
γq −γq
. (302)
In the unitary situation (that is, for q = ±1) only eq. (300) changes; but now different choices for
ω coincide.
3. Since the standard R-matrices are rime as well, we conclude that in dimension 2, all non-unitary
Hecke R-matrices fit into the rime Ansatz. When h1 = 0, the Jordanian R-matrix R̂
(0:h2)
is rime as
well. However, when h1 6= 0, the Jordanian R-matrix R̂
(h1:h2)
cannot be rimed. Indeed, assume that
h1 6= 0 and let A = (T ⊗ T )R̂
(h1:h2)
(T ⊗ T )−1 with some invertible matrix T . Then
(Det(T ))2A1112 = h1 (T
2 (Det(T )− h2 T 11 T 21 ) ,
(Det(T ))2A1121 = −h1 (T 11 )2 (Det(T ) + h2 T 11 T 21 ) ,
(Det(T ))2A2212 = h1 (T
2 (Det(T )− h2 T 11 T 21 ) ,
(Det(T ))2A1121 = −h1 (T 21 )2 (Det(T ) + h2 T 11 T 21 ) .
(303)
For an invertible T , the non-rime entries (303) of A cannot vanish simultaneously.
4. We remark also that all non-standard R-matrices of GL(1|1)-type are uniformly described by the
formula (285). The right quantum spaces for the R-matrix R̂
(285)
(q,ω;γ)
, with γ = 1, read
(R̂− q11⊗ 11)ijkl x
kxl = 0 :
(q + q−1)xy = x2 + y2 ,
(q + q−1)xy = ω−1x2 + ωy2 ;
(304)
(R̂ + q−111⊗ 11)ij
xkxl = 0 :
x2 = 0 ,
y2 = 0 .
(305)
Using the diamond lemma, it is straightforward to verify that the Poincaré series of the quantum
space (304) is of GL(1|1)-type if and only if ω = q−2, 1 or q2.
Appendix C. Rimeless triple
We sketch here a proof that the triple (94) cannot be rimed. Relations xiyj = R̂
kxl, where R̂ is
the R-matrix for the triple (94) with arbitrary multiparameters, read
xiyi = yixi , i = 1, 2, 3, 4 (306)
x1y2 =
y2x1 ,
x1y3 =
y3x1 ,
x1y4 =
y4x1 − rs
y3x2 ,
x2y1 =
y1x2 + (1− q−2)y2x1 ,
x2y3 =
y3x2 ,
x2y4 =
y4x2 ,
(307)
x3y1 =
y1x3 + (1− q−2)y3x1 ,
x3y2 =
y2x3 + (1− q−2)y3x2 ,
x3y4 =
y4x3 ,
x4y1 =
y1x4 + (1− q−2)y4x1 + 1
y2x3 ,
x4y2 =
y2x4 + (1− q−2)y4x2 ,
x4y3 =
y3x4 + (1− q−2)y4x3 .
(308)
The parameter q enters the characteristic equation for R̂, R̂2 = (1− q−2)R̂+ q−211⊗ 11; p, r and s are
the multiparameters. The only needed restriction is q2 6= 1.
Denote by 〈l(1), l(2)〉 a two-dimensional plane spanned by l(1) and l(2). We say that two linear forms
l(1) and l(2) (in four variables) form a rime pair if, for the ordering relations (306) and (307)-(308),
each product l(α)(x)l(β)(y), α = 1, 2, β = 1, 2, is a linear combination of l(1)(y)l(1)(x), l(1)(y)l(2)(x),
l(2)(y)l(1)(x) and l(2)(y)l(2)(x). If, in addition, l(α)(x)l(α)(y) is proportional to l(α)(y)l(α)(x) for α = 1
and 2, we say that l(1) and l(2) form a rime basis in the plane 〈l(1), l(2)〉. We call a plane rime if it
admits a rime basis.
Fork Lemma. Assume that l(1)(x) = x1 + a2x
2 + a3x
3 and l(4)(x) = b2x
2 + b3x
3 + x4 form a rime
pair for some a2, a3, b2 and b3. Then either a3b2 6= 0 and a2 = b3 = 0 or a2b3 6= 0 and a3 = b2 = 0.
If a3b2 6= 0 then
r = s = 1 , l(1)(x) = x1 + wx3 and l(4)(x) = x4 +
q − q−1
x2 , w 6= 0 is arbitrary . (309)
If a2b3 6= 0 then
, r = s , l(1)(x) = x1 + wx2 and l(4)(x) = x4 +
q − q−1
x3 , w 6= 0 is arbitrary . (310)
Moreover, if r = s = 1 and p 6= q−1 then the rime plane 〈l(1), l(4)〉 admits a unique, up to
rescalings, rime basis {l(1), l(4)}; if p = q−1 and r = s 6= 1 then the rime plane 〈l(1), l(4)〉 admits a
unique, up to rescalings, rime basis {l(1), l(4)}; if p = q−1 and r = s = 1 then any two independent
linear combinations of l(1) and l(4) form a rime basis in the plane 〈l(1), l(4)〉.
Proof. A straightforward calculation. �
Assume that a rime basis {x̃i} for the triple (94) exists, x̃i = Aijxj , the matrix Aij is invertible.
Rename the rime variables x̃i in such a way that the minor
A11 A
A41 A
is non-zero and A11A
4 6= 0;
normalize the variables x̃1 and x̃4 to have A11 = A
4 = 1. The plane 〈x̃1, x̃4〉 is, by definition, rime,
with a rime basis {x̃1, x̃4}.
Suppose that r = s = 1 and p 6= q−1 or p = q−1 and r = s 6= 1. Then, by Fork Lemma, the
rime basis in the plane 〈x̃1, x̃4〉 is, up to proportionality, unique, so we know the variables x̃1 and
x̃4. The variables x̃1 and x̃2 form a rime plane. Therefore, if the variable x̃2 contains x4 with a
non-zero coefficient then, by Fork Lemma, x̃2 must be proportional to x̃4, contradicting to the linear
independence of the variables x̃2 and x̃4. Similarly, the variable x̃2 cannot contain x1 with a non-zero
coefficient (the plane 〈x̃2, x̃4〉 is rime). Thus, x̃2 is a linear combination of x2 and x3. Same for x̃3:
it is a linear combination of x2 and x3. One of the variables, x̃2 or x̃3, say, x̃2, contains x2 with a
non-zero coefficient. Writing rime equations for the plane 〈x̃1, x̃2〉 in the case r = s = 1 and p 6= q−1
(for the plane 〈x̃2, x̃4〉 in the case p = q−1 and r = s 6= 1) quickly leads to a contradiction.
Therefore, if the relations (306) and (307)-(307) can be rimed then p = q−1 and r = s = 1. It
follows from Fork Lemma that
x̃4 = (q − q−1)c2c3x1 + c2x2 + c3x3 + x4
for some c2 and c3. The planes 〈x̃a, x̃4〉, a = 1, 2, 3, are rime. Subtracting from the variables x̃a the
variable x̃4 with appropriate coefficients, we find three linearly independent combinations
l(x) = d1x
1 + d2x
2 + d3x
3 , (311)
each forming a rime pair with x̃4. We must have: l(x)l(y) is a linear combination of l(y)l(x), l(y)x̃4,
ỹ4l(x) and ỹ4x̃4. It follows, after a straightforward calculation, that d2d3 = 0. Moreover, d2 = d3 = 0
is excluded by Fork Lemma. In the case d2 6= 0 and d3 = 0 (respectively, d3 6= 0 and d2 = 0), the rime
condition implies that d1 = (q − q−1)c2d3 (respectively, d1 = (q − q−1)c3d2). Thus, only two linearly
independent combinations (311) can form a rime pair with x̃4, the final contradiction.
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	From ice to rime
	Rime Yang–Baxter solutions.25cm
	Non-unitary rime R-matrices
	Unitary rime R-matrices
	Properties
	Rime and Cremmer–Gervais R-matrices
	Classical rime r-matrices.25cm
	Non-skew-symmetric case
	BD triples.
	Skew-symmetric case
	Bézout operators.25cm
	Non-homogeneous associative classical Yang–Baxter equation
	Linear quantization
	Algebraic meaning
	Rota–Baxter operators
	*-multiplication
	Rime Poisson brackets.25cm
	Rime pencil
	Invariance
	Normal form
	Orderable quadratic rime associative algebras.1cm
	Appendix A.  Equations
	Appendix B.  Blocks.25cm
	B.1  Solutions
	B.2  GL(2) and GL(1|1) R-matrices
	B.3  Riming
	Appendix C.  Rimeless triple
	References
