Invariants of Triangular Lie Algebras
Vyacheslav Boyko †, Jiri Patera ‡ and Roman Popovych †§
† Institute of Mathematics of NAS of Ukraine, 3 Tereshchenkivs’ka Str., Kyiv-4, 01601 Ukraine
E-mail: boyko@imath.kiev.ua, rop@imath.kiev.ua
‡ Centre de Recherches Mathématiques, Université de Montréal,
C.P. 6128 succursale Centre-ville, Montréal (Québec), H3C 3J7 Canada
E-mail: patera@CRM.UMontreal.CA
§ Fakultät für Mathematik, Universität Wien, Nordbergstraße 15, A-1090 Wien, Austria
Abstract
Triangular Lie algebras are the Lie algebras which can be faithfully represented by triangular
matrices of any finite size over the real/complex number field. In the paper invariants (‘gen-
eralized Casimir operators’) are found for three classes of Lie algebras, namely those which
are either strictly or non-strictly triangular, and for so-called special upper triangular Lie al-
gebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749; math-ph/0602046],
developed further in [J.Phys. A: Math. Theor., 2007, V.40, 113; math-ph/0606045], is used to
determine the invariants. A conjecture of [J.Phys. A: Math. Gen., 2001, V.34, 9085], concerning
the number of independent invariants and their form, is corroborated.
1 Introduction
The invariants of Lie algebras are one of their defining characteristics. They have numerous appli-
cations in different fields of mathematics and physics, in which Lie algebras arise (representation
theory, integrability of Hamiltonian differential equations, quantum numbers etc). In particular,
the polynomial invariants of a Lie algebra exhaust its set of Casimir operators, i.e., the center of
its universal enveloping algebra. That is why non-polynomial invariants are also called general-
ized Casimir operators, and the usual Casimir operators are seen as ‘trivial’ generalized Casimir
operators. Since the structure of invariants strongly depends on the structure of the algebra and
the classification of all (finite-dimensional) Lie algebras is an inherently difficult problem (actually
unsolvable), it seems to be impossible to elaborate a complete theory for generalized Casimir op-
erators in the general case. Moreover, if the classification of a class of Lie algebras is known, then
the invariants of such algebras can be described exhaustively. These problems have already been
solved for the semi-simple and low-dimensional Lie algebras, and also for the physically relevant
Lie algebras of fixed dimensions (see, e.g., references in [3, 7, 8, 18, 19]).
The actual problem is the investigation of generalized Casimir operators for classes of solvable Lie
algebras or non-solvable Lie algebras with non-trivial radicals of arbitrary finite dimension. There
are a number of papers on the partial classification of such algebras and the subsequent calculation
of their invariants [1, 6, 7, 14, 15, 16, 20, 21, 22, 23]. In particular, Tremblay and Winternitz [22]
classified all the solvable Lie algebras with the nilradicals isomorphic to the nilpotent algebra t0(n)
of strictly upper triangular matrices for any fixed dimension n. Then in [23] invariants of these
algebras were considered. The case n = 4 was investigated exhaustively. After calculating the
invariants for a sufficiently large value of n, Tremblay and Winternitz made conjectures for an
arbitrary n on the number and form of functionally independent invariants of the algebra t0(n),
and the ‘diagonal’ solvable Lie algebras having t0(n) as their nilradicals and possessing either the
maximal (equal to n − 1) or minimal (one) number of nilindependent elements. A statement on a
functional basis of invariants was only proved completely for the algebra t0(n). The infinitesimal
invariant criterion was used for the construction of the invariants. Such an approach entails the
http://arxiv.org/abs/0704.0937v4
http://arxiv.org/abs/math-ph/0602046
http://arxiv.org/abs/math-ph/0606045
necessity of solving a system of ρ first-order linear partial differential equations, where ρ has the
order of the algebra’s dimension. This is why the calculations were very cumbersome and results
were obtained due to the thorough mastery of the method.
In this paper, we use our original algebraic method for the construction of the invariants (‘gen-
eralized Casimir operators’) of Lie algebras via the moving frames approach [3, 4]. The algorithm
makes use of the knowledge of the associated inner automorphism groups and Cartan’s method
of moving frames in its Fels–Olver version [9, 10]. (For modern developments about the moving
frame method and more references, see also [17].) Unlike standard infinitesimal methods, it allows
us to avoid solving systems of differential equations, replacing them instead by algebraic equations.
As a result, the application of the algorithm is simpler. Note that a closed approach was earlier
proposed in [12, 13, 19] for the specific case of inhomogeneous algebras.
The invariants of three classes of triangular Lie algebras are exhaustively investigated (below n
is an arbitrary integer):
• nilpotent Lie algebras t0(n) of n× n strictly upper triangular matrices (Section 3);
• solvable Lie algebras t(n) of n× n upper triangular matrices (Section 4);
• solvable Lie algebras st(n) of n× n special upper triangular matrices (Section 5).
The triangular algebras are especially interesting due to their ‘universality’ properties. More pre-
cisely, any finite-dimensional nilpotent Lie algebra is isomorphic to a subalgebra of t0(n). Similarly,
any finite-dimensional solvable Lie algebra over an algebraically closed field of characteristic 0 (e.g.,
over C) can be embedded as a subalgebra in t(n) (or st(n)).
We have adapted and optimized our algorithm for the specific case of triangular Lie algebras via
special double enumeration of basis elements, individual choice of coordinates in the corresponding
inner automorphism groups and an appropriate modification of the normalization procedure of
the moving frame method. As a result, the problems related to the construction of functional
bases of invariants are reduced for the algebras t0(n) and t(n) to solving linear systems of algebraic
equations! Let us note that due to the natural embedding of st(n) to t(n) and the representation
t(n) = st(n) ⊕ Z(t(n)), where Z(t(n)) is the center of t(n), we can construct a basis in the set
of invariants of st(n) without the usual calculations from a previously found basis in the set of
invariants of t(n).
We re-prove the statement for a basis of invariants of t0(n), which was first constructed in [23]
using the infinitesimal method in a heuristic way, thereafter constructed in [4] using an empiric
technique based on the exclusion of parameters within the framework of the algebraic method. The
aim of this paper in considering t0(n) is to test and better understand the technique of working
with triangular algebras. The calculations for t(n) are similar, albeit more complex, although they
are much clearer and easier than under the standard infinitesimal approach.
As proved in [22], there is a unique algebra with the nilradical t0(n) that contains a maximum
possible number (n− 1) of nilindependent elements. A conjecture on the invariants of this algebra
is formulated in Proposition 1 of [23]. We show that this algebra is isomorphic to st(n). As a result,
the conjecture by Tremblay and Winternitz on its invariants is effectively proved.
2 The algorithm
The applied algebraic algorithm was first proposed in [3] and then developed in [4]. Ibid it was
effectively tested for the low-dimensional Lie algebras and a wide range of solvable Lie algebras
with a fixed structure of nilradicals. The presentation of the algorithm here differs from [3, 4], the
differences being important within the framework of applications.
For convenience of the reader and to introduce some necessary notations, before the description
of the algorithm, we briefly repeat the preliminaries given in [3, 4] about the statement of the
problem of calculating Lie algebra invariants, and on the implementation of the moving frame
method [9, 10]. The comparative analysis of the standard infinitesimal and the presented algebraic
methods, as well as their modifications, is given in the second part of this section.
Consider a Lie algebra g of dimension dim g = n < ∞ over the complex or real field and the
corresponding connected Lie group G. Let g∗ be the dual space of the vector space g. The map
Ad∗ : G → GL(g∗), defined for any g ∈ G by the relation
〈Ad∗gx, u〉 = 〈x,Adg−1u〉 for all x ∈ g
∗ and u ∈ g
is called the coadjoint representation of the Lie group G. Here Ad: G → GL(g) is the usual adjoint
representation of G in g, and the image AdG of G under Ad is the inner automorphism group Int(g)
of the Lie algebra g. The image of G under Ad∗ is a subgroup of GL(g∗) and is denoted by Ad∗G.
A function F ∈ C∞(g∗) is called an invariant of Ad∗G if F (Ad
gx) = F (x) for all g ∈ G and x ∈ g
The set of invariants of Ad∗G is denoted by Inv(Ad
G). The maximal number Ng of functionally
independent invariants in Inv(Ad∗G) coincides with the codimension of the regular orbits of Ad
i.e., it is given by the difference
Ng = dim g− rankAd
Here rankAd∗G denotes the dimension of the regular orbits of Ad
G and will be called the rank of the
coadjoint representation of G (and of g). It is a basis independent characteristic of the algebra g,
the same as dim g and Ng.
To calculate the invariants explicitly, one should fix a basis E = {e1, . . . , en} of the algebra g.
It leads to fixing the dual basis E∗ = {e∗1, . . . , e
n} in the dual space g
∗ and to the identification
of Int(g) and Ad∗G with the associated matrix groups. The basis elements e1, . . . , en satisfy the
commutation relations [ei, ej ] =
k=1 c
ijek, i, j = 1, . . . , n, where c
ij are components of the tensor
of structure constants of g in the basis E .
Let x → x̌ = (x1, . . . , xn) be the coordinates in g
∗ associated with E∗. Given any invariant
F (x1, . . . , xn) of Ad
G, one finds the corresponding invariant of the Lie algebra g by symmetriza-
tion, SymF (e1, . . . , en), of F . It is often called a generalized Casimir operator of g. If F is
a polynomial, SymF (e1, . . . , en) is a usual Casimir operator, i.e., an element of the center of the
universal enveloping algebra of g. More precisely, the symmetrization operator Sym acts only on
the monomials of the forms ei1 · · · eir , where there are non-commuting elements among ei1 , . . . , eir ,
and is defined by the formula
Sym(ei1 · · · eir) =
eiσ1 · · · eiσr ,
where i1, . . . , ir take values from 1 to n, r > 2. The symbol Sr denotes the permutation group
consisting of r elements. The set of invariants of g is denoted by Inv(g).
A set of functionally independent invariants F l(x1, . . . , xn), l = 1, . . . , Ng, forms a functional
basis (fundamental invariant) of Inv(Ad∗G), i.e., any invariant F (x1, . . . , xn) can be uniquely rep-
resented as a function of F l(x1, . . . , xn), l = 1, . . . , Ng. Accordingly the set of SymF
l(e1, . . . , en),
l = 1, . . . , Ng, is called a basis of Inv(g).
Our task here is to determine the basis of the functionally independent invariants for Ad∗G, and
then to transform these invariants into the invariants of the algebra g. Any other invariant of g is
a function of the independent ones.
Let us recall some facts from [9, 10] and adapt them to the particular case of the coadjoint
action of G on g∗. Let G = Ad∗G×g
∗ denote the trivial left principal Ad∗G-bundle over g
∗. The right
regularization R̂ of the coadjoint action of G on g∗ is the diagonal action of Ad∗G on G = Ad
It is provided by the map R̂g(Ad
h, x) = (Ad
h · Ad
,Ad∗gx), g, h ∈ G, x ∈ g
∗, where the action
on the bundle G = Ad∗G × g
∗ is regular and free. We call R̂g the lifted coadjoint action of G.
It projects back to the coadjoint action on g∗ via the Ad∗G-equivariant projection πg∗ : G → g
Any lifted invariant of Ad∗G is a (locally defined) smooth function from G to a manifold, which
is invariant with respect to the lifted coadjoint action of G. The function I : G → g∗ given by
I = I(Ad∗g, x) = Ad
gx is the fundamental lifted invariant of Ad
G, i.e., I is a lifted invariant, and
any lifted invariant can be locally written as a function of I. Using an arbitrary function F (x)
on g∗, we can produce the lifted invariant F ◦ I of Ad∗G by replacing x with I = Ad
gx in the
expression for F . Ordinary invariants are particular cases of lifted invariants, where one identifies
any invariant formed as its composition with the standard projection πg∗. Therefore, ordinary
invariants are particular functional combinations of lifted ones that happen to be independent of
the group parameters of Ad∗G.
The algebraic algorithm for finding invariants of the Lie algebra g is briefly formulated in the
following four steps.
1. Construction of the generic matrix B(θ) of Ad∗G. B(θ) is the matrix of an inner automorphism
of the Lie algebra g in the given basis e1, . . . , en, θ = (θ1, . . . , θr) is a complete tuple of group
parameters (coordinates) of Int(g), and r = dimAd∗G = dim Int(g) = n − dimZ(g), where Z(g) is
the center of g.
2. Representation of the fundamental lifted invariant. The explicit form of the fundamental
lifted invariant I = (I1, . . . ,In) of Ad
G in the chosen coordinates (θ, x̌) in Ad
∗ is I = x̌ ·B(θ),
i.e., (I1, . . . ,In) = (x1, . . . , xn) · B(θ1, . . . , θr).
3. Elimination of parameters by normalization. We choose the maximum possible number ρ of
lifted invariants Ij1 , . . . , Ijρ, constants c1, . . . , cρ and group parameters θk1 , . . . , θkρ such that
the equations Ij1 = c1, . . . , Ijρ = cρ are solvable with respect to θk1 , . . . , θkρ . After substituting
the found values of θk1 , . . . , θkρ into the other lifted invariants, we obtain Ng = n− ρ expressions
F l(x1, . . . , xn) without θ’s.
4. Symmetrization. The functions F l(x1, . . . , xn) necessarily form a basis of Inv(Ad
G). They
are symmetrized to SymF l(e1, . . . , en). It is the desired basis of Inv(g).
Let us give some remarks on the steps of the algorithm, mainly paying attention to the special
features of its variation in this paper, and where it differs from the conventional infinitesimal
method.
Usually, the second canonical coordinate on Int(g) is enough for the first step, although some-
times, the first canonical coordinate on Int(g) is the more appropriate choice. In both the cases, the
matrix B(θ) is calculated by exponentiation from matrices associated with the structure constants.
Often the parameters θ are additionally transformed in a trivial manner (signs, renumbering, re-
denotation etc) for simplification of the final presentation of B(θ). It is also sometimes convenient
for us to introduce ‘virtual’ group parameters corresponding to the center basis elements. Efficient
exploitation of the algorithm imposes certain constrains on the choice of bases for g, in particular,
in the enumeration of their elements; thus automatically yielding simpler expressions for elements
of B(θ) and, therefore, expressions of the lifted invariants. In some cases the simplification is
considerable.
In contrast with the general situation, for the triangular Lie algebras we use special coordinates
for their inner automorphism groups, which naturally harmonize with the canonical matrix rep-
resentations of the corresponding Lie groups and with special ‘matrix’ enumeration of the basis
elements. The application of the individual approach results in the clarification and a substantial
reduction of all calculations. In particular, algebraic systems solved under normalization become
linear with respect to their parameters.
Since B(θ) is a general form matrix from Int(g), it should not be adapted in any way for the
second step.
Indeed, the third step of the algorithm can involve different techniques of elimination of pa-
rameters which are also based on using an explicit form of lifted invariants [3, 4]. The applied
normalization procedure [9, 10] can also be subject to some variations and can applied in a more
involved manner.
As a rule, in complicated cases the main difficulty is created by the determination of the num-
ber ρ, who is actually equal to rankAd∗G, which is equivalent to finding the maximum number Ng
of functionally independent invariants in Inv(Ad∗G), since Ng = dim g − rankAd
G. The rank ρ of
the coadjoint representation Ad∗G can be calculated in different ways, e.g., by the closed formulas
ρ = max
x̌∈Rn
ckijxk
i,j=1
, ρ = max
x̌∈Rn
or with the use of indirect argumentation. The first formula is native to the infinitesimal approach
to invariants (see, e.g., [5, 16, 18, 23] and other references) since it gives the number of algebraically
independent differential equations in the linear system of first-order partial differential equations∑n
j,k=1 c
ijxkFxj = 0, which arises under this approach and is the infinitesimal criterion for invariants
of the algebra g under the fixed basis E . The second formula shows that rankAd∗G coincides
with the maximum dimension of a nonsingular submatrix in the Jacobian matrix ∂I/∂θ. The
tuples of lifted invariants and parameters associated with this submatrix are appropriate for the
normalization procedure, where the constants c1, . . . , cρ are chosen to lie in the range of values of
the corresponding lifted invariants.
If ρ is known then the sufficient number (Ng = dim g − ρ) of functionally independent invari-
ants can be found with various ‘empiric’ techniques in the frameworks of both the infinitesimal
and algebraic approaches. For example, expressions of candidates for invariants can be deduced
from invariants of similar low-dimensional Lie algebras and then tested via substitution to the
infinitesimal criterion for invariants. It is the method used in [23] to describe invariants of the Lie
algebra t0(n) of strictly upper triangular n × n matrices for any fixed n > 2. In the framework of
the algebraic approach, invariants can be constructed via the combination of lifted invariants in
expressions not depending on the group parameters [9, 10]. This method was applied, in particular,
to low-dimensional algebras and the algebra t0(n) [3, 4]. Other empiric techniques, e.g., based on
commutator properties [2] also can be used.
At the same time, a basis of Inv(Ad∗G) may be constructed without first determining the number
of basis elements. Since under such consideration the infinitesimal approach leads to the necessity
of the complete integration of the partial differential equations from the infinitesimal invariant
criterion, the domain of its applicability seems quite narrow (low-dimensional algebras and Lie
algebra of special simple structure). A similar variation of the algebraic method is based on the
following obvious statement.
Proposition 1. Let I = (I1, . . . ,In) be a fundamental lifted invariant, for the lifted invariants
Ij1 , . . . , Ijρ and some constants c1, . . . , cρ the system Ij1 = c1, . . . , Ijρ = cρ be solvable with
respect to the parameters θk1 , . . . , θkρ and substitution of the found values of θk1 , . . . , θkρ into
the other lifted invariants result in m = n− ρ expressions Îl, l = 1, . . . ,m, depending only on x’s.
Then ρ = rankAd∗G, m = Ng and Î1, . . . , Îm form a basis of Inv(Ad
Our experience on the calculation of invariants of a wide range of Lie algebras shows that the
version of the algebraic method, which is based on Proposition 1, is most effective. It is the version
that is used in this paper.
Note that the normalization procedure is difficult to be made algorithmic. There is a big
ambiguity in the choice of the normalization equations. We can take different tuples of ρ lifted
invariants and ρ constants, which lead to systems solvable with respect to ρ parameters. Moreover,
lifted invariants can be additionally combined before forming a system of normalization equations
or substitution of found values of parameters. Another possibility is to use a floating system of
normalization equations (see Section 6.2 of [4]). This means that elements of an invariant basis are
constructed under different normalization constraints. The choice of an optimal method results in
a considerable reduction of calculations and a practical form of constructed invariants.
3 Nilpotent algebra of strictly upper triangular matrices
Consider the nilpotent Lie algebra t0(n) isomorphic to the one of the strictly upper triangular n×n
matrices over the field F, where F is either C or R. t0(n) has dimension n(n − 1)/2. It is the Lie
algebra of the Lie group T0(n) of upper unipotent n × n matrices, i.e., upper triangular matrices
with entries equal to 1 in the diagonal.
As mentioned above, the basis of Inv(t0(n)) was first constructed in a heuristic way in [23]
within the framework of the infinitesimal approach. This result was re-obtained in [4] with the use
of the pure algebraic algorithm first proposed in [3] and developed in [4]. Also, it is the unique
example included among the wide variety of solvable Lie algebras investigated in [4], in which the
‘empiric’ technique of excluding group parameters from lifted invariants was applied. Although
this technique was very effective in constructing a set of functionally independent invariants (calcu-
lations were reduced via a special representation of the coadjoint action to a trivial identity using
matrix determinants, see Note 2), the main difficulty was in proving that the set of invariants is a
basis of Inv(t0(n)), i.e. cardinality of the set equals the maximum possible number of functionally
independent invariants. Under the infinitesimal approach [23] the main difficulty was the same.
In this section we construct a basis of Inv(t0(n)) with the algebraic algorithm but exclude group
parameters from lifted invariants by the normalization procedure. In contrast with the previous
expositions (Section 3 of [23] and Section 8 of [4]), sufficiency of the number of found invariants
for forming a basis of Inv(t0(n)) is proved in the process of calculating them. Investigation of
Inv(t0(n)) in this way gives us a sense of the specific features of the normalization procedure in the
case of Lie algebras having nilradicals isomorphic (or closed) to t0(n).
For the algebra t0(n) we use a ‘matrix’ enumeration of the basis elements with an ‘increasing’
pair of indices, in a similar way to the canonical basis {Enij , i < j} of the isomorphic matrix algebra.
Hereafter Enij (for fixed values of i and j) denotes the n×n matrix (δii′δjj′) with i
′ and j′ running
the numbers of rows and column respectively, i.e., the n × n matrix with a unit element on the
cross of the i-th row and the j-th column, and zero otherwise. En = diag(1, . . . , 1) is the n × n
unity matrix. The indices i, j, k and l run at most from 1 to n. Only additional constraints on the
indices are indicated.
Thus, the basis elements eij ∼ E
ij , i < j, satisfy the commutation relations
[eij , ei′j′] = δi′jeij′− δij′ei′j,
where δij is the Kronecker delta.
Let e∗ji, xji and yij denote the basis element and the coordinate function in the dual space t
and the coordinate function in t0(n), which correspond to the basis element eij , i < j. In particular,
, eij〉 = δii′δjj′. The reverse order of subscripts of the objects associated with the dual space t
is justified by the simplification of a matrix representation of lifted invariants. We complete the
sets of xji and yij in the matrices X and Y with zeros. Hence X is a strictly lower triangular
matrix and Y is a strictly upper triangular one.
We reproduce Lemma 1 from [4] together with its proof, since it is important for further con-
sideration.
Lemma 1. A complete set of independent lifted invariants of Ad∗
T0(n)
is exhaustively given by the
expressions
Iij = xij +
bii′xi′j +
bj′jxij′ +
i<i′, j′<j
bii′ b̂j′jxi′j′ , j < i,
where B = (bij) is an arbitrary matrix from T0(n), and B
−1 = (̂bij) is the inverse matrix of B.
Proof. The adjoint action of B ∈ T0(n) on the matrix Y is AdBY = BYB
−1, i.e.,
yijeij =
(BY B−1)ijeij =
i6i′<j′6j
bii′yi′j′ b̂j′jeij .
After changing eij → xji, yij → e
ji, bij ↔ b̂ij in the latter equality, we obtain the representation of
the coadjoint action of B
i6i′<j′6j
bj′jxjib̂ii′e
j′i′ =
i′<j′
(BXB−1)j′i′e
j′i′ .
Therefore, the elements Iij, j < i, of the matrix I = BXB
−1, B ∈ T0(n), form a complete set of
the independent lifted invariants of Ad∗T0(n).
Note 1. The center of the group T0(n) is Z(T0(n)) = {E
n+b1nE
1n, b1n ∈ F}. The inner automor-
phism group of t0(n) is isomorphic to the factor-group T0(n)/Z(T0(n)) and hence its dimension is
n(n−1)−1. The parameter b1n in the above representation of the lifted invariants is not essential.
Below A
i1,i2
j1,j2
, where i1 6 i2, j1 6 j2, denotes the submatrix (aij)
i=i1,...,i2
j=j1,...,j2
of a matrix A = (aij).
The conjugate value of k with respect to n is denoted by κ, i.e. κ = n − k + 1. The standard
notation |A| = detA is used.
Theorem 1. A basis of Inv(Ad∗
T0(n)
) consists of the polynomials
|, k = 1, . . . ,
Proof. Under normalization we impose the following restriction on the lifted invariants Iij, j < i:
Iij = 0 if j < i, (i, j) 6= (n− j
′ + 1, j′), j′ = 1, . . . ,
It means that we do not only fix the values of the elements of the lifted invariant matrix I, which
are situated on the secondary diagonal, under the main diagonal. The other significant elements
of I are given the value 0. As shown below, the chosen normalization is correct since it provides
satisfying the conditions of Proposition 1.
In view of the (triangular) structure of the matrices B and X the formula I = BXB−1, deter-
mining the lifted invariants implies that BX = IB. This matrix equality is also significant for the
matrix elements underlying the main diagonals of the left and right hand sides, i.e.,
xij +
bii′xi′j = Iij +
Iij′bj′j, j < i.
For convenience we divide the latter system under the chosen normalization conditions into four
sets of subsystems
Sk1 : xκj +
bκi′xi′j = 0, i = κ, j < k, k = 2, . . . ,
Sk2 : xκk +
bκi′xi′k = Iκk, i = κ, j = k, k = 1, . . . ,
Sk3 : xκj +
bκi′xi′j = Iκkbkj, i = κ, k < j < κ, k = 1, . . . ,
Sk4 : xkj +
bki′xi′j = 0, i = k, j < k, k = 2, . . . ,
and solve them one by one. The subsystem S12 consists of the single equation In1 = xn1 which gives
the simplest form of the invariant corresponding to the center of the algebra t0(n). For any fixed
k ∈ {2, . . . , [n/2]} the subsystem Sk1 ∪ S
2 is a well-defined system of linear equations with respect
to bκi′ , i
′ > κ, and Iκk. Solving it, e.g., by the Cramer method, we obtain that bκi′ , i
′ > κ, are
expressions of xi′j , i
′ > κ, j < k, the explicit form of which is not essential in what follows, and
Iκk = (−1)
1,k |
κ+1,n
1,k−1 |
, k = 2, . . . ,
The combination of the found values of Iκk results in the invariants from the statement of the
theorem. The functional independence of these invariants is obvious.
After substituting the expressions of Iκk and bκi′ , i
′ > κ, via x’s, into Sk3 , we trivially resolve
Sk3 with respect to bkj as an uncoupled system of linear equations. In performing the subsequent
substitution of the calculated expressions for bkj to S
4 , for any fixed k, we obtain a well-defined
system of linear equations, e.g., with respect to bki′ , i
′ > κ.
Under the normalization we express the non-normalized lifted invariants via x’s and find a part
of the parameters b’s of the coadjoint action via x’s and the other b’s. No equations involving
only x’s are obtained. In view of Proposition 1, this implies that the choice of the normalization
constraints is correct and, therefore, the number of functionally independent invariants found is
maximal, i.e., they form a basis of Inv(Ad∗T0(n)).
Corollary 1. A basis of Inv(t0(n)) is formed by the Casimir operators
det(eij)
i=1,...,k
j=n−k+1,...,n, k = 1, . . . ,
Proof. Since the basis elements corresponding the coordinate functions from the constructed basis
of Inv(Ad∗T0(n)) commute, the symmetrization procedure is trivial.
Note 2. The set of the invariants from Theorem 1 can be easily found from the equality I = BXB−1
by the following empiric trick used in Lemma 2 from [4]. For any fixed k ∈ {1, . . . , [n/2]} we re-
strict the equality to the submatrix with the row range κ, . . . , n and the column range 1, . . . , k:
1,k = B
1,k (B
1,k. Since |B
κ,n | = |(B
1,k| = 1, we obtain |I
1,k | = |X
1,k |, i.e., |X
1,k |
is an invariant of Ad∗T0(n) in view of the definition of an invariant. Functional independence of
the constructed invariants is obvious. The proof of Nt0(n) = [n/2] is much more difficult (see
Lemma 3 of [4]).
4 Solvable algebra of upper triangular matrices
In a way analogous to the previous section, consider the solvable Lie algebra t(n) isomorphic to the
one of upper triangular n× n matrices. t(n) has dimension n(n+1)/2. It is the Lie algebra of the
Lie group T (n) of nonsingular upper triangular n× n matrices.
Its basis elements are convenient to enumerate with a ‘non-decreasing’ pair of indices in a similar
way to the canonical basis {Enij , i 6 j} of the isomorphic matrix algebra. Thus, the basis elements
eij ∼ E
ij , i 6 j, satisfy the commutation relations
[eij , ei′j′] = δi′jeij′− δij′ei′j,
where δij is the Kronecker delta.
Hereafter the indices i, j, k and l again run at most from 1 to n. Only additional constraints
on the indices are indicated.
The center of t(n) is one-dimensional and coincides with the linear span of the sum e11+ · · ·+enn
corresponding to the unity matrix En. The elements eij , i < j, and e11 + · · ·+ enn form a basis of
the nilradical of t(n), which is isomorphic to t0(n)⊕ a. Here a is the one-dimensional (Abelian) Lie
algebra.
Let e∗ji, xji and yij denote the basis element and the coordinate function in the dual space
t∗(n) and the coordinate function in t(n), which correspond to the basis element eij , i 6 j. Thus,
, eij〉 = δii′δjj′. We complete the sets of xji and yij in the matrices X and Y with zeros. Hence
X is a lower triangular matrix and Y is an upper triangular one.
Lemma 2. A fundamental lifted invariant of Ad∗
T (n) is formed by the expressions
Iij =
i6i′, j′6j
bii′ b̂j′jxi′j′ , j 6 i,
where B = (bij) is an arbitrary matrix from T (n), and B
−1 = (̂bij) is the inverse matrix of B.
Proof. The adjoint action of B ∈ T (n) on the matrix Y is AdBY = BYB
−1, i.e.
yijeij =
(BY B−1)ijeij =
i6i′6j′6j
bii′yi′j′ b̂j′jeij .
After changing eij → xji, yij → e
ji, bij ↔ b̂ij in the latter equality, we obtain the representation
for the coadjoint action of B
i6i′6j′6j
bj′jxjib̂ii′e
j′i′ =
i′6j′
(BXB−1)j′i′e
j′i′ .
Therefore, the elements Iij, j 6 i, of the matrix
I = BXB−1, B ∈ T (n),
form a complete set of the independent lifted invariants of Ad∗T (n).
Note 3. The center of the group T (n) is Z(T (n)) = {βEn | β ∈ F/{0} }. If F = C then the group
T (n) is connected. In the real case the connected component T+(n) of the unity in T (n) is formed
by the matrices from T (n) with positive diagonal elements, i.e., T+(n) ≃ T (n)/Z
2 , where Z
{diag(ε1, . . . , εn) | εi = ±1}. The inner automorphism group Int(t(n)) of t(n) is isomorphic to the
factor-group T (n)/Z(T (n)) (or T+(n)/Z(T (n)) if F is real) and hence its dimension is
n(n+1)−1.
The value of one from the diagonal elements of the matrix B or a homogenous combination of them
in the above representation of lifted invariants can be assumed inessential. It is evident from the
proof of Theorem 2 that in all cases, the invariant sets of the coadjoint representations of Int(t(n))
and t(n) coincide.
Let us remind that A
i1,i2
j1,j2
, where i1 6 i2, j1 6 j2, denotes the submatrix (aij)
i=i1,...,i2
j=j1,...,j2
of a matrix
A = (aij). The conjugate value of k with respect to n is denoted by κ, i.e. κ = n− k + 1.
Under the proof of the below theorem the following technical lemma on matrices is used.
Lemma 3. Suppose 1 < k < n. If |X
κ+1,n
1,k−1 | 6= 0 then for any β ∈ F
1,k−1(X
κ+1,n
1,k−1 )
κ+1,n
j,j =
(−1)k+1
κ+1,n
1,k−1 |
∣∣∣∣∣
1,k−1
κ+1,n
1,k−1
κ+1,n
∣∣∣∣∣ .
In particular, xκk −X
1,k−1(X
κ+1,n
1,k−1 )
κ+1,n
= (−1)k+1|X
κ+1,n
1,k−1 |
1,k |. Analogously
xκj −X
1,k−1
κ+1,n
1,k−1
κ+1,n
xjk −X
1,k−1
κ+1,n
1,k−1
κ+1,n
κ+1,n
1,k−1
∣∣∣∣∣
1,k X
∣∣∣∣∣+
1,k |
κ+1,n
1,k−1
∣∣∣∣∣
1,k−1
κ+1,n
1,k−1 X
κ+1,n
∣∣∣∣∣ .
Theorem 2. A basis of Inv(Ad∗
T (n)) is formed by the rational expressions
1,k |
j=k+1
∣∣∣∣∣
1,k xjj
1,k X
∣∣∣∣∣ , k = 0, . . . ,
where |X
n+1,n
1,0 | := 1.
Proof. We choose the following normalization restriction on the lifted invariants Iij, j 6 i:
In−j+1,j = 1, j = 1, . . . ,
Iij = 0 if j 6 i, (i, j) 6= (j
′, j′), (n − j′ + 1, j′), j′ = 1, . . . ,
This means that we do not only fix the values of the elements of the lifted invariant matrix I, which
are situated on the main diagonal over or on the secondary diagonal. The elements of the secondary
diagonal underlying the main diagonal are given a value of 1. The other significant elements of I
are given a value 0. As shown below, the imposed normalization provides satisfying the conditions
of Proposition 1 and, therefore, is correct.
Similarly to the case of strictly triangular matrices, in view of the (triangular) structure of
the matrices B and X the formula I = BXB−1 determining the lifted invariants implies that
BX = IB. This matrix equality is significant for the matrix elements lying not over the main
diagonals of the left and right hand sides, i.e.,
bii′xi′j =
Iij′bj′j , j 6 i.
For convenience we again divide the latter system under the chosen normalization conditions into
four sets of subsystems
Sk1 :
bκi′xi′j = 0, i = κ, j < k, k = 2, . . . ,
Sk2 :
bκi′xi′j = bkj, i = κ, k 6 j 6 κ, k = 1, . . . ,
Sk3 :
bki′xi′j = 0, i = k, j < k, k = 2, . . . ,
Sk4 :
bki′xi′k = bkkIkk, i = k, j < k, k = 1, . . . ,
and solve them one by one. The subsystem S12 consists of the equations
b1j = bnnxnj
which are already solved with respect to b1j . For any fixed k ∈ {2, . . . , [n/2]} the subsystem S
is a well-defined system of linear equations with respect to bκi′ , i
′ > κ, and bkj, k 6 j 6 κ. We
can solve the subsystem Sk1 with respect to bκi′ , i
′ > κ:
κ+1,n = −bκκX
1,k−1(X
κ+1,n
1,k−1 )
and then substitute the obtained values into the subsystem Sk2 . Another way is to find the expres-
sions for bkj, k 6 j 6 κ, by the Cramer method, from the whole system S
1 ∪ S
2 at once since
only these parameters are further considered. As a result, they have two representations via bκκ
and x’s:
bkj = bκκ
xκj −X
1,k−1
κ+1,n
1,k−1
κ+1,n
(−1)k+1bκκ
κ+1,n
1,k−1
∣∣∣∣∣
1,k−1 xκj
κ+1,n
1,k−1 X
κ+1,n
∣∣∣∣∣ ,
where k 6 j 6 κ. In particular,
bkk = (−1)
k+1bκκ|X
κ+1,n
1,k−1 |
1,k |.
Analogously, for any fixed k ∈ {2, . . . , [(n + 1)/2]} the subsystem Sk3 is a well-defined system of
linear equations with respect to bkj, j > κ, and it implies
κ+1,n = −
k6j6κ
1,k−1(X
κ+1,n
1,k−1 )
Substituting the found expressions for b’s into the equations of the subsystems Sk4 , we completely
exclude the parameters b’s and obtain expressions of Ikk only via x’s. Thus, under k = 1
I11 =
b1ixi1 =
xnixi1 =
xnixi1 =
xi1 xii
xn1 xni
∣∣∣∣+
where the summation range in the first sum can be bounded by 2 and n − 1 since for i = 1 and
i = n the determinants are equal to 0. In the case k ∈ {2, . . . , [(n + 1)/2]}
bkkIkk =
bkixik =
k6j6κ
bkjxjk +
bkixik
k6i6κ
xjk −X
1,k−1(X
κ+1,n
1,k−1 )
κ+1,n
= bκκ
k6i6κ
xκj −X
1,k−1(X
κ+1,n
1,k−1 )
κ+1,n
xjk −X
1,k−1(X
κ+1,n
1,k−1 )
κ+1,n
After using the representation for bnn and manipulations with submatrices of X (see Lemma 3),
we derive that
Ikk =
(−1)k+1
1,k |
k6i6κ
∣∣∣∣∣
∣∣∣∣∣+
(−1)k+1
κ+1,n
1,k−1
k6i6κ
∣∣∣∣∣
1,k−1
κ+1,n
1,k−1
κ+1,n
∣∣∣∣∣ ,
where k = 2, . . . , [(n + 1)/2]. The summation range in the first sum can be taken from k + 1 and
κ − 1 since for i = k and i = κ the determinants are equal to 0.
The combination of the found values of Ikk in the following way
Ĩ00 =
Ijj =
xii, Ĩkk = (−1)
k+1Ikk − Ĩk−1,k−1, k = 1, . . . ,
results in the invariants Ĩk′k′, k
′ = 0, . . . , [(n − 1)/2], from the statement of the theorem. The
functional independence of these invariants is obvious.
Under the normalization we express the non-normalized lifted invariants via x’s and find a part
of the parameters b’s of the coadjoint action via x’s and the other b’s. No equations involving
only x’s are obtained. In view of Proposition 1, this implies that the choice of the normalization
constraints is correct, i.e., the number of the found functionally independent invariant is maximal
and, therefore, they form a basis of Inv(Ad∗
T (n)).
Note 4. An expanded form of the invariants from Theorem 2 is
xj1 xjj
xn1 xnj
∣∣∣∣∣∣
xj1 xj2 xjj
xn−1,1 xn−1,2 xn−1,j
xn1 xn2 xnj
∣∣∣∣∣∣
xn−1,1 xn−1,2
xn1 xn2
, . . . .
The first invariant corresponds to the center of t(n). The invariant tuple ends with
1,n+1
1,n−1
if n is odd and
∣∣∣∣∣∣
∣∣∣∣∣∣
if n is even.
Corollary 2. A basis of Inv(t(n)) consists of the rational invariants
Îk =
j=k+1
∣∣∣∣∣
j,j E
ejj E
∣∣∣∣∣ , k = 0, . . . ,
where E
i1,i2
j1,j2
, i1 6 i2, j1 6 j2, denotes the matrix (eij)
i=i1,...,i2
j=j1,...,j2
n+1,n| := 1, κ = n− k + 1.
Proof. The symmetrization procedure for the tuple of invariants presented in Theorem 2 can be
assumed trivial. To show this, we expand the determinants in each element of the tuple and obtain,
as a result, a rational expression in x’s. Only the monomials from the numerator, which do not
contain the ‘diagonal’ elements xjj, include coordinate functions associated with noncommuting
basis elements of the algebra t(n). More precisely, each of the monomials includes a single pair of
such coordinate functions, namely, xji′ and xj′j for some values i
′ ∈ {1, . . . , k}, j′ ∈ {κ, . . . , n} and
j ∈ {k + 1, . . . ,κ − 1}. Hence, it is sufficient to symmetrize only the corresponding pairs of basis
elements.
After the symmetrization and the transposition of the matrices, we construct the following
expressions for the invariants of t(n) associated with the invariants from Theorem 2:
(−1)k
j=k+1
ejj +
j=k+1
ei′jejj′+ ejj′ei′j
(−1)i
∣∣E1,k;̂i
κ,n;ĵ′
∣∣E1,k;̂i
κ,n;ĵ′
∣∣ denotes the minor of the matrix E1,kκ,n complementary to the element ei′j′. Since
ei′ieij′ = eij′ei′i + ei′j′ then
ei′ieij′+ eij′ei′i
(−1)i
∣∣E1,k;̂i
κ,n;ĵ′
∣∣∣∣∣
i,i E
∣∣∣∣∣±
|E1,k
κ,n|,
where the sign ‘+’ (resp. ‘−’) have to be taken if the elements of E
i,i are placed after (resp. before)
the elements of E
κ,n in all the relevant monomials. Up to constant summands, we obtain the
expressions for the elements of an invariant basis, which are adduced in the statement and formally
derived from the corresponding expressions given in Theorem 2 by the replacement xij → eji and
the transposition of all matrices. That is why the symmetrization procedure can be assumed trivial
in the sense described. The transposition is necessary in order to improve the representation of
invariants since xij ∼ eji, j 6 i.
Note 5. The invariants from Corollary 2 can be rewritten as
Îk =
j=k+1
∣∣∣∣∣
j,j E
∣∣∣∣∣+ (−1)
j=k+1
ejj, k = 0, . . . ,
In particular, Î0 =
j ejj .
Note 6. Let us emphasize that a uniform order of elements from E
and E
κ,n has to be fixed in all
the monomials under usage of the ‘non-symmetrized’ forms of invariants presented in Corollary 2,
Note 5 and Theorem 4 (see below).
5 Solvable algebra of special upper triangular matrices
The Lie algebra st(n) of the special (i.e., having zero traces) upper triangular n × n matrices is
imbedded in a natural way in t(n) as an ideal. dim st(n) = 1
n(n+ 1)− 1. Moreover,
t(n) = st(n)⊕ Z(t(n)),
where Z(t(n)) = 〈e11 + · · · + enn〉 is the center of t(n), which corresponds to the one-dimensional
Abelian Lie algebra of the matrices proportional to En. Due to this fact we can construct a basis of
Inv(st(n)) without the usual calculations involved in finding the basis of Inv(t(n)). It is well known
that if the Lie algebra g is decomposable into the direct sum of Lie algebras g1 and g2 then the
union of bases of Inv(g1) and Inv(g2) is a basis of Inv(g). A basis of Inv(Z(t(n))) obviously consists
of only one element, e.g., e11 + · · · + enn. Therefore, the cardinality of the basis of Inv(st(n)) is
equal to the cardinality of the basis of Inv(t(n)) minus 1, i.e., [(n − 1)/2]. To construct a basis
of Inv(st(n)), it is enough for us to rewrite [(n − 1)/2] functionally independent combinations of
elements from a basis of Inv(t(n)) via elements of st(n) and to exclude the central element from
the basis.
The following basis in st(n) is chosen as a subalgebra of t(n):
eij , i < j, fk =
eii −
i=k+1
eii, k = 1, . . . , n− 1.
(Usage of this basis allows for the presentation of our results in such a form that their identity with
Proposition 1 from [23] becomes absolutely evident.) The commutation relations of st(n) in the
chosen basis are
[eij , ei′j′] = δi′jeij′− δij′ei′j, i < j, i
′ < j′;
[fk, fk′ ] = 0, k, k
′ = 1, . . . , n− 1;
[fk, eij ] = 0, i < j 6 k or k 6 i < j;
[fk, eij ] = eij , i 6 k 6 j, i < j
and, therefore, coincide with those of the algebra L(n, n−1) from [22], i.e., L(n, n−1) is isomorphic
to st(n). Combining this observation with Lemma 6 of [22] results in the following theorem.
Theorem 3. The Lie algebra st(n) has the maximal number of dimensions (equal to 1
n(n+1)−1)
among the solvable Lie algebras which have nilradicals isomorphic to t0(n). It is the unique algebra
with such a property.
Theorem 4. A basis of Inv(st(n)) consists of the rational invariants
Ǐk =
(−1)k+1
j=k+1
∣∣∣∣∣
j,j E
∣∣∣∣∣+ fk − fn−k, k = 1, . . . ,
where E
i1,i2
j1,j2
, i1 6 i2, j1 6 j2, denotes the matrix (eij)
i=i1,...,i2
j=j1,...,j2
n+1,n| := 1, κ = n− k + 1.
Proof. It is enough to observe (see Note 5) that
Ǐk = (−1)
k+1Îk +
n− 2k
Î0, k = 1, . . . ,
These combinations of elements from a basis of Inv(t(n)) are functionally independent. They are
expressed via elements of st(n). Their number is [(n − 1)/2]. Therefore, they form a basis of
Inv(st(n)).
6 Conclusion and discussion
In this paper we extend our purely algebraic approach for computing invariants of Lie algebras by
means of moving frames [3, 4] to the classes of Lie algebras t0(n), t(n) and st(n) of strictly, non-
strictly and special upper triangular matrices of an arbitrary fixed dimension n. In contrast to the
conventional infinitesimal method which involves solving an associated system of PDEs, the main
steps of the applied algorithm are the construction of the matrix B(θ) of inner automorphisms of the
Lie algebra under consideration, and the exclusion of the parameters θ from the algebraic system
I = x̌ · B(θ) in some way. The version of the algorithm, applied in this paper, is distinguished in
that a special usage of the normalization procedure when the number, and a form of elements in a
functional basis of an invariant set, are determined under excluding the parameters simultaneously.
A basis of Inv(t0(n)) was already known and constructed by both the infinitesimal method [23]
and the algebraic algorithm with an elegant but empiric technique of excluding the parameters [4].
Note that the proof introduced in [23] is very sophisticated and was completed only due to the
thorough mastery of the used infinitesimal method. A form of elements from a functional basis of
Inv(t0(n)) was guessed via calculation of bases for a number of small n’s and then justified with
the infinitesimal method, and both the testing steps (on invariance and on sufficiency of number)
were quite complicated.
Invariants of t0(n) are considered in this paper in order to demonstrate the advantages of the
normalization technique and to pave the way for further applications of this technique to the more
complicated algebras t(n) and st(n), being too complex for the infinitesimal method (only the
lowest few were completely investigated there). First the invariants of the algebras t(n) and st(n)
are exhaustively studied in this paper. The performed calculations are simple and clear since the
normalization procedure is reduced by the choice of natural coordinates on the inner automorphism
groups and by the use of a special normalization technique to solving a linear system of algebraic
equations. The results obtained for Inv(st(n)) in Theorem 4 completely agree with the conjecture
formulated as Proposition 1 in [23] on the number and form of basis elements of this invariant set.
A direct extension of the present investigation is to describe the invariants of the subalgebras
of st(n), which contain t0(n). Such subalgebras exhaust the set of solvable Lie algebras which can be
embedded in the matrix Lie algebra gl(n) and have the nilradicals isomorphic to t0(n). A technique
similar to that used in this paper can be applied. The main difficulties will be created by breaking
in symmetry and complication of coadjoint representations. The question on ways of investigation
of the other solvable Lie algebras with the nilradicals isomorphic to t0(n) remains open. (See, e.g.,
[22] for classification of the algebras of such type.)
A more general problem is to circumscribe an applicability domain of the developed algebraic
method. It has been already applied only to the low-dimensional Lie algebras and a wide range of
classes of solvable Lie algebras in [3, 4] and this paper. The next step which should be performed
is the extension of the method to classes of unsolvable Lie algebras of arbitrary dimensions, e.g.,
with fixed structures of radicals or Levi factors. An adjoining problem is the implementation of
the algorithm with symbolic calculation systems. Similar work has already began in the framework
of the general method of moving frames, e.g., in the case of rational invariants for rational actions
of algebraic groups [11]. Some other possibilities on the applications of the algorithm are outlined
in [4].
Acknowledgments. The work was partially supported by the National Science and Engineering
Research Council of Canada, by MITACS. The research of R.P. was supported by Austrian Science
Fund (FWF), Lise Meitner project M923-N13. V. B. is grateful for the hospitality the Centre de
Recherches Mathématiques, Université de Montréal.
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http://arxiv.org/abs/math-ph/0511027
http://arxiv.org/abs/math-ph/0602046
http://arxiv.org/abs/math-ph/0606045
http://arxiv.org/abs/math-ph/0602065
http://arxiv.org/abs/math-ph/0411023
http://arxiv.org/abs/0709.3581
http://arxiv.org/abs/0709.3116
	Introduction
	The algorithm
	Nilpotent algebra of strictly upper triangular matrices
	Solvable algebra of upper triangular matrices
	Solvable algebra of special upper triangular matrices
	Conclusion and discussion

  Triangular Lie algebras are the Lie algebras which can be faithfully
represented by triangular matrices of any finite size over the real/complex
number field. In the paper invariants ('generalized Casimir operators') are
found for three classes of Lie algebras, namely those which are either strictly
or non-strictly triangular, and for so-called special upper triangular Lie
algebras. Algebraic algorithm of [J. Phys. A: Math. Gen., 2006, V.39, 5749;
math-ph/0602046], developed further in [J. Phys. A: Math. Theor., 2007, V.40,
113; math-ph/0606045], is used to determine the invariants. A conjecture of [J.
Phys. A: Math. Gen., 2001, V.34, 9085], concerning the number of independent
invariants and their form, is corroborated.

Introduction
The invariants of Lie algebras are one of their defining characteristics. They have numerous appli-
cations in different fields of mathematics and physics, in which Lie algebras arise (representation
theory, integrability of Hamiltonian differential equations, quantum numbers etc). In particular,
the polynomial invariants of a Lie algebra exhaust its set of Casimir operators, i.e., the center of
its universal enveloping algebra. That is why non-polynomial invariants are also called general-
ized Casimir operators, and the usual Casimir operators are seen as ‘trivial’ generalized Casimir
operators. Since the structure of invariants strongly depends on the structure of the algebra and
the classification of all (finite-dimensional) Lie algebras is an inherently difficult problem (actually
unsolvable), it seems to be impossible to elaborate a complete theory for generalized Casimir op-
erators in the general case. Moreover, if the classification of a class of Lie algebras is known, then
the invariants of such algebras can be described exhaustively. These problems have already been
solved for the semi-simple and low-dimensional Lie algebras, and also for the physically relevant
Lie algebras of fixed dimensions (see, e.g., references in [3, 7, 8, 18, 19]).
The actual problem is the investigation of generalized Casimir operators for classes of solvable Lie
algebras or non-solvable Lie algebras with non-trivial radicals of arbitrary finite dimension. There
are a number of papers on the partial classification of such algebras and the subsequent calculation
of their invariants [1, 6, 7, 14, 15, 16, 20, 21, 22, 23]. In particular, Tremblay and Winternitz [22]
classified all the solvable Lie algebras with the nilradicals isomorphic to the nilpotent algebra t0(n)
of strictly upper triangular matrices for any fixed dimension n. Then in [23] invariants of these
algebras were considered. The case n = 4 was investigated exhaustively. After calculating the
invariants for a sufficiently large value of n, Tremblay and Winternitz made conjectures for an
arbitrary n on the number and form of functionally independent invariants of the algebra t0(n),
and the ‘diagonal’ solvable Lie algebras having t0(n) as their nilradicals and possessing either the
maximal (equal to n − 1) or minimal (one) number of nilindependent elements. A statement on a
functional basis of invariants was only proved completely for the algebra t0(n). The infinitesimal
invariant criterion was used for the construction of the invariants. Such an approach entails the
http://arxiv.org/abs/0704.0937v4
http://arxiv.org/abs/math-ph/0602046
http://arxiv.org/abs/math-ph/0606045
necessity of solving a system of ρ first-order linear partial differential equations, where ρ has the
order of the algebra’s dimension. This is why the calculations were very cumbersome and results
were obtained due to the thorough mastery of the method.
In this paper, we use our original algebraic method for the construction of the invariants (‘gen-
eralized Casimir operators’) of Lie algebras via the moving frames approach [3, 4]. The algorithm
makes use of the knowledge of the associated inner automorphism groups and Cartan’s method
of moving frames in its Fels–Olver version [9, 10]. (For modern developments about the moving
frame method and more references, see also [17].) Unlike standard infinitesimal methods, it allows
us to avoid solving systems of differential equations, replacing them instead by algebraic equations.
As a result, the application of the algorithm is simpler. Note that a closed approach was earlier
proposed in [12, 13, 19] for the specific case of inhomogeneous algebras.
The invariants of three classes of triangular Lie algebras are exhaustively investigated (below n
is an arbitrary integer):
• nilpotent Lie algebras t0(n) of n× n strictly upper triangular matrices (Section 3);
• solvable Lie algebras t(n) of n× n upper triangular matrices (Section 4);
• solvable Lie algebras st(n) of n× n special upper triangular matrices (Section 5).
The triangular algebras are especially interesting due to their ‘universality’ properties. More pre-
cisely, any finite-dimensional nilpotent Lie algebra is isomorphic to a subalgebra of t0(n). Similarly,
any finite-dimensional solvable Lie algebra over an algebraically closed field of characteristic 0 (e.g.,
over C) can be embedded as a subalgebra in t(n) (or st(n)).
We have adapted and optimized our algorithm for the specific case of triangular Lie algebras via
special double enumeration of basis elements, individual choice of coordinates in the corresponding
inner automorphism groups and an appropriate modification of the normalization procedure of
the moving frame method. As a result, the problems related to the construction of functional
bases of invariants are reduced for the algebras t0(n) and t(n) to solving linear systems of algebraic
equations! Let us note that due to the natural embedding of st(n) to t(n) and the representation
t(n) = st(n) ⊕ Z(t(n)), where Z(t(n)) is the center of t(n), we can construct a basis in the set
of invariants of st(n) without the usual calculations from a previously found basis in the set of
invariants of t(n).
We re-prove the statement for a basis of invariants of t0(n), which was first constructed in [23]
using the infinitesimal method in a heuristic way, thereafter constructed in [4] using an empiric
technique based on the exclusion of parameters within the framework of the algebraic method. The
aim of this paper in considering t0(n) is to test and better understand the technique of working
with triangular algebras. The calculations for t(n) are similar, albeit more complex, although they
are much clearer and easier than under the standard infinitesimal approach.
As proved in [22], there is a unique algebra with the nilradical t0(n) that contains a maximum
possible number (n− 1) of nilindependent elements. A conjecture on the invariants of this algebra
is formulated in Proposition 1 of [23]. We show that this algebra is isomorphic to st(n). As a result,
the conjecture by Tremblay and Winternitz on its invariants is effectively proved.
2 The algorithm
The applied algebraic algorithm was first proposed in [3] and then developed in [4]. Ibid it was
effectively tested for the low-dimensional Lie algebras and a wide range of solvable Lie algebras
with a fixed structure of nilradicals. The presentation of the algorithm here differs from [3, 4], the
differences being important within the framework of applications.
For convenience of the reader and to introduce some necessary notations, before the description
of the algorithm, we briefly repeat the preliminaries given in [3, 4] about the statement of the
problem of calculating Lie algebra invariants, and on the implementation of the moving frame
method [9, 10]. The comparative analysis of the standard infinitesimal and the presented algebraic
methods, as well as their modifications, is given in the second part of this section.
Consider a Lie algebra g of dimension dim g = n < ∞ over the complex or real field and the
corresponding connected Lie group G. Let g∗ be the dual space of the vector space g. The map
Ad∗ : G → GL(g∗), defined for any g ∈ G by the relation
〈Ad∗gx, u〉 = 〈x,Adg−1u〉 for all x ∈ g
∗ and u ∈ g
is called the coadjoint representation of the Lie group G. Here Ad: G → GL(g) is the usual adjoint
representation of G in g, and the image AdG of G under Ad is the inner automorphism group Int(g)
of the Lie algebra g. The image of G under Ad∗ is a subgroup of GL(g∗) and is denoted by Ad∗G.
A function F ∈ C∞(g∗) is called an invariant of Ad∗G if F (Ad
gx) = F (x) for all g ∈ G and x ∈ g
The set of invariants of Ad∗G is denoted by Inv(Ad
G). The maximal number Ng of functionally
independent invariants in Inv(Ad∗G) coincides with the codimension of the regular orbits of Ad
i.e., it is given by the difference
Ng = dim g− rankAd
Here rankAd∗G denotes the dimension of the regular orbits of Ad
G and will be called the rank of the
coadjoint representation of G (and of g). It is a basis independent characteristic of the algebra g,
the same as dim g and Ng.
To calculate the invariants explicitly, one should fix a basis E = {e1, . . . , en} of the algebra g.
It leads to fixing the dual basis E∗ = {e∗1, . . . , e
n} in the dual space g
∗ and to the identification
of Int(g) and Ad∗G with the associated matrix groups. The basis elements e1, . . . , en satisfy the
commutation relations [ei, ej ] =
k=1 c
ijek, i, j = 1, . . . , n, where c
ij are components of the tensor
of structure constants of g in the basis E .
Let x → x̌ = (x1, . . . , xn) be the coordinates in g
∗ associated with E∗. Given any invariant
F (x1, . . . , xn) of Ad
G, one finds the corresponding invariant of the Lie algebra g by symmetriza-
tion, SymF (e1, . . . , en), of F . It is often called a generalized Casimir operator of g. If F is
a polynomial, SymF (e1, . . . , en) is a usual Casimir operator, i.e., an element of the center of the
universal enveloping algebra of g. More precisely, the symmetrization operator Sym acts only on
the monomials of the forms ei1 · · · eir , where there are non-commuting elements among ei1 , . . . , eir ,
and is defined by the formula
Sym(ei1 · · · eir) =
eiσ1 · · · eiσr ,
where i1, . . . , ir take values from 1 to n, r > 2. The symbol Sr denotes the permutation group
consisting of r elements. The set of invariants of g is denoted by Inv(g).
A set of functionally independent invariants F l(x1, . . . , xn), l = 1, . . . , Ng, forms a functional
basis (fundamental invariant) of Inv(Ad∗G), i.e., any invariant F (x1, . . . , xn) can be uniquely rep-
resented as a function of F l(x1, . . . , xn), l = 1, . . . , Ng. Accordingly the set of SymF
l(e1, . . . , en),
l = 1, . . . , Ng, is called a basis of Inv(g).
Our task here is to determine the basis of the functionally independent invariants for Ad∗G, and
then to transform these invariants into the invariants of the algebra g. Any other invariant of g is
a function of the independent ones.
Let us recall some facts from [9, 10] and adapt them to the particular case of the coadjoint
action of G on g∗. Let G = Ad∗G×g
∗ denote the trivial left principal Ad∗G-bundle over g
∗. The right
regularization R̂ of the coadjoint action of G on g∗ is the diagonal action of Ad∗G on G = Ad
It is provided by the map R̂g(Ad
h, x) = (Ad
h · Ad
,Ad∗gx), g, h ∈ G, x ∈ g
∗, where the action
on the bundle G = Ad∗G × g
∗ is regular and free. We call R̂g the lifted coadjoint action of G.
It projects back to the coadjoint action on g∗ via the Ad∗G-equivariant projection πg∗ : G → g
Any lifted invariant of Ad∗G is a (locally defined) smooth function from G to a manifold, which
is invariant with respect to the lifted coadjoint action of G. The function I : G → g∗ given by
I = I(Ad∗g, x) = Ad
gx is the fundamental lifted invariant of Ad
G, i.e., I is a lifted invariant, and
any lifted invariant can be locally written as a function of I. Using an arbitrary function F (x)
on g∗, we can produce the lifted invariant F ◦ I of Ad∗G by replacing x with I = Ad
gx in the
expression for F . Ordinary invariants are particular cases of lifted invariants, where one identifies
any invariant formed as its composition with the standard projection πg∗. Therefore, ordinary
invariants are particular functional combinations of lifted ones that happen to be independent of
the group parameters of Ad∗G.
The algebraic algorithm for finding invariants of the Lie algebra g is briefly formulated in the
following four steps.
1. Construction of the generic matrix B(θ) of Ad∗G. B(θ) is the matrix of an inner automorphism
of the Lie algebra g in the given basis e1, . . . , en, θ = (θ1, . . . , θr) is a complete tuple of group
parameters (coordinates) of Int(g), and r = dimAd∗G = dim Int(g) = n − dimZ(g), where Z(g) is
the center of g.
2. Representation of the fundamental lifted invariant. The explicit form of the fundamental
lifted invariant I = (I1, . . . ,In) of Ad
G in the chosen coordinates (θ, x̌) in Ad
∗ is I = x̌ ·B(θ),
i.e., (I1, . . . ,In) = (x1, . . . , xn) · B(θ1, . . . , θr).
3. Elimination of parameters by normalization. We choose the maximum possible number ρ of
lifted invariants Ij1 , . . . , Ijρ, constants c1, . . . , cρ and group parameters θk1 , . . . , θkρ such that
the equations Ij1 = c1, . . . , Ijρ = cρ are solvable with respect to θk1 , . . . , θkρ . After substituting
the found values of θk1 , . . . , θkρ into the other lifted invariants, we obtain Ng = n− ρ expressions
F l(x1, . . . , xn) without θ’s.
4. Symmetrization. The functions F l(x1, . . . , xn) necessarily form a basis of Inv(Ad
G). They
are symmetrized to SymF l(e1, . . . , en). It is the desired basis of Inv(g).
Let us give some remarks on the steps of the algorithm, mainly paying attention to the special
features of its variation in this paper, and where it differs from the conventional infinitesimal
method.
Usually, the second canonical coordinate on Int(g) is enough for the first step, although some-
times, the first canonical coordinate on Int(g) is the more appropriate choice. In both the cases, the
matrix B(θ) is calculated by exponentiation from matrices associated with the structure constants.
Often the parameters θ are additionally transformed in a trivial manner (signs, renumbering, re-
denotation etc) for simplification of the final presentation of B(θ). It is also sometimes convenient
for us to introduce ‘virtual’ group parameters corresponding to the center basis elements. Efficient
exploitation of the algorithm imposes certain constrains on the choice of bases for g, in particular,
in the enumeration of their elements; thus automatically yielding simpler expressions for elements
of B(θ) and, therefore, expressions of the lifted invariants. In some cases the simplification is
considerable.
In contrast with the general situation, for the triangular Lie algebras we use special coordinates
for their inner automorphism groups, which naturally harmonize with the canonical matrix rep-
resentations of the corresponding Lie groups and with special ‘matrix’ enumeration of the basis
elements. The application of the individual approach results in the clarification and a substantial
reduction of all calculations. In particular, algebraic systems solved under normalization become
linear with respect to their parameters.
Since B(θ) is a general form matrix from Int(g), it should not be adapted in any way for the
second step.
Indeed, the third step of the algorithm can involve different techniques of elimination of pa-
rameters which are also based on using an explicit form of lifted invariants [3, 4]. The applied
normalization procedure [9, 10] can also be subject to some variations and can applied in a more
involved manner.
As a rule, in complicated cases the main difficulty is created by the determination of the num-
ber ρ, who is actually equal to rankAd∗G, which is equivalent to finding the maximum number Ng
of functionally independent invariants in Inv(Ad∗G), since Ng = dim g − rankAd
G. The rank ρ of
the coadjoint representation Ad∗G can be calculated in different ways, e.g., by the closed formulas
ρ = max
x̌∈Rn
ckijxk
i,j=1
, ρ = max
x̌∈Rn
or with the use of indirect argumentation. The first formula is native to the infinitesimal approach
to invariants (see, e.g., [5, 16, 18, 23] and other references) since it gives the number of algebraically
independent differential equations in the linear system of first-order partial differential equations∑n
j,k=1 c
ijxkFxj = 0, which arises under this approach and is the infinitesimal criterion for invariants
of the algebra g under the fixed basis E . The second formula shows that rankAd∗G coincides
with the maximum dimension of a nonsingular submatrix in the Jacobian matrix ∂I/∂θ. The
tuples of lifted invariants and parameters associated with this submatrix are appropriate for the
normalization procedure, where the constants c1, . . . , cρ are chosen to lie in the range of values of
the corresponding lifted invariants.
If ρ is known then the sufficient number (Ng = dim g − ρ) of functionally independent invari-
ants can be found with various ‘empiric’ techniques in the frameworks of both the infinitesimal
and algebraic approaches. For example, expressions of candidates for invariants can be deduced
from invariants of similar low-dimensional Lie algebras and then tested via substitution to the
infinitesimal criterion for invariants. It is the method used in [23] to describe invariants of the Lie
algebra t0(n) of strictly upper triangular n × n matrices for any fixed n > 2. In the framework of
the algebraic approach, invariants can be constructed via the combination of lifted invariants in
expressions not depending on the group parameters [9, 10]. This method was applied, in particular,
to low-dimensional algebras and the algebra t0(n) [3, 4]. Other empiric techniques, e.g., based on
commutator properties [2] also can be used.
At the same time, a basis of Inv(Ad∗G) may be constructed without first determining the number
of basis elements. Since under such consideration the infinitesimal approach leads to the necessity
of the complete integration of the partial differential equations from the infinitesimal invariant
criterion, the domain of its applicability seems quite narrow (low-dimensional algebras and Lie
algebra of special simple structure). A similar variation of the algebraic method is based on the
following obvious statement.
Proposition 1. Let I = (I1, . . . ,In) be a fundamental lifted invariant, for the lifted invariants
Ij1 , . . . , Ijρ and some constants c1, . . . , cρ the system Ij1 = c1, . . . , Ijρ = cρ be solvable with
respect to the parameters θk1 , . . . , θkρ and substitution of the found values of θk1 , . . . , θkρ into
the other lifted invariants result in m = n− ρ expressions Îl, l = 1, . . . ,m, depending only on x’s.
Then ρ = rankAd∗G, m = Ng and Î1, . . . , Îm form a basis of Inv(Ad
Our experience on the calculation of invariants of a wide range of Lie algebras shows that the
version of the algebraic method, which is based on Proposition 1, is most effective. It is the version
that is used in this paper.
Note that the normalization procedure is difficult to be made algorithmic. There is a big
ambiguity in the choice of the normalization equations. We can take different tuples of ρ lifted
invariants and ρ constants, which lead to systems solvable with respect to ρ parameters. Moreover,
lifted invariants can be additionally combined before forming a system of normalization equations
or substitution of found values of parameters. Another possibility is to use a floating system of
normalization equations (see Section 6.2 of [4]). This means that elements of an invariant basis are
constructed under different normalization constraints. The choice of an optimal method results in
a considerable reduction of calculations and a practical form of constructed invariants.
3 Nilpotent algebra of strictly upper triangular matrices
Consider the nilpotent Lie algebra t0(n) isomorphic to the one of the strictly upper triangular n×n
matrices over the field F, where F is either C or R. t0(n) has dimension n(n − 1)/2. It is the Lie
algebra of the Lie group T0(n) of upper unipotent n × n matrices, i.e., upper triangular matrices
with entries equal to 1 in the diagonal.
As mentioned above, the basis of Inv(t0(n)) was first constructed in a heuristic way in [23]
within the framework of the infinitesimal approach. This result was re-obtained in [4] with the use
of the pure algebraic algorithm first proposed in [3] and developed in [4]. Also, it is the unique
example included among the wide variety of solvable Lie algebras investigated in [4], in which the
‘empiric’ technique of excluding group parameters from lifted invariants was applied. Although
this technique was very effective in constructing a set of functionally independent invariants (calcu-
lations were reduced via a special representation of the coadjoint action to a trivial identity using
matrix determinants, see Note 2), the main difficulty was in proving that the set of invariants is a
basis of Inv(t0(n)), i.e. cardinality of the set equals the maximum possible number of functionally
independent invariants. Under the infinitesimal approach [23] the main difficulty was the same.
In this section we construct a basis of Inv(t0(n)) with the algebraic algorithm but exclude group
parameters from lifted invariants by the normalization procedure. In contrast with the previous
expositions (Section 3 of [23] and Section 8 of [4]), sufficiency of the number of found invariants
for forming a basis of Inv(t0(n)) is proved in the process of calculating them. Investigation of
Inv(t0(n)) in this way gives us a sense of the specific features of the normalization procedure in the
case of Lie algebras having nilradicals isomorphic (or closed) to t0(n).
For the algebra t0(n) we use a ‘matrix’ enumeration of the basis elements with an ‘increasing’
pair of indices, in a similar way to the canonical basis {Enij , i < j} of the isomorphic matrix algebra.
Hereafter Enij (for fixed values of i and j) denotes the n×n matrix (δii′δjj′) with i
′ and j′ running
the numbers of rows and column respectively, i.e., the n × n matrix with a unit element on the
cross of the i-th row and the j-th column, and zero otherwise. En = diag(1, . . . , 1) is the n × n
unity matrix. The indices i, j, k and l run at most from 1 to n. Only additional constraints on the
indices are indicated.
Thus, the basis elements eij ∼ E
ij , i < j, satisfy the commutation relations
[eij , ei′j′] = δi′jeij′− δij′ei′j,
where δij is the Kronecker delta.
Let e∗ji, xji and yij denote the basis element and the coordinate function in the dual space t
and the coordinate function in t0(n), which correspond to the basis element eij , i < j. In particular,
, eij〉 = δii′δjj′. The reverse order of subscripts of the objects associated with the dual space t
is justified by the simplification of a matrix representation of lifted invariants. We complete the
sets of xji and yij in the matrices X and Y with zeros. Hence X is a strictly lower triangular
matrix and Y is a strictly upper triangular one.
We reproduce Lemma 1 from [4] together with its proof, since it is important for further con-
sideration.
Lemma 1. A complete set of independent lifted invariants of Ad∗
T0(n)
is exhaustively given by the
expressions
Iij = xij +
bii′xi′j +
bj′jxij′ +
i<i′, j′<j
bii′ b̂j′jxi′j′ , j < i,
where B = (bij) is an arbitrary matrix from T0(n), and B
−1 = (̂bij) is the inverse matrix of B.
Proof. The adjoint action of B ∈ T0(n) on the matrix Y is AdBY = BYB
−1, i.e.,
yijeij =
(BY B−1)ijeij =
i6i′<j′6j
bii′yi′j′ b̂j′jeij .
After changing eij → xji, yij → e
ji, bij ↔ b̂ij in the latter equality, we obtain the representation of
the coadjoint action of B
i6i′<j′6j
bj′jxjib̂ii′e
j′i′ =
i′<j′
(BXB−1)j′i′e
j′i′ .
Therefore, the elements Iij, j < i, of the matrix I = BXB
−1, B ∈ T0(n), form a complete set of
the independent lifted invariants of Ad∗T0(n).
Note 1. The center of the group T0(n) is Z(T0(n)) = {E
n+b1nE
1n, b1n ∈ F}. The inner automor-
phism group of t0(n) is isomorphic to the factor-group T0(n)/Z(T0(n)) and hence its dimension is
n(n−1)−1. The parameter b1n in the above representation of the lifted invariants is not essential.
Below A
i1,i2
j1,j2
, where i1 6 i2, j1 6 j2, denotes the submatrix (aij)
i=i1,...,i2
j=j1,...,j2
of a matrix A = (aij).
The conjugate value of k with respect to n is denoted by κ, i.e. κ = n − k + 1. The standard
notation |A| = detA is used.
Theorem 1. A basis of Inv(Ad∗
T0(n)
) consists of the polynomials
|, k = 1, . . . ,
Proof. Under normalization we impose the following restriction on the lifted invariants Iij, j < i:
Iij = 0 if j < i, (i, j) 6= (n− j
′ + 1, j′), j′ = 1, . . . ,
It means that we do not only fix the values of the elements of the lifted invariant matrix I, which
are situated on the secondary diagonal, under the main diagonal. The other significant elements
of I are given the value 0. As shown below, the chosen normalization is correct since it provides
satisfying the conditions of Proposition 1.
In view of the (triangular) structure of the matrices B and X the formula I = BXB−1, deter-
mining the lifted invariants implies that BX = IB. This matrix equality is also significant for the
matrix elements underlying the main diagonals of the left and right hand sides, i.e.,
xij +
bii′xi′j = Iij +
Iij′bj′j, j < i.
For convenience we divide the latter system under the chosen normalization conditions into four
sets of subsystems
Sk1 : xκj +
bκi′xi′j = 0, i = κ, j < k, k = 2, . . . ,
Sk2 : xκk +
bκi′xi′k = Iκk, i = κ, j = k, k = 1, . . . ,
Sk3 : xκj +
bκi′xi′j = Iκkbkj, i = κ, k < j < κ, k = 1, . . . ,
Sk4 : xkj +
bki′xi′j = 0, i = k, j < k, k = 2, . . . ,
and solve them one by one. The subsystem S12 consists of the single equation In1 = xn1 which gives
the simplest form of the invariant corresponding to the center of the algebra t0(n). For any fixed
k ∈ {2, . . . , [n/2]} the subsystem Sk1 ∪ S
2 is a well-defined system of linear equations with respect
to bκi′ , i
′ > κ, and Iκk. Solving it, e.g., by the Cramer method, we obtain that bκi′ , i
′ > κ, are
expressions of xi′j , i
′ > κ, j < k, the explicit form of which is not essential in what follows, and
Iκk = (−1)
1,k |
κ+1,n
1,k−1 |
, k = 2, . . . ,
The combination of the found values of Iκk results in the invariants from the statement of the
theorem. The functional independence of these invariants is obvious.
After substituting the expressions of Iκk and bκi′ , i
′ > κ, via x’s, into Sk3 , we trivially resolve
Sk3 with respect to bkj as an uncoupled system of linear equations. In performing the subsequent
substitution of the calculated expressions for bkj to S
4 , for any fixed k, we obtain a well-defined
system of linear equations, e.g., with respect to bki′ , i
′ > κ.
Under the normalization we express the non-normalized lifted invariants via x’s and find a part
of the parameters b’s of the coadjoint action via x’s and the other b’s. No equations involving
only x’s are obtained. In view of Proposition 1, this implies that the choice of the normalization
constraints is correct and, therefore, the number of functionally independent invariants found is
maximal, i.e., they form a basis of Inv(Ad∗T0(n)).
Corollary 1. A basis of Inv(t0(n)) is formed by the Casimir operators
det(eij)
i=1,...,k
j=n−k+1,...,n, k = 1, . . . ,
Proof. Since the basis elements corresponding the coordinate functions from the constructed basis
of Inv(Ad∗T0(n)) commute, the symmetrization procedure is trivial.
Note 2. The set of the invariants from Theorem 1 can be easily found from the equality I = BXB−1
by the following empiric trick used in Lemma 2 from [4]. For any fixed k ∈ {1, . . . , [n/2]} we re-
strict the equality to the submatrix with the row range κ, . . . , n and the column range 1, . . . , k:
1,k = B
1,k (B
1,k. Since |B
κ,n | = |(B
1,k| = 1, we obtain |I
1,k | = |X
1,k |, i.e., |X
1,k |
is an invariant of Ad∗T0(n) in view of the definition of an invariant. Functional independence of
the constructed invariants is obvious. The proof of Nt0(n) = [n/2] is much more difficult (see
Lemma 3 of [4]).
4 Solvable algebra of upper triangular matrices
In a way analogous to the previous section, consider the solvable Lie algebra t(n) isomorphic to the
one of upper triangular n× n matrices. t(n) has dimension n(n+1)/2. It is the Lie algebra of the
Lie group T (n) of nonsingular upper triangular n× n matrices.
Its basis elements are convenient to enumerate with a ‘non-decreasing’ pair of indices in a similar
way to the canonical basis {Enij , i 6 j} of the isomorphic matrix algebra. Thus, the basis elements
eij ∼ E
ij , i 6 j, satisfy the commutation relations
[eij , ei′j′] = δi′jeij′− δij′ei′j,
where δij is the Kronecker delta.
Hereafter the indices i, j, k and l again run at most from 1 to n. Only additional constraints
on the indices are indicated.
The center of t(n) is one-dimensional and coincides with the linear span of the sum e11+ · · ·+enn
corresponding to the unity matrix En. The elements eij , i < j, and e11 + · · ·+ enn form a basis of
the nilradical of t(n), which is isomorphic to t0(n)⊕ a. Here a is the one-dimensional (Abelian) Lie
algebra.
Let e∗ji, xji and yij denote the basis element and the coordinate function in the dual space
t∗(n) and the coordinate function in t(n), which correspond to the basis element eij , i 6 j. Thus,
, eij〉 = δii′δjj′. We complete the sets of xji and yij in the matrices X and Y with zeros. Hence
X is a lower triangular matrix and Y is an upper triangular one.
Lemma 2. A fundamental lifted invariant of Ad∗
T (n) is formed by the expressions
Iij =
i6i′, j′6j
bii′ b̂j′jxi′j′ , j 6 i,
where B = (bij) is an arbitrary matrix from T (n), and B
−1 = (̂bij) is the inverse matrix of B.
Proof. The adjoint action of B ∈ T (n) on the matrix Y is AdBY = BYB
−1, i.e.
yijeij =
(BY B−1)ijeij =
i6i′6j′6j
bii′yi′j′ b̂j′jeij .
After changing eij → xji, yij → e
ji, bij ↔ b̂ij in the latter equality, we obtain the representation
for the coadjoint action of B
i6i′6j′6j
bj′jxjib̂ii′e
j′i′ =
i′6j′
(BXB−1)j′i′e
j′i′ .
Therefore, the elements Iij, j 6 i, of the matrix
I = BXB−1, B ∈ T (n),
form a complete set of the independent lifted invariants of Ad∗T (n).
Note 3. The center of the group T (n) is Z(T (n)) = {βEn | β ∈ F/{0} }. If F = C then the group
T (n) is connected. In the real case the connected component T+(n) of the unity in T (n) is formed
by the matrices from T (n) with positive diagonal elements, i.e., T+(n) ≃ T (n)/Z
2 , where Z
{diag(ε1, . . . , εn) | εi = ±1}. The inner automorphism group Int(t(n)) of t(n) is isomorphic to the
factor-group T (n)/Z(T (n)) (or T+(n)/Z(T (n)) if F is real) and hence its dimension is
n(n+1)−1.
The value of one from the diagonal elements of the matrix B or a homogenous combination of them
in the above representation of lifted invariants can be assumed inessential. It is evident from the
proof of Theorem 2 that in all cases, the invariant sets of the coadjoint representations of Int(t(n))
and t(n) coincide.
Let us remind that A
i1,i2
j1,j2
, where i1 6 i2, j1 6 j2, denotes the submatrix (aij)
i=i1,...,i2
j=j1,...,j2
of a matrix
A = (aij). The conjugate value of k with respect to n is denoted by κ, i.e. κ = n− k + 1.
Under the proof of the below theorem the following technical lemma on matrices is used.
Lemma 3. Suppose 1 < k < n. If |X
κ+1,n
1,k−1 | 6= 0 then for any β ∈ F
1,k−1(X
κ+1,n
1,k−1 )
κ+1,n
j,j =
(−1)k+1
κ+1,n
1,k−1 |
∣∣∣∣∣
1,k−1
κ+1,n
1,k−1
κ+1,n
∣∣∣∣∣ .
In particular, xκk −X
1,k−1(X
κ+1,n
1,k−1 )
κ+1,n
= (−1)k+1|X
κ+1,n
1,k−1 |
1,k |. Analogously
xκj −X
1,k−1
κ+1,n
1,k−1
κ+1,n
xjk −X
1,k−1
κ+1,n
1,k−1
κ+1,n
κ+1,n
1,k−1
∣∣∣∣∣
1,k X
∣∣∣∣∣+
1,k |
κ+1,n
1,k−1
∣∣∣∣∣
1,k−1
κ+1,n
1,k−1 X
κ+1,n
∣∣∣∣∣ .
Theorem 2. A basis of Inv(Ad∗
T (n)) is formed by the rational expressions
1,k |
j=k+1
∣∣∣∣∣
1,k xjj
1,k X
∣∣∣∣∣ , k = 0, . . . ,
where |X
n+1,n
1,0 | := 1.
Proof. We choose the following normalization restriction on the lifted invariants Iij, j 6 i:
In−j+1,j = 1, j = 1, . . . ,
Iij = 0 if j 6 i, (i, j) 6= (j
′, j′), (n − j′ + 1, j′), j′ = 1, . . . ,
This means that we do not only fix the values of the elements of the lifted invariant matrix I, which
are situated on the main diagonal over or on the secondary diagonal. The elements of the secondary
diagonal underlying the main diagonal are given a value of 1. The other significant elements of I
are given a value 0. As shown below, the imposed normalization provides satisfying the conditions
of Proposition 1 and, therefore, is correct.
Similarly to the case of strictly triangular matrices, in view of the (triangular) structure of
the matrices B and X the formula I = BXB−1 determining the lifted invariants implies that
BX = IB. This matrix equality is significant for the matrix elements lying not over the main
diagonals of the left and right hand sides, i.e.,
bii′xi′j =
Iij′bj′j , j 6 i.
For convenience we again divide the latter system under the chosen normalization conditions into
four sets of subsystems
Sk1 :
bκi′xi′j = 0, i = κ, j < k, k = 2, . . . ,
Sk2 :
bκi′xi′j = bkj, i = κ, k 6 j 6 κ, k = 1, . . . ,
Sk3 :
bki′xi′j = 0, i = k, j < k, k = 2, . . . ,
Sk4 :
bki′xi′k = bkkIkk, i = k, j < k, k = 1, . . . ,
and solve them one by one. The subsystem S12 consists of the equations
b1j = bnnxnj
which are already solved with respect to b1j . For any fixed k ∈ {2, . . . , [n/2]} the subsystem S
is a well-defined system of linear equations with respect to bκi′ , i
′ > κ, and bkj, k 6 j 6 κ. We
can solve the subsystem Sk1 with respect to bκi′ , i
′ > κ:
κ+1,n = −bκκX
1,k−1(X
κ+1,n
1,k−1 )
and then substitute the obtained values into the subsystem Sk2 . Another way is to find the expres-
sions for bkj, k 6 j 6 κ, by the Cramer method, from the whole system S
1 ∪ S
2 at once since
only these parameters are further considered. As a result, they have two representations via bκκ
and x’s:
bkj = bκκ
xκj −X
1,k−1
κ+1,n
1,k−1
κ+1,n
(−1)k+1bκκ
κ+1,n
1,k−1
∣∣∣∣∣
1,k−1 xκj
κ+1,n
1,k−1 X
κ+1,n
∣∣∣∣∣ ,
where k 6 j 6 κ. In particular,
bkk = (−1)
k+1bκκ|X
κ+1,n
1,k−1 |
1,k |.
Analogously, for any fixed k ∈ {2, . . . , [(n + 1)/2]} the subsystem Sk3 is a well-defined system of
linear equations with respect to bkj, j > κ, and it implies
κ+1,n = −
k6j6κ
1,k−1(X
κ+1,n
1,k−1 )
Substituting the found expressions for b’s into the equations of the subsystems Sk4 , we completely
exclude the parameters b’s and obtain expressions of Ikk only via x’s. Thus, under k = 1
I11 =
b1ixi1 =
xnixi1 =
xnixi1 =
xi1 xii
xn1 xni
∣∣∣∣+
where the summation range in the first sum can be bounded by 2 and n − 1 since for i = 1 and
i = n the determinants are equal to 0. In the case k ∈ {2, . . . , [(n + 1)/2]}
bkkIkk =
bkixik =
k6j6κ
bkjxjk +
bkixik
k6i6κ
xjk −X
1,k−1(X
κ+1,n
1,k−1 )
κ+1,n
= bκκ
k6i6κ
xκj −X
1,k−1(X
κ+1,n
1,k−1 )
κ+1,n
xjk −X
1,k−1(X
κ+1,n
1,k−1 )
κ+1,n
After using the representation for bnn and manipulations with submatrices of X (see Lemma 3),
we derive that
Ikk =
(−1)k+1
1,k |
k6i6κ
∣∣∣∣∣
∣∣∣∣∣+
(−1)k+1
κ+1,n
1,k−1
k6i6κ
∣∣∣∣∣
1,k−1
κ+1,n
1,k−1
κ+1,n
∣∣∣∣∣ ,
where k = 2, . . . , [(n + 1)/2]. The summation range in the first sum can be taken from k + 1 and
κ − 1 since for i = k and i = κ the determinants are equal to 0.
The combination of the found values of Ikk in the following way
Ĩ00 =
Ijj =
xii, Ĩkk = (−1)
k+1Ikk − Ĩk−1,k−1, k = 1, . . . ,
results in the invariants Ĩk′k′, k
′ = 0, . . . , [(n − 1)/2], from the statement of the theorem. The
functional independence of these invariants is obvious.
Under the normalization we express the non-normalized lifted invariants via x’s and find a part
of the parameters b’s of the coadjoint action via x’s and the other b’s. No equations involving
only x’s are obtained. In view of Proposition 1, this implies that the choice of the normalization
constraints is correct, i.e., the number of the found functionally independent invariant is maximal
and, therefore, they form a basis of Inv(Ad∗
T (n)).
Note 4. An expanded form of the invariants from Theorem 2 is
xj1 xjj
xn1 xnj
∣∣∣∣∣∣
xj1 xj2 xjj
xn−1,1 xn−1,2 xn−1,j
xn1 xn2 xnj
∣∣∣∣∣∣
xn−1,1 xn−1,2
xn1 xn2
, . . . .
The first invariant corresponds to the center of t(n). The invariant tuple ends with
1,n+1
1,n−1
if n is odd and
∣∣∣∣∣∣
∣∣∣∣∣∣
if n is even.
Corollary 2. A basis of Inv(t(n)) consists of the rational invariants
Îk =
j=k+1
∣∣∣∣∣
j,j E
ejj E
∣∣∣∣∣ , k = 0, . . . ,
where E
i1,i2
j1,j2
, i1 6 i2, j1 6 j2, denotes the matrix (eij)
i=i1,...,i2
j=j1,...,j2
n+1,n| := 1, κ = n− k + 1.
Proof. The symmetrization procedure for the tuple of invariants presented in Theorem 2 can be
assumed trivial. To show this, we expand the determinants in each element of the tuple and obtain,
as a result, a rational expression in x’s. Only the monomials from the numerator, which do not
contain the ‘diagonal’ elements xjj, include coordinate functions associated with noncommuting
basis elements of the algebra t(n). More precisely, each of the monomials includes a single pair of
such coordinate functions, namely, xji′ and xj′j for some values i
′ ∈ {1, . . . , k}, j′ ∈ {κ, . . . , n} and
j ∈ {k + 1, . . . ,κ − 1}. Hence, it is sufficient to symmetrize only the corresponding pairs of basis
elements.
After the symmetrization and the transposition of the matrices, we construct the following
expressions for the invariants of t(n) associated with the invariants from Theorem 2:
(−1)k
j=k+1
ejj +
j=k+1
ei′jejj′+ ejj′ei′j
(−1)i
∣∣E1,k;̂i
κ,n;ĵ′
∣∣E1,k;̂i
κ,n;ĵ′
∣∣ denotes the minor of the matrix E1,kκ,n complementary to the element ei′j′. Since
ei′ieij′ = eij′ei′i + ei′j′ then
ei′ieij′+ eij′ei′i
(−1)i
∣∣E1,k;̂i
κ,n;ĵ′
∣∣∣∣∣
i,i E
∣∣∣∣∣±
|E1,k
κ,n|,
where the sign ‘+’ (resp. ‘−’) have to be taken if the elements of E
i,i are placed after (resp. before)
the elements of E
κ,n in all the relevant monomials. Up to constant summands, we obtain the
expressions for the elements of an invariant basis, which are adduced in the statement and formally
derived from the corresponding expressions given in Theorem 2 by the replacement xij → eji and
the transposition of all matrices. That is why the symmetrization procedure can be assumed trivial
in the sense described. The transposition is necessary in order to improve the representation of
invariants since xij ∼ eji, j 6 i.
Note 5. The invariants from Corollary 2 can be rewritten as
Îk =
j=k+1
∣∣∣∣∣
j,j E
∣∣∣∣∣+ (−1)
j=k+1
ejj, k = 0, . . . ,
In particular, Î0 =
j ejj .
Note 6. Let us emphasize that a uniform order of elements from E
and E
κ,n has to be fixed in all
the monomials under usage of the ‘non-symmetrized’ forms of invariants presented in Corollary 2,
Note 5 and Theorem 4 (see below).
5 Solvable algebra of special upper triangular matrices
The Lie algebra st(n) of the special (i.e., having zero traces) upper triangular n × n matrices is
imbedded in a natural way in t(n) as an ideal. dim st(n) = 1
n(n+ 1)− 1. Moreover,
t(n) = st(n)⊕ Z(t(n)),
where Z(t(n)) = 〈e11 + · · · + enn〉 is the center of t(n), which corresponds to the one-dimensional
Abelian Lie algebra of the matrices proportional to En. Due to this fact we can construct a basis of
Inv(st(n)) without the usual calculations involved in finding the basis of Inv(t(n)). It is well known
that if the Lie algebra g is decomposable into the direct sum of Lie algebras g1 and g2 then the
union of bases of Inv(g1) and Inv(g2) is a basis of Inv(g). A basis of Inv(Z(t(n))) obviously consists
of only one element, e.g., e11 + · · · + enn. Therefore, the cardinality of the basis of Inv(st(n)) is
equal to the cardinality of the basis of Inv(t(n)) minus 1, i.e., [(n − 1)/2]. To construct a basis
of Inv(st(n)), it is enough for us to rewrite [(n − 1)/2] functionally independent combinations of
elements from a basis of Inv(t(n)) via elements of st(n) and to exclude the central element from
the basis.
The following basis in st(n) is chosen as a subalgebra of t(n):
eij , i < j, fk =
eii −
i=k+1
eii, k = 1, . . . , n− 1.
(Usage of this basis allows for the presentation of our results in such a form that their identity with
Proposition 1 from [23] becomes absolutely evident.) The commutation relations of st(n) in the
chosen basis are
[eij , ei′j′] = δi′jeij′− δij′ei′j, i < j, i
′ < j′;
[fk, fk′ ] = 0, k, k
′ = 1, . . . , n− 1;
[fk, eij ] = 0, i < j 6 k or k 6 i < j;
[fk, eij ] = eij , i 6 k 6 j, i < j
and, therefore, coincide with those of the algebra L(n, n−1) from [22], i.e., L(n, n−1) is isomorphic
to st(n). Combining this observation with Lemma 6 of [22] results in the following theorem.
Theorem 3. The Lie algebra st(n) has the maximal number of dimensions (equal to 1
n(n+1)−1)
among the solvable Lie algebras which have nilradicals isomorphic to t0(n). It is the unique algebra
with such a property.
Theorem 4. A basis of Inv(st(n)) consists of the rational invariants
Ǐk =
(−1)k+1
j=k+1
∣∣∣∣∣
j,j E
∣∣∣∣∣+ fk − fn−k, k = 1, . . . ,
where E
i1,i2
j1,j2
, i1 6 i2, j1 6 j2, denotes the matrix (eij)
i=i1,...,i2
j=j1,...,j2
n+1,n| := 1, κ = n− k + 1.
Proof. It is enough to observe (see Note 5) that
Ǐk = (−1)
k+1Îk +
n− 2k
Î0, k = 1, . . . ,
These combinations of elements from a basis of Inv(t(n)) are functionally independent. They are
expressed via elements of st(n). Their number is [(n − 1)/2]. Therefore, they form a basis of
Inv(st(n)).
6 Conclusion and discussion
In this paper we extend our purely algebraic approach for computing invariants of Lie algebras by
means of moving frames [3, 4] to the classes of Lie algebras t0(n), t(n) and st(n) of strictly, non-
strictly and special upper triangular matrices of an arbitrary fixed dimension n. In contrast to the
conventional infinitesimal method which involves solving an associated system of PDEs, the main
steps of the applied algorithm are the construction of the matrix B(θ) of inner automorphisms of the
Lie algebra under consideration, and the exclusion of the parameters θ from the algebraic system
I = x̌ · B(θ) in some way. The version of the algorithm, applied in this paper, is distinguished in
that a special usage of the normalization procedure when the number, and a form of elements in a
functional basis of an invariant set, are determined under excluding the parameters simultaneously.
A basis of Inv(t0(n)) was already known and constructed by both the infinitesimal method [23]
and the algebraic algorithm with an elegant but empiric technique of excluding the parameters [4].
Note that the proof introduced in [23] is very sophisticated and was completed only due to the
thorough mastery of the used infinitesimal method. A form of elements from a functional basis of
Inv(t0(n)) was guessed via calculation of bases for a number of small n’s and then justified with
the infinitesimal method, and both the testing steps (on invariance and on sufficiency of number)
were quite complicated.
Invariants of t0(n) are considered in this paper in order to demonstrate the advantages of the
normalization technique and to pave the way for further applications of this technique to the more
complicated algebras t(n) and st(n), being too complex for the infinitesimal method (only the
lowest few were completely investigated there). First the invariants of the algebras t(n) and st(n)
are exhaustively studied in this paper. The performed calculations are simple and clear since the
normalization procedure is reduced by the choice of natural coordinates on the inner automorphism
groups and by the use of a special normalization technique to solving a linear system of algebraic
equations. The results obtained for Inv(st(n)) in Theorem 4 completely agree with the conjecture
formulated as Proposition 1 in [23] on the number and form of basis elements of this invariant set.
A direct extension of the present investigation is to describe the invariants of the subalgebras
of st(n), which contain t0(n). Such subalgebras exhaust the set of solvable Lie algebras which can be
embedded in the matrix Lie algebra gl(n) and have the nilradicals isomorphic to t0(n). A technique
similar to that used in this paper can be applied. The main difficulties will be created by breaking
in symmetry and complication of coadjoint representations. The question on ways of investigation
of the other solvable Lie algebras with the nilradicals isomorphic to t0(n) remains open. (See, e.g.,
[22] for classification of the algebras of such type.)
A more general problem is to circumscribe an applicability domain of the developed algebraic
method. It has been already applied only to the low-dimensional Lie algebras and a wide range of
classes of solvable Lie algebras in [3, 4] and this paper. The next step which should be performed
is the extension of the method to classes of unsolvable Lie algebras of arbitrary dimensions, e.g.,
with fixed structures of radicals or Levi factors. An adjoining problem is the implementation of
the algorithm with symbolic calculation systems. Similar work has already began in the framework
of the general method of moving frames, e.g., in the case of rational invariants for rational actions
of algebraic groups [11]. Some other possibilities on the applications of the algorithm are outlined
in [4].
Acknowledgments. The work was partially supported by the National Science and Engineering
Research Council of Canada, by MITACS. The research of R.P. was supported by Austrian Science
Fund (FWF), Lise Meitner project M923-N13. V. B. is grateful for the hospitality the Centre de
Recherches Mathématiques, Université de Montréal.
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http://arxiv.org/abs/math-ph/0511027
http://arxiv.org/abs/math-ph/0602046
http://arxiv.org/abs/math-ph/0606045
http://arxiv.org/abs/math-ph/0602065
http://arxiv.org/abs/math-ph/0411023
http://arxiv.org/abs/0709.3581
http://arxiv.org/abs/0709.3116
	Introduction
	The algorithm
	Nilpotent algebra of strictly upper triangular matrices
	Solvable algebra of upper triangular matrices
	Solvable algebra of special upper triangular matrices
	Conclusion and discussion
