N -HOMOGENEOUS SUPERALGEBRAS
PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
ABSTRACT. We develop the theory of N -homogeneous algebras in a super setting, with par-
ticular emphasis on the Koszul property. To any Hecke operator R on a vector superspace,
we associate certain superalgebras SR,N and ΛR,N generalizing the ordinary symmetric and
Grassmann algebra, respectively. We prove that these algebras are N -Koszul. For the special
case where R is the ordinary supersymmetry, we derive an N -generalized super-version of
MacMahon’s classical “master theorem”.
CONTENTS
Introduction 1
1. Review of linear superalgebra 5
2. The supercharacter 10
3. Homogeneous superalgebras 17
4. N -Koszul superalgebras 27
5. Koszul duality and master theorem 34
Appendix 38
References 41
INTRODUCTION
0.1. The theory of N -homogeneous algebras owes its existence primarily to the concerns of
noncommutative geometry. In fact, as has been expounded by Manin in his landmark pub-
lications [36], [37], quadratic algebras (the case N = 2) provide a convenient framework
for the investigation of quantum group actions on noncommutative spaces. Moreover, certain
Artin-Schelter regular algebras [1], natural noncommutative analogs of ordinary polynomial
algebras, can be presented as associative algebras defined by cubic relations (N = 3). The
latter algebras, as well as many of the quadratic algebras studied by Manin, enjoy the addi-
tional “Koszul property” which will be of central importance in the present article; it will be
reviewed in detail in 0.6 below.
Motivated by these examples and others, Berger [5] initiated the systematic investigation
of N -homogeneous algebras for all N ≥ 2, introducing in particular a natural extension of
the notion of Koszul algebra from the familiar quadratic setting to general N -homogeneous
2000 Mathematics Subject Classification. Primary 16S37, 05A19.
PHH is supported by the DFG through a Heisenberg-Fellowship.
ML’s research is supported in part by NSA Grants H98230-05-1-0025 and H98230-07-1-0008 and by Lever-
hulme Research Interchange Grant F/00158/X.
http://arxiv.org/abs/0704.1888v2
2 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
algebras. Article [5] gives examples of N -Koszul algebras for all N ≥ 2; these are the
so-called N -symmetric algebras, the special case N = 2 being the ordinary symmetric (poly-
nomial) algebra. Following the general outline of Manin’s lecture notes [37] on the case of
quadratic algebras, Berger, Dubois-Violette and Wambst developed the categorical aspects of
N -homogeneous algebras in [7].
0.2. Current interest in N -homogeneous algebras is fueled in part by the fact that they do
occur naturally in mathematical physics and in combinatorics. Indeed, Connes and Dubois-
Violette [10], [11] introduced a class of 3-homogeneous algebras, called Yang-Mills algebras,
which are in fact 3-Koszul. There are two versions of Yang-Mills algebras: in the language of
linear superalgebra, the first kind has even (parity 0̄) algebra generators while the second kind
is generated by odd (parity 1̄) elements.
Combinatorics enters the picture via MacMahon’s celebrated “master theorem” [35], specif-
ically the recent quantum generalization of the master theorem due to Garoufalidis, Lê and
Zeilberger [20]. As has been pointed out by two of the present authors in [28], the yoga of
(quadratic) Koszul algebras leads to a rather effortless and conceptual proof of the quantum
master theorem based on the fact that a certain quadratic algebra, known as quantum affine
space, is Koszul. Further quantum generalizations and super versions of the master theorem
have been obtained by several authors using a variety of approaches; see Foata and Han [17],
[18], [19], Konvalinka and Pak [33], Etingof and Pak [16].
0.3. From an algebraic point of view, MacMahon’s master theorem (MT) in its various incar-
nations finds its most natural explanation by the phenomenon of “Koszul duality”. Indeed, all
versions of MT can be expressed in the form that, for some algebra B, an equation Σ1 ·Σ2 = 1
holds for suitable power series Σ1,Σ2 ∈ BJtK. Here is a brief outline how one can arrive at
such an equation starting with a given N -Koszul algebra A. Associated with A, there is a
graded complex, K(A), which is exact in positive degrees, and a certain endomorphism bial-
gebra, endA, which coacts on all components of K(A). These components therefore define
elements of the representation ring RendA of endA, and exactness of K(A) in positive de-
grees yields an equation in the power series ring RendAJtK. Due to the specific form of K(A),
which is constructed from A together with its so-called dual algebra A!, the equation in ques-
tion does indeed state that ρ1 · ρ2 = 1 holds for suitable ρ1, ρ2 ∈ RendAJtK. The last step
in deriving a MT for A consists in using (super-)characters to transport the abstract duality
equation ρ1 · ρ2 = 1 from RendAJtK to the power series ring over the algebra endA, where it
takes a more explicit and useable form. Here then is the flow chart of our approach:
N -Koszul algebra
// exact Koszul complex
// duality equation
in RendAJtK
// MT for A
The actual labor involved in this process consists in the explicit evaluation of (super-)characters
at the last arrow above. This step is often facilitated by specializing the bialgebra endA, which
is highly noncommutative, to a more familiar algebra B via a homomorphism endA → B.
For example:
• MacMahon’s original MT [35] follows in the manner described above by starting with
A = O(kd) = k[x1, . . . , xd], the ordinary polynomial algebra or “affine space”, and
N -HOMOGENEOUS SUPERALGEBRAS 3
restricting the resulting MT over endO(kd) to the coordinate ring of d × d-matrices,
O(Matd(k)) = k[x
j | 1 ≤ i, j ≤ d].
• As was explained in [28], taking “quantum affine space” Oq(k
d) as the point of de-
parture one arrives at the quantum MT of Garoufalidis, Lê and Zeilberger [20] (and
Konvalinka and Pak [33] in the multi-parameter case). The endomorphism bialgebra
of Oq(k
d) is exactly the algebra of right-quantum matrices as defined in [20].
• Berger’s N -symmetric algebra [5] leads to the N -generalization of the MT proved by
Etingof and Pak [16] using the above approach, again after restricting to O(Matd(k)).
0.4. The present article aims to set forth an extension of the existing theory of N -homogeneous
algebras to the category Vects
of vector superspaces over some base field k. While this does
not give rise to principal obstacles given that [37] and [7] are at hand as guiding references, the
setting of superalgebra requires careful consideration of the order of terms and the so-called
“rule of signs” will be ubiquitous in our formulæ. In view of the potential interdisciplinary
interest of this material, we have opted to keep our presentation reasonably self-contained and
complete.
Therefore, in Sections 1 and 2, we deploy the requisite background material from superal-
gebra in some detail before turning to N -homogeneous superalgebras in Section 3. The latter
section, while following the general outline of [37] and [7] rather closely, also offers explicit
discussions of a number of important examples. We interpolate the pure even and pure odd
Yang-Mills algebras defined by Connes and Dubois-Violette [10], [11] by a family of super-
algebras YMp|q and give a unified treatment of these algebras. (It turns out, however, that the
mixed algebras YMp|q, with p and q both nonzero, are less well-behaved than the pure cases.)
Moreover, we discuss a superized version of the N -symmetric algebras of Berger [5]. Finally,
in Example 3.4, we introduce new N -homogeneous superversions of the symmetric algebra
and the Grassmann algebra of a vector superspace V ; these are associated with any Hecke
operator R : V ⊗2 → V ⊗2 and will be denoted by SR,N and ΛR,N , respectively.
Sections 4 and 5 contain our main results: Theorem 4.5 shows that the superalgebras SR,N
and ΛR,N are in fact N -Koszul, and Theorem 5.4 is superized version of the aforementioned
N -generalized MT of Etingof and Pak [16, Theorem 2]. The special case N = 2 of The-
orem 5.4 is a superization of the original master theorem of MacMahon [35]. The present
article was motivated in part by a comment in Konvalinka and Pak [33, 13.4] asking for a
“real” super-analog of the classical MT.
0.5. A considerable amount of research has been done by mathematical physicists on various
quantum matrix identities. Some of these investigations have been carried out in a super set-
ting; see, e.g., Gurevich, Pyatov and Saponov [23], [24] and the references therein. However,
the techniques employed in these articles appear to be quite different from ours.
After submitting this article, we also learned of recent work of Konvalinka [31], [32] which
not only concerns MacMahon’s MT but also other matrix identities such as the determinantal
identity of Sylvester. These identities are proved in [31], [32] by combinatorial means in
various noncommutative settings including the right-quantum matrix algebra endOq(k
0.6. We conclude this Introduction by reviewing the precise definitions of N -homogeneous
and N -Koszul algebras. Our basic reference is Berger [5]; see also [2], [7], [21].
4 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Let A be a connected Z≥0-graded algebra over a field k; so A =
n≥0An for k-subspaces
An with A0 = k and AnAm ⊆ An+m. Choose a minimal generating set for the algebra A
consisting of homogeneous elements of positive degree; this amounts to choosing a graded
basis for a graded subspace V ⊆ A+ =
n>0An such that A+ = A
+ ⊕ V . The grading
of V imparts a grading to the tensor algebra T(V ) of the space V , and we have a graded
presentation
T(V )/I
for some graded ideal I of T(V ), the ideal of relations of A.
Recall that a graded vector space M =
n∈ZMn is said to live in degrees ≥ n0 if Mn = 0
for all n < n0. Note that the relation ideal I lives in degrees ≥ 2, because T(V )0 ⊕T(V )1 ⊆
k⊕ V and k⊕ V injects into A. Fix an integer N ≥ 2 and define the jump function
νN (i) =
N if i is even
N + 1 if i is odd
(0.1)
The following proposition is identical with [8, Proposition 2.1] except for the fact that we do
not a priori assume A to be generated in degree 1. A proof is given in the Appendix.
Proposition 0.1. The ideal I of relations of A lives in degrees ≥ N if and only if TorAi (k, k)
lives in degrees ≥ νN (i) for all i ≥ 0.
Following Berger [5], the graded algebra A is said to be N -Koszul if TorAi (k, k) is con-
centrated in degree νN (i) for all i ≥ 0. This implies that the space of algebra generators V
is concentrated in degree νN (1) = 1; so the algebra A is 1-generated. Moreover, choosing a
minimal set of homogeneous ideal generators for the relation ideal I amounts to choosing a
graded basis for a graded subspace R ⊆ I such that
I = R⊕ (V ⊗ I + I ⊗ V ) (0.2)
Then TorA2 (k, k)
∼= R and so R must be concentrated in degree νN (2) = N when A is
N -Koszul. To summarize, all N -Koszul algebras are necessarily 1-generated and they have
defining relations in degree N ; so there is a graded isomorphism
A ∼= T(V )/(R) with R ⊆ V ⊗N
Such algebras are called N -homogeneous.
We remark that Green et al. [21] have studied N -Koszul algebras in the more general con-
text where the grading A =
n≥0An is not necessarily connected (A0 = k). In [21, The-
orem 4.1], it is shown that an N -homogeneous algebra A with A0 split semisimple over k is
N -Koszul if and only if the Yoneda Ext-algebra E(A) =
n≥0 Ext
A(A0,A0) is generated
in degrees ≤ 2.
Any N -homogeneous algebra A whose generating space V carries a Z2-grading and whose
defining relations R are Z2-graded is naturally a k-superalgebra, that is, A has a Z2-grading
(“parity”) besides the basic Z≥0-grading (“degree”). As will be reviewed below, this extra
structure provides us with additional functions on Grothendieck rings, namely superdimension
and supercharacters, which lead to natural formulations of the MT in a superized context.
Note, however, that the defining property of N -Koszul algebras makes no reference to the
Z2-grading of A. Thus, an N -homogeneous superalgebra is Koszul precisely if it is Koszul
as an ordinary N -homogeneous algebra (forgetting the Z2-grading).
N -HOMOGENEOUS SUPERALGEBRAS 5
0.7. Throughout k is a commutative field and ⊗ stands for ⊗
. Scalar multiplication in
k-vector spaces will often, but not always, be written on the right while linear maps will
act from the left. We tacitly assume throughout that char k 6= 2; further restrictions on the
characteristic of k will be stated when required.
1. REVIEW OF LINEAR SUPERALGEBRA
1.1. Vector superspaces. A vector superspace over k is a k-vector space V equipped with a
grading by the group Z2 = Z/2Z = {0̄, 1̄}. Thus, we have a decomposition V = V0̄ ⊕ V1̄
with k-subspaces V0̄ and V1̄ whose elements are called even and odd, respectively. In general,
the Z2-degree of a homogeneous element a ∈ V is also called its parity; it will be denoted by
â ∈ Z2. Vector superspaces over k form a category Vect
whose morphisms are given by the
linear maps preserving the Z2-grading; such maps are also called even linear maps.
The dimension of an object V of Vects
is the usual k-linear dimension. We shall use the
notation
d = dim
V , p = dim
V0̄ and q = dimk V1̄
So d = p+ q. The superdimension of a vector superspace V with d < ∞ is defined by
sdimV = p− q ∈ Z
When working with a fixed basis {xi} of a given V in Vect
we shall assume that each xi is
homogeneous; the parity of xi will be denoted by î. The basis x1, x2 . . . is called standard if
î = 0̄ (i ≤ p) and î = 1̄ (i > p).
1.2. Tensors. The tensor product U ⊗V of vector superspaces U and V in Vects
is the usual
tensor product over k of the underlying vector spaces equipped with the natural Z2-grading:
if a, b are homogeneous elements then the parity of a ⊗ b is â + b̂ ∈ Z2. Instead of the usual
symmetry isomorphism U ⊗ V
−→ V ⊗ U for interchanging terms in a tensor product we
shall use the rule of signs, that is, the following functorial supersymmetry isomorphism in
Vects
cU,V : U ⊗ V
−→ V ⊗ U , u⊗ v 7→ (−1)bubvv ⊗ u (1.1)
for u, v homogeneous. (All formulas stated for homogeneous elements only are to be ex-
tended to arbitrary elements by linearity.) The supersymmetry isomorphisms cU,V satisfy
cV,U ◦ cU,V = IdU⊗V , and they are compatible with the usual associativity isomorphims
aU,V,W : (U ⊗V )⊗W ∼= U ⊗ (V ⊗W ) in Vect
, that is, they satisfy the “Hexagon Axiom”;
see [29, Def. XIII.1.1]. Therefore, Vects
is a symmetric tensor category; the unit object is the
field k, with parity 0̄. See [29, Chap. XIII] or [12] for background on tensor categories.
1.3. Homomorphisms. The space Hom
(V,U) of all k-linear maps between vector super-
spaces V and U is again an object of Vects
, with grading Hom
(V,U)0̄ = Homk(V0̄, U0̄) ⊕
(V1̄, U1̄) and Homk(V,U)1̄ = Homk(V0̄, U1̄)⊕Homk(V1̄, U0̄); so
(V,U)0̄ = HomVects
(V,U)
In particular, the linear dual V ∗ = Hom
(V, k) belongs to Vects
. Given homogeneous bases
{xj} of V and {yi} of U we can describe any f ∈ Hom
(V,U) by its matrix F = (F ij ):
f(xj) =
j (1.2)
6 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
When f is an even map then F ij = 0 unless î+ ĵ = 0̄.
For finite-dimensional vector superspaces, we have the following functorial isomorphisms
in Vects
(see, e.g., [43, I.8]):
U ⊗ V ∗ ∼= Hom
(V,U) (1.3)
via (u⊗ f)(v) = u〈f, v〉, and
V ∗1 ⊗ . . . ⊗ V
∼= (Vm ⊗ . . . ⊗ V1)
∗ (1.4)
via 〈f1 ⊗ . . .⊗ fm, vm ⊗ . . .⊗ v1〉 =
i〈fi, vi〉. Here, we use the notation 〈f, v〉 = f(v) for
the evaluation pairing
evV = 〈 . , . 〉 : V
∗ ⊗ V → k
in Vects
. Similarly, we have a pairing
V ⊗ V ∗
cV,V ∗
−→ V ∗ ⊗ V
which yields an isomorphism
−→ V ∗∗ (1.5)
in Vects
The isomorphism (1.3) (which is valid as long as one of U or V is finite-dimensional) has
the following explicit description. Fix homogeneous bases {xj} of V and {yi} of U and let
F = (F ij ) be the matrix of a given f ∈ Homk(V,U) with respect to these bases, as in (1.2).
Let {xj} be the dual basis of V ∗, defined by 〈xj , xℓ〉 = δ
(Kronecker delta). Then the image
of f in U ⊗ V ∗ is given by
i,j yi ⊗ x
jF ij . Note also that xi and x
i have the same parity.
Finally, if U , V and W are vector superspaces, with U finite-dimensional, then the isomor-
phism Id⊗cW,U∗ : V ⊗W ⊗U
∗ ∼−→V ⊗U∗ ⊗W together with (1.3) yields an isomorphism
(U, V ⊗W )
−→ Hom
(U, V )⊗W (1.6)
in Vects
which is explicitly given by (f ⊗ w)(u) = (−1)bwbuf(u) ⊗ w. Similarly, for vector
superspaces U , U ′, V , V ′ with U , U ′ finite-dimensional, there is an isomorphism
(U ⊗ U ′, V ⊗ V ′)
−→ Hom
(U, V )⊗Hom
(U ′, V ′) (1.7)
in Vects
given by (f ⊗ g)(u ⊗ v) = (−1)bgbuf(u)⊗ g(v).
1.4. Supertrace. Let V be a finite-dimensional object of Vects
. The supertrace is the map
str : End
(1.3)
V ⊗ V ∗ −→
(1.3)
k (1.8)
in Vects
. In order to describe the supertrace in terms of matrices, fix a basis {xi} of V
consisting of homogeneous elements and let F = (F ij ) be the matrix of f ∈ Endk(V ) as in
(1.2). Then
str(f) =
biF ii
where î is the parity of xi (and of the dual basis vector x
i ∈ V ∗) as in §1.1. Thus,
str(IdV ) = sdimV.1
N -HOMOGENEOUS SUPERALGEBRAS 7
1.5. Action of the symmetric group. Given vector superspaces V1, . . . , Vn, we can consider
the morphism
ci : V1 ⊗ · · · ⊗ Vi ⊗ Vi+1 ⊗ · · · ⊗ Vn −→ V1 ⊗ · · · ⊗ Vi+1 ⊗ Vi ⊗ · · · ⊗ Vn
in Vects
which interchanges the factors Vi and Vi+1 via cVi,Vi+1 and is the identity on all other
factors. More generally, for any σ ∈ Sn, the symmetric group consisting of all permutations
of {1, 2, . . . , n}, one can define a morphism
cσ : V1 ⊗ · · · ⊗ Vn −→ Vσ−1(1) ⊗ · · · ⊗ Vσ−1(n)
in Vects
as follows. Recall that Sn is generated by the transpositions σ1, . . . , σn−1 where σi
interchanges i and i + 1 and leaves all other elements of {1, 2, . . . , n} fixed. The minimal
length of a product in the σi’s which expresses a given element σ ∈ Sn is called the length of
σ and denoted ℓ(σ); it is given by
ℓ(σ) = #inv(σ) with inv(σ) = {(i, j) | i < j but σ(i) > σ(j)}
Writing σ ∈ Sn as a product of certain σi, the analogous product of the maps ci yields a
morphism cσ as above. This morphism is independent of the way σ is expressed in terms of the
transpositions σi; see [43, I.4.13] or [29, Theorem XIII.1.3]. If all vi ∈ Vi are homogeneous
cσ(v1 ⊗ · · · ⊗ vn) = (−1)
(i,j)∈inv(σ) bvi bvjvσ−1(1) ⊗ · · · ⊗ vσ−1(n) (1.9)
For example, if all vi are even then the ±-sign on the right is +, and if all vi are odd then it is
sgn(σ), the signature of σ.
Taking all Vi = V we obtain a representation c : Sn −→ AutVects
(V ⊗n) where V ⊗n =
V ⊗ · · · ⊗ V (n factors). Letting k[Sn] denote the group algebra of the symmetric group, this
extends uniquely to an algebra map
c : k[Sn] −→ EndVects
(V ⊗n) (1.10)
We will write ca := c(a) for a ∈ k[SN ].
For the dual superspace V ∗, besides the above representation c : k[Sn] −→ EndVects
(V ∗⊗n),
we also have the contragredient representation
c∗ : k[Sn] −→ EndVects
(V ∗⊗n)
for the pairing 〈 . , . 〉 : V ∗⊗n ⊗ V ⊗n → k in (1.4). Explicitly,
〈c∗a(x), y〉 = 〈x, ca∗(y)〉
for all a ∈ k[Sn], x ∈ V
∗⊗n and y ∈ V ⊗n. Here, . ∗ : k[Sn] → k[Sn] is the involution
sending σ ∈ Sn to σ
−1. These two representations are related by
c∗a = cτaτ (1.11)
where τ = (1, n)(2, n−1) · · · ∈ Sn is the order reversal involution. One only needs to check
(1.11) for the transpositions a = σi, which is straightforward.
8 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
1.6. Hecke algebras. We recall some standard facts concerning Hecke algebras; these are
suitable deformations of the group algebra k[Sn] considered above. For details, see [13],
[14].
Fix 0 6= q ∈ k. The Hecke algebra Hn,q is generated as k-algebra by elements T1, . . . , Tn−1
subject to the relations
(Ti + 1)(Ti − q) = 0
TiTi+1Ti = Ti+1TiTi+1
TiTj = TjTi if |i− j| ≥ 2
(1.12)
When q = 1, one has an isomorphism Hn,1
−→ k[Sn], Ti 7→ σi where σi is the transposition
(i, i + 1) as in §1.5. The algebra Hn,q has a k-basis {Tσ | σ ∈ Sn} so that
(i) TId = 1 and Tσi = Ti;
(ii) TσTσi =
Tσσi if ℓ(σσi) = ℓ(σ) + 1;
qTσσi + (q − 1)Tσ otherwise
By k-linear extension of the rule
T ∗σ := Tσ−1 (σ ∈ Sn)
one obtains an involution . ∗ : Hn,q → Hn,q. Moreover, the elements T
i := −qT
q − 1− Ti also satisfy relations (1.12). Therefore,
α(Ti) := −qT
i (1.13)
defines an algebra automorphism α : Hn,q → Hn,q of order 2.
The Hecke algebra Hn,q is always a symmetric algebra, and Hn,q is a split semisimple
k-algebra iff the following condition is satisfied:
[n]q! :=
[i]q 6= 0 where [i]q := 1 + q + · · ·+ q
i−1 (1.14)
More precisely, if (1.14) holds then
Matdλ×dλ(k) (1.15)
where λ runs over all partitions of n and dλ denotes the number of standard λ-tableaux. The
only partitions λ with dλ = 1 are λ = (n) and λ = (1
n). The central primitive idempotents
of Hn,q for these partitions are given by
Xn :=
[n]q!
Tσ (1.16)
Yn :=
[n]q−1 !
(−q)−ℓ(σ)Tσ (1.17)
These idempotents are usually called the q-symmetrizer and the q-antisymmetrizer, respec-
tively. One has
XnTσ = TσXn = q
ℓ(σ)Xn and YnTσ = TσYn = (−1)
ℓ(σ)Yn (1.18)
for σ ∈ Sn. Furthermore, α(Xn) = Yn.
N -HOMOGENEOUS SUPERALGEBRAS 9
For later use, we note the following well-known consequence of (1.18). If M is any Hn,q-
module, with corresponding representation µ : Hn,q → End
(M), then
Im(µ(Xn)) =
Im(µ(Ti) + 1) (1.19)
Indeed, (1.18) implies that Xn = [2]
q (Ti + 1)Xn, which yields the inclusion ⊆. On the
other hand, any m ∈
i=1 Im(µ(Ti) + 1) satisfies (µ(Ti) − q)(m) = 0 for all i, by
(1.12). Therefore, µ(Tσ)(m) = q
ℓ(σ)m holds for all σ ∈ Sn, and hence µ(Xn)(m) =
[n]q!
qℓ(σ)m = m. This proves ⊇.
1.7. Hecke operators. Again, let 0 6= q ∈ k. A Hecke operator (associated to q) on a vector
superspace V is a morphism R : V ⊗2 → V ⊗2 in Vects
satisfying the Hecke equation
(R + 1)(R − q) = 0
and the Yang-Baxter equation
R1R2R1 = R2R1R2
where R1 := R ⊗ IdV : V
⊗3 → V ⊗3 and similarly R2 := IdV ⊗R.
The Hecke equation implies that R is invertible. Moreover, if R is a Hecke operator
associated to q then so is −qR−1.
Defining ρ(Ti) := Id
V ⊗R ⊗ Id
⊗n−i−1
V , one obtains a representation
ρ = ρn,R : Hn,q −→ EndVects
(V ⊗n) (1.20)
The representations ρn,R and ρn,−qR−1 are related by ρn,−qR−1 = ρn,R ◦ α, where α is the
automorphism of Hn,q defined in (1.13).
Example 1.1. The supersymmetry operator cV,V : V ⊗2 → V ⊗2 in (1.1) is a Hecke operator
associated to q = 1, as is its negative, −cV,V . The representation ρcV,V of Hn,1 = k[Sn] in
(1.20) is identical with (1.10).
Example 1.2 (superized Drinfel’d-Jimbo [38], [27]). Let x1, . . . , xd be a standard basis of V
as in §1.1. The super analog R = RDJ of the standard Drinfel’d-Jimbo Hecke operator is
defined as follows. Writing
R(xi ⊗ xj) =
xk ⊗ xlR
the matrix components Rk,li,j ∈ k are given by
i,j =
q2 − q2εi,j
1 + q2εi,j
i,j + (−1)
bibj q
εi,j(q2 + 1)
1 + q2εi,j
Here, εi,j = sgn(i− j). Thus,
ii = q
2 if î = 0̄
ii = −1 if î = 1̄
ij = q
2 − 1 if i < j
ij = (−1)
bibjq if i 6= j
(1.21)
10 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
and Rk,li,j = 0 in all other cases. One checks that R is a Hecke operator that is associated to
2. THE SUPERCHARACTER
2.1. Superalgebras, supercoalgebras etc. An algebra A in Vects
is called a superalgebra
over k; this is just an ordinary k-algebra such that the unit map k → A and the multiplication
µ : A⊗A → A
are morphisms in Vects
. In other words, A is a Z2-graded k-algebra in the usual sense:
A = A0̄ ⊕ A1̄ with k-subspaces A0̄ and A1̄ such that Ar̄As̄ ⊆ Ar+s. Homomorphisms of
superalgebras, by definition, are algebra maps in Vects
, that is, they preserve the Z2-grading.
If V is a vector superspace in Vects
then the tensor algebra T(V ) =
n≥0 V
⊗n is a
superalgebra via the Z2-grading of each V
⊗n as in §1.2. In general, if A is any superalgebra,
then by selecting a Z2-graded subspace V ⊆ A which generates the algebra A, we obtain a
canonical isomorphism of superalgebras
T(V )/(R)
−→ A (2.1)
where (R) is the two-sided ideal of T(V ) that is generated by a Z2-graded linear subspace
R ⊆ T(V ).
Given superalgebras A and B, the tensor product A⊗ B is the superalgebra with the usual
additive structure and grading and with multiplication µA⊗B defined by using the supersym-
metry map (1.1): µA⊗B = (µA ⊗ µB) ◦ (IdA⊗cB,A ⊗ IdB) or, explicitly,
(a⊗ b)(a′ ⊗ b′) = (−1)
ba′bbaa′ ⊗ bb′
for homogeneous a′ ∈ A and b ∈ B. In other words, the canonical images of A and B in
A⊗ B supercommute, in the sense that the supercommutator
[a, b] = ab− (−1)ba
bbba (2.2)
vanishes for any pair of homogeneous elements a ∈ A and b ∈ B.
Supercoalgebras, superbialgebras etc. are defined similarly as suitable objects of Vects
such that all structure maps are maps in Vects
. The compatibility between the comultiplica-
tion ∆ and the multiplication of a superbialgebra B amounts to the following rule:
∆(ab) =
(a),(b)
(−1)ba(2)
bb(1)a(1)b(1) ⊗ a(2)b(2)
for homogeneous elements a, b ∈ B. Here we use the Sweedler notation ∆(a) =
(a) a(1) ⊗
a(2) and a(1), a(2) are chosen homogeneous with â(1) + â(2) = â.
Example 2.1 (Symmetric superalgebra [40, 3.2.5]). The symmetric superalgebra of a given
V in Vects
is defined by
S(V ) = T(V )/ ([v,w]⊗ | v,w ∈ V )
where [v,w]⊗ is the supercommutator (2.2) in T(V ). Ignoring parity, S(V ) is isomorphic to
S(V0̄)⊗Λ(V1̄), where S( . ) and Λ( . ) denote the ordinary symmetric and exterior (Grassmann)
algebras, respectively. The symmetric superalgebra is a Hopf superalgebra: comultiplication
∆: S(V ) → S(V )⊗ S(V ) is given by ∆(v) = v ⊗ 1 + 1⊗ v for v ∈ V and extension to all
N -HOMOGENEOUS SUPERALGEBRAS 11
of S(V ) by multiplicativity. Similarly, the counit ε : S(V ) → k is given by ε(v) = 0 and the
antipode S : S(V ) → S(V ) by S(v) = −v for v ∈ V .
2.2. Comodules. We refer to [29, Chap. III] for background on comodules, comodule alge-
bras etc.
Given a superbialgebra B, we let ComodsB denote the category of all right B-comodules
and B-comodule maps in Vects
. Thus, for any object V in ComodsB, we have a “coaction”
morphism
δV : V → V ⊗B
in Vects
. If x1, . . . , xd is a fixed basis of V consisting of homogeneous elements, with î
denoting the parity of xi as before, then we will write
δV (xj) =
xi ⊗ b
j with b
j ∈ Bbi+bj (2.3)
The tensor product of vector superspaces makes ComodsB into a tensor category: if U and
V are in ComodsB then B coacts on U ⊗ V by
δU⊗V : U ⊗ V
δU⊗δV
−→ U ⊗ B ⊗ V ⊗ B
−→ U ⊗ V ⊗ B ⊗ B
Id⊗µB
−→ U ⊗ V ⊗ B (2.4)
If B is supercommutative as a superalgebra then the supersymmetry cU,V is a B-comodule
morphism, i.e., δV⊗U ◦ cU,V = (cU,V ⊗ IdB) ◦ δU⊗V . Therefore Comod
B is a symmetric
tensor category in this case .
2.3. The supercharacter map. Let B denote a superbialgebra and let V be a finite dimen-
sional object in ComodsB. The coaction δV is an even map in Homk(V, V ⊗B). Consider the
following morphism in Vects
χs : End
δV ◦( . )
−→ Hom
(V, V ⊗ B)
(1.6)
(V )⊗ B
str⊗Id
−→ k⊗ B = B (2.5)
where str is the supertrace as in (1.8). This map will be called the supercharacter map of
V . Forgetting parity and viewing all elements as even, the supertrace becomes the ordinary
trace and the supercharacter becomes the usual character. These will be denoted by tr and χ,
respectively.
In particular, we have the element
χsV := χ
s(IdV ) ∈ B0̄
To obtain explicit formulas, fix a basis x1, . . . , xd of V consisting of homogeneous elements
and let (F ij ) and (b
j) be the matrices of f ∈ Endk(V ) and of δV with respect to this basis as
in (1.2), (2.3). Then
χs(f) =
bibjbijF
i (2.6)
Let ε : B → k denote the counit of B. Then xj =
i xiε(b
j) holds in (2.3). Hence
ε(bij) = δ
j .1k and (2.6) gives
ε(χs(f)) = str(f) (2.7)
12 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
When f is even formula (2.6) becomes χs(f) =
i,j(−1)
bibijF
i , because F
i = 0 unless
î+ ĵ = 0̄. In particular,
χsV =
bibii (2.8)
In the following, we let comodsB denote the full subcategory of Comod
B consisting of all
objects that are finite-dimensional over k. The supercharacter has the following properties
analogous to standard properties of the ordinary character.
Lemma 2.2. Let B denote a superbialgebra and let U , V and W be objects of comodsB .
(a) If f : V → U and g : U → V are B-comodule maps (not necessarily even) then
χs(f ◦ g) = (−1)
bfbgχs(g ◦ f)
(b) For f ∈ End
(V ), g ∈ End
(U) view f ⊗ g ∈ End
(V ⊗ U) as in (1.7). Then
χs(f ⊗ g) = χs(f)χs(g)
(c) Given an exact sequence 0 → U
−→ W → 0 in comodsB, let f ∈ Endk(V )
be such that f(µ(U)) ⊆ µ(U), and let g ∈ End
(U), h ∈ End
(W ) be the maps
induced by f . Then
χs(f) = χs(g) + χs(h)
In particular, χsV = χ
U + χ
W . Moreover, if f ∈ EndcomodsB(V ) is a projection (i.e.,
f2 = f ) then χs(f) = χsIm f .
Proof. (a) Let TV denote the map Hom
(V, V ⊗ B) −→ B in (2.5); so χs(f) = TV (δV ◦ f).
Since f and g are comodule maps, we have δU ◦ f = (f ⊗ IdB) ◦ δV and similarly for g.
Putting h = δU ◦ f ∈ Hom
(V,U ⊗ B) we obtain χs(f ◦ g) = TU (δU ◦ f ◦ g) = TU (h ◦ g)
and χs(g ◦ f) = TV (δV ◦ g ◦ f) = TV ((g ⊗ IdB) ◦ h). Therefore, we must show that
TU (h ◦ g) = (−1)
bfbgTV ((g ⊗ IdB) ◦ h)
Using the identification Hom
(V,U⊗B) ∼= Hom
(V,U)⊗B as in (1.6), write h =
i fi⊗bi
with fi ∈ Hom
(V,U), bi ∈ B, and f̂i + b̂i = ĥ = f̂ . Then h ◦ g ∈ Hom
(U,U ⊗ B)
becomes the element (
i fi ⊗ bi) ◦ g =
i(−1)
bbibg(fi ◦ g) ⊗ bi ∈ End
(U) ⊗ B, and
(g ⊗ IdB) ◦ hHom
(V, V ⊗B) becomes
i(g ◦ fi)⊗ bi. The standard identity str(fi ◦ g) =
bfibgstr(g ◦ fi) (cf., e.g., [40, p. 165 §3(b)]) now yields
TU (h ◦ g) =
bbibgstr(fi ◦ g)⊗ bi
bbibg+bfibgstr(g ◦ fi)⊗ bi
= (−1)
bfbgTV ((g ⊗ IdB) ◦ h)
as desired.
(b) Fix homogeneous k-bases {xi} and {yℓ} of V and U , respectively, and write x̂i = î,
ŷℓ = ℓ̂ as usual. Moreover, let (F
j ) and (G
m) be the matrices of f and g for these bases, as
N -HOMOGENEOUS SUPERALGEBRAS 13
in (1.2). Then {xi ⊗ yℓ} is a basis of V ⊗ U , with xi ⊗ yℓ having parity î+ ℓ̂. Moreover,
(f ⊗ g)(xj ⊗ ym) = (−1)
bgbjf(xj)⊗ g(ym)
= (−1)bg
xi ⊗ yℓΦ
j,m with Φ
j,m = (−1)
(bℓ+ bm)bjF ijG
because Gmℓ = 0 unless ℓ̂+ m̂ = ĝ. Similarly, writing δV (xj) =
i xi ⊗ b
j with b
j ∈ Bbi+bj
and δU (ym) =
ℓ yℓ ⊗ c
m with c
m ∈ Bbℓ+bm, one obtains using (2.4)
δV⊗U (xj ⊗ ym) =
xi ⊗ yℓ ⊗Ψ
j,m with Ψ
j,m = (−1)
(bi+bj)bℓbijc
Therefore, formula (2.6) becomes
χs(f ⊗ g) =
i,ℓ,j,m
(−1)(
bi+bℓ)(bj+ bm)Ψ
i,ℓ,j,m
bibj+bℓbmbijF
= χs(f)χs(g)
(c) Choose a basis {xi} of V consisting of homogeneous elements so that xi = µ(yi) for
i ≤ dimU and let (F ij ) be the matrix of f for this basis. Then F
j = 0 for i > dimU ,
j ≤ dimU . Moreover, the yi form a basis of U and the zi = π(xi) form a basis of W ,
and the matrices of g and h for these bases are (F ij )i,j≤dimU and (F
j )i,j>dimU , respectively.
Similarly, if (bij) is the matrix of δV with respect to the basis basis {xi} as in (2.3) then b
j = 0
for i > dimU , j ≤ dimU , and the matrices of δU and δW for the given bases are (b
j)i,j≤dimU
and (bij)i,j>dimU , respectively. Therefore,
χs(f) =
bibjbijF
i,j≤dimU
bibjbijF
i,j>dimU
bibjbijF
= χs(g) + χs(h)
The remaining assertions are clear. �
2.4. The Grothendieck ring. Let B be a superbialgebra and let
RB = K0(comod
denote the Grothendieck group of the category comodsB. Thus, for each V in comod
B, there
is an element [V ] ∈ RB and each short exact sequence 0 → U → V → W → 0 in comod
gives rise to an equation [V ] = [U ] + [W ] in RB. The group RB is in fact a ring with
multiplication given by the tensor product of B-comodules. If B is supercommutative as a
superalgebra then the ring RB is commutative; see §2.2.
14 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Both the ordinary dimension and the superdimension are additive on short exact sequences
and multiplicative on tensor products. Hence they yield ring homomorphisms
dim, sdim: RB → Z
Parts (b) and (c) of Lemma 2.2 and formula (2.7) have the following immediate consequence:
Corollary 2.3. The map [V ] 7→ χsV yields a well-defined ring homomorphism χ
s : RB → B0̄.
Furthermore, the following diagram commutes
Z can.
Forgetting the Z2-grading, the corollary also gives the more familiar version with χ and
dim in place of χs and sdim, respectively.
2.5. General linear supergroup and Berezinian. Let V in Vects
be finite-dimensional and
fix a standard basis x1, . . . , xd with î = 0̄ (i ≤ p) and î = 1̄ (i > p).
2.5.1. For each supercommutative k-superalgebra R we denote by E(V )(R) the set of all
R-linear maps V ⊗ R → V ⊗ R in Vects
. Using the identification EndR(V ⊗ R) ∼=
(V, V ⊗R) ∼= End
(V )⊗R (see (1.6)), we may view E(V )(R) as the even subspace
of End
(V )⊗R:
E(V )(R) = (End
(V )⊗R)0̄
This defines a functor E(V ) from the category of supercommutative k-superalgebras to the
category of semigroups.
2.5.2. Tensoring the supertrace str : End
(V ) → k of (1.8) with IdR, we obtain an R-linear
supertrace map str : End
(V )⊗R → R in Vects
which restricts to a map E(V )(R) → R0̄.
The given standard basis x1, . . . , xd of V is an R-basis of V ⊗ R. In terms of this basis, an
element φ ∈ E(V )(R) is given by
φ(xj) =
j with Φ
j ∈ Rbi+bj (2.9)
Thus φ is described by a supermatrix Φ =
in standard form over R:
(2.10)
where A =
i,j≤p
and D =
i,j>p
are square matrices with entries in R0̄ while C,D
are matrices over R1̄. The supertrace of φ is given by
str(φ) =
biΦii = tr(A)− tr(D) =: str(Φ)
N -HOMOGENEOUS SUPERALGEBRAS 15
2.5.3. The functor E(V ) is represented by a supercommutative k-superbialgebra which coacts
on V ; this algebra will be denoted by
B = O(E(V ))
Thus, there is a natural isomorphism of E(V ) with the functor Hom(B, ?) of parity preserving
algebra homorphisms. In particular, the identity map on B corresponds to an element ξ ∈
E(V )(B). Let X = (xij)d×d be the matrix of ξ, as in (2.9). The elements x
j have parity
î + ĵ and they form a set of supercommuting algebraically independent generators of B. In
fact, B is isomorphic to the symmetric superalgebra S(V ∗ ⊗ V ), with xij 7→ x
i ⊗ xj , where
{xi} ⊆ V ∗ is the dual basis for the given basis of V .
We can think of X as the generic supermatrix with respect to the given basis of V : any
supermatrix Φ =
as in (2.9) comes from an algebra map B → R via xij 7→ Φ
j . The
canonical coaction δ : V → V ⊗B, the comultiplication ∆ and the counit ε of B are given by
δ(xj) =
xi ⊗ x
∆(xij) =
xik ⊗ x
ε(xij) = δ
(2.11)
These formulas can also be written as δ(x1, . . . , xd) = (x1, . . . , xd) ⊗X, ∆(X) = X ⊗X
and ε(X) = 1.
2.5.4. Similarly, GL(V )(R) is defined, for any supercommutative k-superalgebra R, as the
set of all invertible R-linear endomorphism of V ⊗R in Vects
. The condition for a superma-
trix Φ in standard form (as in (2.10)) to be invertible is that A and D are invertible as ordinary
matrices over R0̄. In this case, the inverse of Φ is given by
Φ−1 =
(A−BD−1C)−1 −A−1B(D − CA−1B)−1
−D−1C(A−BD−1C)−1 (D − CA−1B)−1
See Berezin [3, Theorem 3.1 and Lemma 3.2]. The element
ber(Φ) := det(A) det(D − CA−1B)−1 = det(D)−1 det(A−BD−1C) (2.12)
is called the superdeterminant or Berezinian of Φ; it is an invertible element of R0̄.
The functor GL(V ) is represented by a supercommutative Hopf superalgebra O(GL(V ))
which is generated over B = O(E(V )) by det(X11)−1 and det(X22)−1, where X11 =(
i,j≤p
and X22 =
i,j>p
are the even blocks of the generic supermatrix X. By [3,
Theorem 3.3], the Berezinian ber(X) is a group-like element in O(GL(V )).
2.6. Supersymmetric functions and exterior powers. Throughout this section, V will de-
note a finite-dimensional vector superspace over k. We assume that the characteristic of k is
zero.
16 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
2.6.1. Let
sgn(σ)σ ∈ k[Sn]
be the antisymmetrizer idempotent of the group algebra k[Sn] and define
nV := Im cYn ⊆ V
⊗n (2.13)
where c : k[Sn] → EndVects
(V ⊗n) is as in (1.10). Thus, ΛnV is the space of antisymmetric
n-tensors,
nV = {y ∈ V ⊗n | cσ(y) = sgn(σ)y for all σ ∈ Sn}
For later use, we describe an explicit basis of ΛnV . To this end, fix a standard basis
x1, . . . , xd of V , with î = 0̄ for i ≤ p and î = 1̄ for i > p. Then the products xi =
xi1 ⊗ xi2 ⊗ · · · ⊗ xin for sequences i = (i1, i2, . . . , in) ∈ {1, 2, . . . , d}
n form a graded basis
of V ⊗n that is permuted up to a ±-sign by the action of Sn on V
⊗n; see formula (1.9):
cσ(xi) = sgni(σ)xσ(i) (2.14)
sgni(σ) = (−1)
(p,q)∈inv(σ)
bip biq and σ(i) = (iσ−1(1), iσ−1(2), . . . , iσ−1(n))
Therefore, by elementary properties of monomial group representations, a k-basis of ΛnV
is given by the nonzero elements cYn(xi) where i ranges over a transversal for the Sn-
action on {1, 2, . . . , d}n. Such a transversal is provided by the weakly increasing sequences
i ∈ {1, 2, . . . , d}n. Moreover, for a weakly increasing i, it is easily seen from (2.14) that
cYn(xi) = 0 holds precisely if iℓ = iℓ+1 ≤ p for some ℓ. Therefore, a basis of Λ
nV is given
by the elements cYn(xi) with i = (i1 < i2 < · · · < im < im+1 ≤ · · · ≤ in) ∈ {1, 2, . . . , d}
and im ≤ p < im+1.
In particular,
m+m′=n
q +m′ − 1
(2.15)
where p = dim
V0̄ and q = dimk V1̄. Equivalently, the generating power series in ZJtK for
the sequence dim
nV is given by
nV tn =
(1 + t)p
(1− t)q
(2.16)
When q > 0 then all ΛnV are nonzero. For additional details on exterior powers, see, e.g.,
[43, Sections I.5 and I.7].
2.6.2. Consider the super bialgebra B = O(E(V )) as defined in §2.5.3 and recall that V is
in comodsB. The representation c : k[Sn] → EndVects
(V ⊗n) of (1.10) actually has image in
EndcomodsB(V
⊗n), since B is supercommutative. Therefore, ΛnV also belongs to comodsB
and we can define the nth elementary supersymmetric function by
en := χ
ΛnV = χ
s(cYn) ∈ B0̄
Here, the equality χs
ΛnV = χ
s(cYn) holds by Lemma 2.2(c).
Similarly, one defines the nth super power sum by
pn := χ
s(c(1,2,...,n)) ∈ B0̄
N -HOMOGENEOUS SUPERALGEBRAS 17
where (1, 2, . . . , n) ∈ Sn the cyclic permutation mapping 1 7→ 2 7→ 3 7→ . . . 7→ n 7→ 1. In
terms of the generic supermatrix X from §2.5.3, one has
pn = str(X
Modulo the space spanned by the Lie commutators fg− gf with f, g ∈ k[Sn], the follow-
ing relation is easily seen to hold in k[Sn]:
nYn ≡
(−1)i−1(1, 2, . . . , i)Yn−i
(with Y0 = 1). Applying the function χ
s ◦ c : k[Sn] → B0̄ to this relation and using
Lemma 2.2(a),(b), one obtains the Newton relations:
nen =
(−1)i−1pien−i
Let t be a formal parameter (of parity 0̄) and consider the generating functions P (t) =∑
n≥1 pnt
n−1 and E(t) =
n≥0 ent
n in B0̄JtK. The Newton relations can be written in
the form P (−t) = d
logE(t); see, e.g., [34, p. 23]. Combining this with the identity
ber(exp(tX)) = exp(str(tX))
due to Berezin ([3, Chapter 3] or [40, p. 167]), one obtains the following expansion for the
characteristic function ber(1 + tX) of generic supermatrix X:
Proposition 2.4. ber(1 + tX) =
n≥0 ent
This proposition is known; see, e.g., Khudaverdian and Voronov [30, Prop. 1].
3. HOMOGENEOUS SUPERALGEBRAS
3.1. N -homogeneous superalgebras. Let N be an integer with N ≥ 2. A homogeneous
superalgebra of degree N or N -homogeneous superalgebra is an algebra A of the form (2.1)
with V finite-dimensional and R ⊆ V ⊗N :
A = A(V,R) ∼= T(V )/(R)
The assumption R ⊆ V ⊗N implies that, besides the usual Z2-grading (“parity”), A also has a
connected Z+-grading (“degree”),
The algebra A is generated by A1 = V and all homogeneous components An are finite-
dimensional objects of Vects
. In fact,
An ∼= V
⊗n/Rn with Rn := (R) ∩ V
i+j+N=n
V ⊗i ⊗R⊗ V ⊗j (3.1)
Note that Rn = 0 for n < N ; so An ∼= V
⊗n if n < N .
Morphisms of N -homogeneous superalgebras f : A = A(V,R) → A′ = A(V ′, R′) are
morphism of superalgebras which also respect the Z+-grading. Equivalently, by restricting to
degree 1, we have a morphism f1 : A1 = V → A
1 = V
′ in Vects
whose N th tensor power sat-
isfies f⊗N1 (R) ⊆ R
′. Thus, one has a category HNAlg
of N -homogeneous k-superalgebras.
18 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Finally, N -homogeneous superalgebras with N = 2 are called quadratic superalgebras; for
N = 3, they are called cubic, etc..
3.2. Some examples. In order to explicitly describe a certain N -homogeneous superalgebra
A = A(V,R), we will usually fix a Z2-graded k-basis x1, . . . , xd of V = A1 and denote the
the parity of xi by î, as in §1.1. The xi form a set of algebra generators for A. Following
Manin [38],[39], the d-tuple f = (1̂, . . . , d̂) ∈ Zd2 is called the format of the basis {xi}.
Example 3.1 (Quantum superspace [39]). For a fixed family q of scalars 0 6= qij ∈ k (1 ≤
i < j ≤ d) and a given format f = (1̂, . . . , d̂) ∈ Zd2 of the basis x1, . . . , xd, the quadratic
superalgebra A = Sfq is defined as the factor of T(V ) modulo the ideal generated by the
elements
ri := xi ⊗ xi ∈ (V
⊗2)0̄ (̂i = 1̄) (3.2)
rij := xj ⊗ xi − qij(−1)
bibjxi ⊗ xj ∈ (V
bi+bj
(i < j) (3.3)
Thus, the algebra Sfq is generated by x1, . . . , xd subject to the defining relations
xixi = 0 (̂i = 1̄)
xjxi = qij(−1)
bibjxixj (i < j).
In the special case where all qij = 1, the algebra S
q is the symmetric superalgebra S(V ) of V
as in Example 2.1.
The ordered monomials of the form xm11 x
2 . . . x
, with
i mi = n, mi ≥ 0 for all i
and mi ≤ 1 if î = 1̄, form a k-basis of the n
th homogeneous component of Sfq. Therefore,
(Sfq)n =
r+s=n
r + p− 1
(3.4)
where dimV0̄ = p and dimV1̄ = q as usual.Thus, the generating series of the dimensions is
(Sfq)nt
(1 + t)q
(1− t)p
Example 3.2 (Yang-Mills algebras [11],[10]). Fix a collection of elements x1, . . . , xd (d ≥
2), numbered so as to have parity î = 0̄ for i ≤ p and î = 1̄ for i > p. Let G = (gij) ∈ GLd(k)
be an invertible symmetric d × d-matrix satisfying gij = 0 if î 6= ĵ and consider the cubic
superalgebra A that is generated by elements x1, . . . , xd subject to the relations
gij [xi, [xj , xk]] = 0 (k = 1, . . . , d) (3.5)
Here [ . , . ] is the supercommutator (2.2). The algebra A will be denoted by YMp|q (q = d−p).
In particular, the pure even algebra YMd|0 is the ordinary Yang-Mills algebra introduced in
[10] while YM0|d is the super Yang-Mills algebra as in [11].
As usual, put V =
i kxi and let [ . , . ]⊗ denote the supercommutator in T(V ). Further-
more, put rk =
i,j gij [xi, [xj , xk]⊗]⊗ and R =
k krk ⊆ V
⊗3; so YMp|q = T(V )/(R).
Using the symmetry of G, we may replace the rk by simpler relations as follows. Choose an
N -HOMOGENEOUS SUPERALGEBRAS 19
invertible d × d-matrix C = (cij) with cij = 0 if î 6= ĵ and such that C
trGC is diagonal,
i,j cirgijcjs = gsδ
s . Replace the bases {xi} of V and {rk} of R by the new bases
ijxj and sk =
kℓrℓ where C
−1 = (cij). Note that yi has parity î and sk has
parity k̂, the parity of rk. A simple calculation shows that sk =
i 6=k gi[yi, [yi, yk]⊗]⊗. Thus
we obtain the following defining relations for the generators y1, . . . yd of YM
i 6=k
gi[yi, [yi, yk]] = 0 (k = 1, . . . , d) (3.6)
The resulting algebras for d = 2 are as follows. Putting x = y1 and y = y2 we have two
defining relations: [x, [x, y]] = 0 and [y, [y, x]] = 0. In the pure even case (x̂ = ŷ = 0̄),
the supercommutators are the ordinary Lie commutators. So YM2|0 is the enveloping algebra
of the Heisenberg Lie algebra; see [1, (0.4)]. In the pure odd case (x̂ = ŷ = 1̄), the two
relations can be written as x2y = yx2 and yx2 = x2y. The resulting algebra YM0|2 is a cubic
Artin-Schelter algebra of type S1 [1, (8.6)]. Thus, both unmixed algebras are Artin-Schelter
regular of global dimension 3. In the mixed case, however (x̂ = 0̄, ŷ = 1̄), the relations say
that x commutes with the Lie commutator [x, y] while y anticommutes: y[x, y] = −[x, y]y.
Thus, [x, y] is a normal element of YM1|1 and YM1|1/([x, y]) is a polynomial algebra in two
variables over k. Moreover, the calculation
[x, y]2 = [x, [x, y]y] = −[x, y[x, y]] = −[x, y]2
shows that [x, y]2 = 0. Thus, the algebra YM1|1 is noetherian with Gelfand-Kirillov dimen-
sion 2 and infinite global dimension.
Returning to the case of general d ≥ 2, we now concentrate on the unmixed algebras intro-
duced by Connes and Dubois-Violette. We will denote these algebras by YM+ = YMd|0 and
YM− = YM0|d. In all formulas below, + applies to YM+ and − to YM−. The generators
i 6=k gi[yi, [yi, yk]⊗]⊗ of the space of relations R can be written as sk =
ℓ yℓ⊗mℓk =
ℓmkℓ ⊗ yℓ with
mℓk =
gℓ (yℓ ⊗ yk − (1± 1)yk ⊗ yℓ) for ℓ 6= k
i 6=k giyi ⊗ yi for ℓ = k
Thus, putting Y = (y1, . . . , yd) and letting M denote the d × d-matrix over YM
± whose
(ℓ, k)-entry is the image of mℓk, the defining relations (3.6) can be written as
YM = 0 or MY tr = 0 (3.7)
The defining relations (3.6) for A = YM− amount to the even element
i giy
i ∈ A2 being
central in A.
Example 3.3 (N -symmetric superalgebra; cf. [5]). Let N ≥ 2 be given and let V be a vector
superspace V over a field k with char k = 0 or char k > N . Define
SN (V ) = A(V,R) with R = Λ
NV = cYN
⊆ V ⊗N
where YN is the antisymmetrizer idempotent of the group algebra k[SN ]; see (2.13). This
defines a functor SN ( . ) : Vect
→ HNAlg
. Since 2cY2 is the supercommutator in T(V ), the
algebra S2(V ) is just the symmetric superalgebra S(V ) of V ; see Example 3.1. The algebra
SN (V ), for a pure even space V = V0̄ and general N ≥ 2, has been introduced in [5].
20 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
If 2 ≤ M ≤ N then, viewing k[SM ] as a subalgebra of k[SN ] as usual, the antisymmetriz-
ers of k[SN ] and k[SM ] satisfy YN = YMa for some a ∈ k[SN ]. Therefore,
R = cYN
⊆ cYM
= cYM
⊗ V ⊗(N−M)
This shows that the identity map on V extends to an epimorphism of superalgebras SN(V ) ։
SM (V ).
Now assume that dimk V = d and fix a standard basis x1, . . . , xd of V , with î = 0̄ for
i ≤ p and î = 1̄ for i > p. From the basis for ΛNV exhibited in §2.6.1 we obtain that the
algebra SN (V ) is generated by x1, . . . , xd subject to the relations
(p,q)∈inv(σ) 1+
bip biqxi
σ−1(1)
σ−1(2)
. . . xi
σ−1(N)
with 1 ≤ i1 < i2 < · · · < im ≤ p = dim
V0̄ < im+1 ≤ · · · ≤ iN ≤ d = dimk V ; see
formula (2.14).
Example 3.4. The following construction generalizes Example 3.3. Fix N ≥ 2 and 0 6= q ∈ k
and assume that condition (1.14) is satisfied. Given a Hecke operator R : V ⊗2 → V ⊗2 on a
vector superspace V we define the N -homogeneous superalgebra
ΛR,N := A(V,R) with R = Im ρR(XN ) ⊆ V
⊗N (3.8)
where XN ∈ HN,q is the q-symmetrizer (1.16) and ρR is the representation (1.20) of HN,q.
We also put
SR,N := Λ−qR−1,N = A(V,R) with R = Im ρR(YN ) ⊆ V
⊗N (3.9)
where YN ∈ HN,q is the antisymmetrizer (1.17). The algebra SN(V ) in Example 3.3 is
identical with ScV,V ,N (q = 1).
3.3. The dual of a homogeneous superalgebra. Let A = A(V,R) be an N -homogeneous
superalgebra. The dual A! of A is defined by
A! = A(V ∗, R⊥)
where, R⊥ ⊆ V ∗⊗N is the (homogeneous) subspace consisting of all elements that vanish on
R ⊆ V ⊗N , using (1.4) in order to evaluate elements of V ∗⊗N on V ⊗N . Thus, (3.1) takes the
A!n = V
∗⊗n/R⊥n with R
i+j+N=n
V ∗⊗i ⊗R⊥ ⊗ V ∗⊗j (3.10)
Identifying V ∗⊗n with the linear dual of V ⊗n via (1.4), we have V ∗⊗i ⊗ R⊥ ⊗ V ∗⊗j =(
V ⊗j ⊗R⊗ V ⊗i
. Hence,
R⊥n =
i+j+N=n
V ⊗j ⊗R⊗ V ⊗i
(3.11)
The canonical isomorphism V
−→ V ∗∗ in (1.5) leads to an isomorphism V ⊗N
V ∗∗⊗N which maps R onto R⊥⊥. Hence,
A! ! ∼= A (3.12)
N -HOMOGENEOUS SUPERALGEBRAS 21
Moreover, if f : A = A(V,R) → A′ = A(V ′, R′) is any morphism in HNAlg
then the
transpose of f1 : V → V
′ induces a morphism f ! : (A′)! → A! in HNAlg
. Thus, we have a
contravariant quasi-involutive dualization functor A 7→ A!, f 7→ f ! on HNAlg
Example 3.5. The dual of A(V, 0) = T(V ) is A(V ∗, V ∗⊗N ); so
T(V )! = T(V ∗)/
V ∗⊗N
In particular, letting V = k be the unit object of Vects
, we have A(k, 0) = k[t] (polynomial
algebra) and A(k, 0)! = k[d]/(dN ), with t and d both having degree 1 and parity 0̄.
Example 3.6 (Dual of quantum superspace). We will describe the dual A! of quantum su-
perspace A = Sfq; see Example 3.1. Fix a homogeneous k-basis x1, . . . , xd with format f
for V , and let x1, . . . , xd denote the dual basis of V ∗; this basis also has format f . Evaluat-
ing an arbitrary element f =
ℓ,m fℓmx
ℓ ⊗ xm ∈ V ∗⊗2 on one of the generating relations
ri, rij ∈ R in (3.2), (3.3) we obtain 〈f, ri〉 = fii and 〈f, rij〉 = fij − qij(−1)
bibjfji. Therefore,
the space R⊥ ⊆ V ∗⊗2 has a basis consisting of the elements sℓ := xℓ ⊗ xℓ (ℓ̂ = 0̄) and
sℓ,k := xℓ ⊗ xk + qkℓ(−1)
bkbℓxk ⊗ xℓ (k < ℓ). In summary, A! is generated by x1, . . . , xd
subject to the defining relations
xℓxℓ = 0 (ℓ̂ = 0̄)
xℓxk = −qkℓ(−1)
bkbℓxkxℓ (k < ℓ).
Thus, A! is isomorphic to quantum superspace Sf
with q′ij = (−1)
bi+bjqij and f
′ = f +
(1̄, . . . , 1̄) the format obtained from f by parity reversal in all components.
Example 3.7 (Duals of the Yang-Mills algebras). Continuing with the notation of Exam-
ple 3.2, we now desribe the algebra A! for A = YMp|q. We assume that char k = 0 and work
with generators y1, . . . , yd of A satisfying (3.6).
Let y1, . . . , yd denote the basis of V ∗ given by 〈yi, yj〉 = δ
j and put γ =
yi ∈ V ∗⊗2. Then, for the generators sk =
i 6=k gi[yi, [yi, yk]⊗]⊗ of R as in Example 3.2,
one computes
〈ya ⊗ yb ⊗ yc, sk〉 = gcδ
k + (−1)
bbgbδ
k − (−1)
babk(1 + (−1)ba)gaδ
〈yi ⊗ γ, sk〉 = δ
(3.13)
Therefore, the map ϕ 7→ ϕ−
k〈ϕ, sk〉y
k ⊗ γ is an epimorphism V ∗⊗3 ։ R⊥ ⊂ V ∗⊗3. We
obtain that the algebra A! is generated by y1, . . . , yd subject to the relations
yaybyc = (gcδ
a + (−1)
bbgbδ
c − (−1)ba
bb(1 + (−1)ba)gaδ
b)g (3.14)
where g = 1
iyi is the image of γ in A.
Since A! is 3-homogeneous, we clearly have A!0 = k, A
i = V ∗ and A!2 =⊕
i,j ky
iyj ∼= V ∗⊗2. By (3.13), the elements yag form a k-basis of A!3 = V
∗⊗3/R⊥ ∼= R∗.
Using the defining relations (3.14) it is not hard to see that A!4 = kg
2 and A!n = 0 for n ≥ 5.
If A = YMp|q is of mixed type (i.e., p 6= 0 and q 6= 0) then g2 = 0.
22 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Example 3.8 (Dual of the N -symmetric superalgebra). Recall from Example 3.3 that SN (V ) =
A(V,R) with R = cYN
. Since YN is central in k[SN ] and stable under the inversion
involution ∗ of k[SN ], it follows from (1.11) that
〈x, cYN (y)〉 = 〈cYN (x), y〉
holds for all x ∈ V ∗⊗N and y ∈ V ⊗N . Therefore,
R⊥ = KerV ∗⊗N (cYN ) = (1− cYN )
V ∗⊗N
and so
SN (V )
! = A
V ∗, (1 − cYN )(V
∗⊗N )
Note that ⋂
i+j+N=n
V ⊗i ⊗R⊗ V ⊗j = cYn
(3.15)
holds for all n ≥ N . This follows from (1.19). Alternatively, as has been noted in Exam-
ple 3.3, we have cYn (V
⊗n) ⊆ R ⊗ V ⊗(n−N). In the same way, one sees that cYn (V
⊗n) ⊆
V ⊗i ⊗ R ⊗ V ⊗j whenever i + j + N = n. For the reverse inclusion, note that each
x ∈ V ⊗i ⊗ R ⊗ V ⊗j satisfies cσℓ(x) = −x for all transpositions σℓ = (ℓ, ℓ + 1) ∈ Sn
with i < ℓ < i+N . Hence, the left hand side of (3.15) is contained in the space of antisym-
metric n-tensors, ΛnV = cYn (V
⊗n), thereby proving (3.15). We deduce from (3.10), (3.11)
and (2.15) that
SN (V )
dn if n < N
r+s=n
q+s−1
if n ≥ N
(3.16)
where d = dim
V , p = dim
V0̄ and q = dimk V1̄.
3.4. The operations ◦ and • on HNAlg
. Let A = A(V,R) and A′ = A(V ′, R′) be N -
homogeneous superalgebras. Following [37] and [7] we define the white and black products
A ◦ A′ and A • A′ by
A ◦A′ = A
V ⊗ V ′, cπN
R⊗ V ′⊗N + V ⊗N ⊗R′
A •A′ = A
V ⊗ V ′, cπN
where πN ∈ S2N is the inverse of the permutation
(1, 2, . . . , 2N) 7→ (1, N + 1, 2, N + 2, . . . , k,N + k, . . . ,N, 2N)
Explicitly, cπN : V
⊗N ⊗ V ′⊗N −→ (V ⊗ V ′)⊗N is the morphism in Vects
that is given by
v1 ⊗ . . . vN ⊗ v
1 ⊗ . . . v
= (−1)
bvj (v1 ⊗ v
1)⊗ . . . (vN ⊗ v
N ) (3.17)
Hence, cπN (R⊗R
′) and cπN
R⊗ V ′⊗N + V ⊗N ⊗R′
are homogeneous subspaces of (V⊗
V ′)⊗N and so A ◦ A′ and A • A′ belong to HNAlg
Under the isomorphism (V ′∗⊗V ∗)⊗N
−→ (V ⊗V ′)∗⊗N which comes from (1.4), the rela-
tions cπN
R′⊥ ⊗R⊥
of A′! • A! map onto the relations
R⊗ V ′⊗N + V ⊗N ⊗R′
of (A ◦ A′)!. In fact, by (1.11) we have c∗πN = cπN , because πN τ = τπN , and so 〈x, y〉 =
〈cπN (x), cπN (y)〉 holds for all x ∈ V
′∗ ⊗N ⊗ V ∗⊗N and y ∈ V ⊗N ⊗ V ′⊗N . Therefore,
canonically,
(A ◦A′)! ∼= A
′! • A! and (A • A′)! ∼= A′! ◦ A! (3.18)
N -HOMOGENEOUS SUPERALGEBRAS 23
the two identities being equivalent by (3.12).
By definition of ◦, the canonical isomorphisms k⊗V ∼= V ∼= V ⊗ k in Vects
give isomor-
phisms A(k, 0) ◦ A ∼= A ∼= A ◦ A(k, 0) in HNAlg
, and (3.18) yields similar isomorphisms
for •, with A(k, 0)! = k[d]/(dN ) replacing A(k, 0) = k[t]; see Example 3.5.
The supersymmetry isomorphism cV,V ′ : V ⊗ V
′ ∼−→ V ′ ⊗ V in Vects
(see (1.1)) yields
isomorphisms
A ◦ A′ ∼= A
′ ◦ A and A • A′ ∼= A′ • A (3.19)
in HNAlg
. To see this, note that the following diagram of isomorphisms in Vects
commutes:
V ⊗N ⊗ V ′⊗N
cπN //
V⊗N,V ′⊗N
(V ⊗ V ′)
V,V ′
V ′⊗N ⊗ V ⊗N cπN
// (V ′ ⊗ V )
with v1⊗. . . vN⊗v
1⊗. . . v
N 7→ (−1)
bvj (v′1⊗v1)⊗. . . (v
N⊗vN ) in both composites.
Therefore, putting RA◦A′ = cπN
R⊗ V ′⊗N + V ⊗N ⊗R′
and similarly for RA′◦A etc., we
V,V ′
(RA◦A′) =
cπN ◦ cV ⊗N ,V ′⊗N
R⊗ V ′⊗N + V ⊗N ⊗R′
= cπN
R′ ⊗ V ⊗N + V ′⊗N ⊗R
= RA′◦A
In the same way, one sees that c⊗N
V,V ′
(RA•A′) = RA′•A. This proves (3.19).
Similarly, the associativity isomorphism aV,V ′,V ′′ : (V ⊗ V
′) ⊗ V ′′ ∼= V ⊗ (V ′ ⊗ V ′′) in
Vects
leads to isomorphisms
(A ◦ A′) ◦ A′′ ∼= A ◦ (A
′ ◦ A′′) and (A •A′) • A′′ ∼= A • (A′ • A′′) (3.20)
in HNAlg
. This is a consequence of the following commutative diagram of isomorphisms in
Vects
V ⊗N ⊗ V ′⊗N
⊗ V ′′⊗N
cπN⊗Id
V ⊗N,V ′⊗N,V ′′⊗N
(V ⊗ V ′)
⊗ V ′′⊗N cπN
// ((V ⊗ V ′)⊗ V ′′)
V,V ′,V ′′
V ⊗N ⊗
V ′⊗N ⊗ V ′′⊗N
Id⊗cπN
// V ⊗N ⊗ (V ′ ⊗ V ′′)⊗N cπN
// (V ⊗ (V ′ ⊗ V ′′))
Finally, the compatibility between the isomorphisms cV,V ′ and aV,V ′,V ′′ (see §1.2) is inher-
ited by the isomorphisms (3.19) and (3.20) in HNAlg
. To summarize:
Proposition 3.9. The operations ◦ and • both make the category HNAlg
of N -homogeneous
k-superalgebras into a symmetric tensor category, with unit objects A(k, 0) = k[t] for ◦ and
A(k, 0)! = k[d]/(dN ) for •.
24 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
3.5. The superalgebra map i : A◦A′ → A⊗A′. Let A = A(V,R) and A′ = A(V ′, R′) be
objects of HNAlg
. The superalgebra A⊗A′ is generated by V ⊕ V ′ subject to the relations
R+R′ ⊆ (V ⊕ V ′)⊗N and [V, V ′]⊗ ⊆ (V ⊕ V
where [ . , . ]⊗ is the supercommutator (2.2) in the tensor algebra, as usual. Thus, A ⊗ A
not N -homogeneous when N ≥ 3. Nonetheless, there always is an injective superalgebra
homomorphism i : A ◦ A′ → A ⊗ A′ which is defined as follows. The linear embedding
V ⊗ V ′ →֒ T(V )⊗ T(V ′) extends uniquely to a superalgebra map
ι̃ : T(V ⊗ V ′) → T(V )⊗ T(V ′) (3.21)
which doubles degrees: the restriction of ι̃ to degree n is the embedding
T(V ⊗ V ′)n = (V ⊗ V
−→ V ⊗n ⊗ V ′⊗n ⊆ (T(V )⊗ T(V ′))2n
in Vects
, where cπn is as in (3.17). Thus, ι̃ identifies the superalgebra T(V ⊗ V
′) with the
(super) Segre product
n≥0 V
⊗n ⊗ V ′⊗n of T(V ) and T(V ′).
The map ι̃ sends RA◦A′ = cπN
R⊗ V ′⊗N + V ⊗N ⊗R′
⊆ (V ⊗V ′)⊗N to R⊗V ′⊗N +
V ⊗N ⊗R′, the kernel of the canonical epimorphism V ⊗N ⊗ V ′⊗N ։ AN ⊗A
N . Thus:
Proposition 3.10. The algebra map ι̃ in (3.21) passes down to yield an injective homomor-
phism k-superalgebras i : A ◦ A′ ֌ A ⊗ A′ which doubles degree. The image of i is the
super Segre product
n≥0An ⊗A
n of A and A
3.6. Internal Hom. The isomorphisms (1.3) and (1.4) together with associativity lead to a
functorial isomorphism
(U ⊗ V,W ∗) ∼= Hom
(U, (V ⊗W )∗)
in Vects
. Explicitly, if g ∈ Hom
(U ⊗ V,W ∗) and g′ ∈ Hom
(U, (V ⊗W )∗) correspond to
each other under the above isomorphism then
〈g(u ⊗ v), w〉 = 〈g′(u), v ⊗ w〉 (3.22)
holds for all u ∈ U , v ∈ V and w ∈ W .
In particular, by restricting to 0̄-components, we have a k-linear isomorphism
HomVects
(U ⊗ V,W ∗) ∼= HomVects
(U, (V ⊗W )∗) (3.23)
This isomorphism leads to
Proposition 3.11. There is a functorial isomorphism
HomHNAlgs
(A • B, C) ∼= HomHNAlgs
(A, C ◦ B!)
Proof. We follow Manin [37, 4.2]. Let A = A(U,R), B = A(V, S) and C = A(W,T ) be
N -homogeneous superalgebras. We will prove the proposition in the following equivalent
form; see (3.12) and (3.18):
HomHNAlgs
(A • B, C!) ∼= HomHNAlgs
(A, (B • C)!)
Recall that C! = A(W ∗, T⊥) and (B•C)! = A((V ⊗W )∗, (cπN (S⊗T ))
⊥). Let g : U⊗V →
W ∗ be a morphism in Vects
and let g′ : U → (V ⊗ W )∗ be the morphism in Vects
N -HOMOGENEOUS SUPERALGEBRAS 25
corresponds to g under (3.23). We must show that, for homogeneous subspaces R ⊆ U⊗N ,
S ⊆ V ⊗N and T ⊆ W⊗N ,
g⊗N (cπN (R ⊗ S)) ⊆ T
⊥ ⇔ g′⊗N (R) (cπN (S ⊗ T ))
Identifying T⊥⊥ with T as in §3.3, the first inclusion is equivalent to
〈g⊗N (cπN (R ⊗ S)) , T 〉 = 0 (3.24)
while the second inclusion states that
〈g′⊗N (R), cπN (S ⊗ T )〉 = 0 (3.25)
But (3.22) shows that (3.24) and (3.25) are equivalent, which proves the proposition. �
Proposition 3.11 says that the tensor category (HNAlg
, •) has an internal Hom which is
given by
Hom(A,B) = B ◦ A!
Explicitly, Hom(A,B) is an object of HNAlg
which represents the functor (HNAlg
)op →
Sets, X 7→ HomHNAlgs
(X • A,B); so there is an isomorphism of functors
HomHNAlgs
(? • A,B) ∼= HomHNAlgs
(?,Hom(A,B))
By general properties of Hom (see [12, Def. 1.6]), the morphism IdHom(A,B) corresponds to
a morphism
µ : Hom(A,B) • A → B (3.26)
in HNAlg
satisfying the following universal property: for any morphism f : X • A → B in
HNAlg
there exists a unique morphism g : X → Hom(A,B) such that the following diagram
commutes:
X • A
g•IdA
Hom(A,B) • A
In degree 1, the map µ is simply IdV ⊗evU : V ⊗ U
∗ ⊗ U −→ V ⊗ k = V .
From Hom(B, C) • Hom(A,B) • A
Id •µ
−→ Hom(B, C) • B
−→ C one obtains in this way a
composition morphism
m : Hom(B, C) •Hom(A,B) → Hom(A, C) (3.27)
in HNAlg
. The morphisms µ and m satisfy the obvious associativity properties.
3.7. The superbialgebra endA. Following Manin [37, 4.2] we define
hom(A,B) = Hom(A!,B!)! = A! • B
for A, B in HNAlg
. Applying the dualization functor to (3.26), (3.27) and recalling (3.18),
we obtain morphisms
δ◦ : A → B ◦ hom(B,A)
∆◦ : hom(A, C) → hom(A,B) ◦ hom(B, C)
26 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
in HNAlg
. The associativity properties of µ and m translate into corresponding coassociativ-
ity properties for δ◦ and ∆◦. Following δ◦ and ∆◦ by the algebra map i of Proposition 3.10,
we obtain superalgebra maps
δ : A → B ⊗ hom(B,A) (3.28)
∆: hom(A, C) → hom(A,B)⊗ hom(B, C) (3.29)
Now take A = B = C = A(V,R) and put endA = hom(A,A); so
endA = A! • A = A(V ∗ ⊗ V, cπN (R
⊥ ⊗R)) (3.30)
Then (3.29) yields a coassociative superalgebra map
∆: endA → endA⊗ endA
Moreover, by Proposition 3.11, the morphism A!
−→ A! ∼= k[t] ◦ A! corresponds to a mor-
phism endA = A! • A → k[t] in HNAlg
. Following this morphism by the map t 7→ 1 we
obtain a superalgebra map
ε : endA → k
which in degree 1 is the usual evaluation pairing evV : V
∗ ⊗ V → k in Vects
. Finally, (3.28)
provides us with a superalgebra map
δA : A → A⊗ endA (3.31)
Note that δA maps the degree n-component of A according to
−→ (A ◦ endA)n
−→ An ⊗ (endA)n →֒ An ⊗ endA (3.32)
Fixing a graded k-basis x1, . . . , xd of V and denoting the dual basis of V
∗ by x1, . . . , xd as
before, endA has algebra generators
zij := x
i ⊗ xj (3.33)
of degree-1 and parity î+ ĵ. In terms of these generators, the maps ε, δA and ∆ are given by
i ) = δ
i or ε(Z) = 1
δA(xj) =
xi ⊗ z
j or δA(x1, . . . , xd) = (x1, . . . , xd)⊗ Z
∆(zij) =
zik ⊗ z
j or ∆(Z) = Z ⊗ Z
(3.34)
where Z = (zij)d×d.
Proposition 3.12. Let A = A(V,R) be an N -homogeneous k-superalgebra.
(a) With ∆ as comultiplication and ε as counit, the superalgebra endA becomes a super-
bialgebra. Moreover, δA makes A into a graded right endA-comodule superalgebra.
N -HOMOGENEOUS SUPERALGEBRAS 27
(b) Given any k-superalgebra B and a morphism of superalgebras δ : A → A⊗ B satis-
fying δ(V ) ⊆ V ⊗ B, there is a unique morphism of superalgebras ϕ : endA → B
such that the following diagram commutes:
δA $$J
A⊗ endA
IdA ⊗ϕ
The proposition is proved as in [37, §5] or [7, Theorem 3].
Example 3.13. When A = A(V, 0) = T(V ), we have endA = A(V ∗⊗V, 0) = T(V ∗⊗V );
endT(V ) = T(V ∗ ⊗ V )
the free superalgebra generated by the elements zij in (3.33).
Example 3.14. By Examples 3.3 and 3.8, we have
endSN (V ) = A
V ∗ ⊗ V, cπN
(1− cYN )(V
∗⊗N )⊗ cYN (V
For example, the algebra end S2(V ) is generated by the elements z
j with parity î+ ĵ subject
to the relations
] + (−1)
bi1 bi2+(bi1+bi2)bj1 [z
] = 0
where [ . , . ] is the supercommutator (2.2). This algebra is highly noncommutative, even for a
pure even space V .
Let O(E(V )) = S(V ∗ ⊗ V ) be the supercommutative superbialgebra as in §2.5.3, with
generators xij . There is a map of superbialgebras
ϕ : endSN (V ) → O(E(V )) , z
j 7→ x
j (3.35)
Indeed, write B = O(E(V )) for brevity and recall the coaction δ : V → V ⊗ B, xj 7→∑
i xi ⊗ x
j from (2.11). Since cYN ∈ EndcomodsB(V
⊗N ) (see §2.6.2), the map δ extends to a
map of superalgebras
δ : SN (V ) → SN (V )⊗ B
Therefore, Proposition 3.12(b) yields the desired ϕ. Note that the coaction of end SN (V ) on
V , when restricted along ϕ, becomes the canonical coaction of O(E(V )) on V ; see (2.11) and
(3.34).
4. N -KOSZUL SUPERALGEBRAS
Throughout this section, we fix an N -homogeneous superalgebra A = A(V,R).
28 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
4.1. The graded dual A!∗. The graded dual
A!∗ =
A! ∗n
of A! has a natural structure of a graded right endA-comodule. Indeed, the linear dual A! ∗n
of the degree n-component of A! embeds into V ⊗n as follows. Recall from (3.11) that
A! ∗n =
V ⊗n if n < N
i+j+N=n V
⊗i ⊗R⊗ V ⊗j if n ≥ N
(4.1)
This identification makes the graded dual A!∗ into a graded right endA-comodule. For, by
(3.32) the coaction δA restricts in degree 1 to a map V → V ⊗endA which makes T(V ) into a
graded right endA-comodule superalgebra. The structure map T(V ) → T(V )⊗endA sends
R → R ⊗ endA. Therefore, each V ⊗i ⊗ R ⊗ V ⊗j is a endA-subcomodule of V ⊗(i+j+N),
and hence A! ∗n is a endA-subcomodule of V
⊗n. Finally, for all n ≥ 0,
A! ∗n+1 ⊆ V ⊗A
n and A
n+N ⊆ V
⊗N ⊗A! ∗n ∩R⊗ V
⊗n = R⊗A! ∗n (4.2)
4.2. The Koszul complex. The map
A⊗ V ⊗(i+1) → A⊗ V ⊗i
a⊗ (v1 ⊗ · · · ⊗ vi+1) 7→ av1 ⊗ (v2 ⊗ · · · ⊗ vi+1)
is a morphism in the category ComodsendA of right endA-comodules, because the endA-
coaction δA in (3.31) is a superalgebra map. Furthermore, this map is a left A-module map
which preserves total degree, and it restricts to a map of endA-subcomodules
d : A⊗A! ∗i+1 → AV ⊗A
i →֒ A ⊗A
which is the A-linear extension of the embedding (4.2). The map dN sends A! ∗i+N to AR ⊗
A! ∗i = 0; so d
N = 0. In other words, we have an N -complex
K(A) : . . .
−→ A⊗A! ∗i+1
−→ A⊗A! ∗i
−→ . . .
−→ A −→ 0 (4.3)
in ComodsendA consisting of graded-free left A-modules and A-module maps which pre-
serve total degree. Therefore, K(A) splits into a direct sum of N -complexes K(A)n =⊕
i+j=nAi ⊗A
j in comod
endA.
Following [7], the Koszul complex K(A) defined by Berger in [5] can be described as the
following contraction of K(A):
K(A) : . . .
−→ A⊗A! ∗N+1
−→ A⊗A! ∗N
−→ A⊗A! ∗1
−→ A −→ 0 (4.4)
This is an ordinary complex in ComodsendA which splits into a direct sum of complexes
K(A)n in comodsendA. The i
th components of K(A) and of K(A)n are given by
K(A)i = A⊗A
ν(i) and K(A)
i = An−ν(i) ⊗A
with ν(i) = νN (i) as in (0.1). The differential on K(A) is
δi : K(A)i → K(A)i−1 where δi =
dN−1 for i even
d for i odd
N -HOMOGENEOUS SUPERALGEBRAS 29
Writing A+ =
n>0 An = AV as usual, we have
Ker δi ⊆ A+K(A)i
for all i. Indeed, this is clear for odd i, since δi = d is injective on A
. For even i, the
restriction of δi = d
N−1 to A! ∗
is given by dN−1 : A! ∗
= A! ∗
ν(i−1)+N−1
→֒ V ⊗(N−1) ⊗
ν(i−1)
−→ AN−1 ⊗A
ν(i−1)
→֒ A ⊗A! ∗
ν(i−1)
where the first embedding comes from (4.2).
Since A! ∗
= A! ∗1 = V and A
= A! ∗N = R by (4.1), the start of the Koszul complex,
augmented by the canonical map
A ։ k = A/A+
is as follows:
−→ A⊗ V
δ1=mult
−→ A −→ k −→ 0 (4.5)
This piece is easily seen to be exact: writing A = T(V )/I with I = (R) = I⊗V +T(V )⊗R
as in (0.2), the map T(V )+ = T(V ) ⊗ V ։ A ⊗ V
։ A+ has kernel I . Thus, Ker δ1 =
I/I ⊗ V = Im δ2. Hence (4.5) is the start of the minimal graded-free resolution of the left
A-module k.
4.3. N -homogeneous Koszul superalgebras. Recall from the Introduction that an N -homoge-
neous superalgebra A is called N -Koszul if TorAi (k, k) is concentrated in degree νN (i) for
all i ≥ 0. By [5, Proposition 2.12] or [8, Theorem 2.4], this happens exactly if the Koszul
complex K(A) is exact in degrees i > 0 and in view of (4.5), this amounts to exactness of
K(A) in degrees i ≥ 2. In this case,
K(A) −→ k −→ 0
is the minimal graded-free resolution of the trivial left A-module k.
The Yoneda Ext-algebra E(A) =
i≥0 Ext
A(k, k) of an N -Koszul superalgebra A has
the following description in terms of the dual algebra A!:
ExtiA(k, k)
ν(i) (i ≥ 0)
Moreover, identifying ExtiA(k, k) and A
, the Yoneda product f · g and the A!-product fg
for f ∈ ExtiA(k, k) = A
and g ∈ ExtjA(k, k) = A
are related by f · g = (−1)ijfg
when N = 2, and
f · g =
fg if i or j is even
0 if i and j are both odd
for N > 2; see [21, Theorem 9.1], [8, Proposition 3.1].
Example 4.1. Quadratic algebras having a PBW-basis are 2-Koszul; see, e.g., [41, Chap. 4,
Theorem 3.1]. This applies in particular to quantum superspace A = Afq; see Example 3.1. A
PBW-basis in this case is given by the collection of ordered monomials xm11 x
2 . . . x
d with
mi ≥ 0 for all i and mi ≤ 1 if î = 1̄, as in Example 3.1. For a more general result, see [41,
Chap. 4, Theorem 8.1].
30 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Example 4.2. The unmixed Yang-Mills algebras A = YM± (see Example 3.2) were shown
to be 3-Koszul in [10], [11]. Indeed, letting A[ℓ] denotes the shift of A that is defined by
A[ℓ]n = Aℓ+n, the defining relations for A in the form (3.7) imply that the following complex
of graded-free left A-modules is exact:
0 −→ A[−4]
−→ A[−3]d
−→ A[−1]d
·Y tr
−→ A −→ k −→ 0 (4.6)
The piece A[−3]d
−→ A[−1]d
·Y tr
−→ A −→ k −→ 0 is identical with (4.5). Therefore, (4.6)
is the minimal graded-free resolution of k. The resolution shows that each TorAi (k, k) is con-
centrated in degree ν3(i), and hence A is 3-Koszul. It also follows that (4.6) is isomorphic to
K(A) → k → 0. In particular, (4.6) confirms the dimensions of the corresponding compo-
nents A!n in Example 3.7. As has been pointed out in [10], [11], it follows from (4.6) that the
Hilbert series HA(t) =
n≥0 dimkAn t
n of A = YM± has the form
HA(t) =
1− dt+ dt3 − t4
(1− t2)(1− dt+ t2)
If d > 2 then the series has a pole in the interval (0, 1), and hence dim
An grows exponen-
tially with n. Therefore, A is not noetherian in this case; see Stephenson and Zhang [42].
The mixed Yang-Mills algebras A = YMp|q with p 6= 0 and q 6= 0, on the other hand, are
never 3-Koszul. For YM1|1 this follows from the description given in Example 3.2: this alge-
bra has infinite global dimension. In general, one can check that the so-called extra condition
(see (4.10) below) fails for A, and so A cannot be Koszul, by [5, Prop. 2.7].
Example 4.3. It has been shown in [5, Theorem 3.13] that the N -symmetric algebra SN(V )
of a pure even space V over a field of characteristic 0 is N -Koszul. An extension of this result
will be offered in Theorem 4.5 below.
4.4. Confluence and Koszulity. For the convenience of the reader, we recall the notions of
reduction operators and confluence and their relation to the Koszul property. Complete details
can be found in Berger [4], [5].
Let V in Vects
be given along with a graded basis X = {x1, . . . xd} that is ordered by
x1 > x2 > · · · > xd. The tensors (“monomials”) xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xiN for i =
(i1, i2, . . . , iN ) ∈ {1, 2, . . . , d}
N form a basis of V ⊗N which will be given the lexicographical
ordering. An X-reduction operator on V ⊗N is a projection S ∈ EndVects
(V ⊗N ) such that
either S(xi) = xi or S(xi) < xi holds for each i, where the latter inequality means that
S(xi) is a linear combination (possibly 0) of monomials < xi. The monomials xi satisfying
S(xi) = xi are called S-reduced, all other monomials are S-nonreduced. We denote by
Red(S) and NRed(S)) the (super) subspaces of V ⊗N that are generated by the S-reduced
monomials and the S-nonreduced monomials, respectively; so V ⊗N = Red(S) ⊕ NRed(S)
and Im(S) = Red(S).
Let LX(V
⊗N ) denote the collection of all X-reduction operators on V ⊗N . The proof of
[4, Theorem 2.3] shows that the application S 7→ Ker(S) is a bijection between LX(V
and the set of all super subspaces of V ⊗N . Hence LX(V
⊗N ) inherits a lattice structure: for
S, S′ ∈ LX(V
⊗N ) one has X-reduction operators S ∧ S′ and S ∨ S′ on V ⊗N which are
N -HOMOGENEOUS SUPERALGEBRAS 31
defined by
Ker(S ∧ S′) = Ker(S) + Ker(S′)
Ker(S ∨ S′) = Ker(S) ∩Ker(S′)
A pair (S, S′) of X-reduction operators is said to be confluent if
Red(S ∨ S′) = Red(S) + Red(S′)
Since the inclusion ⊇ is always true, confluence of (S, S′) is equivalent to the inequality
Im(S ∨ S′) ≤ dim
(Im(S) + Im(S′)) (4.7)
Let n ≥ N . Any X-reduction operator S on V ⊗N gives rise to X-reduction operators Sn,i
on V ⊗n which are defined by
Sn,i := IdV ⊗i ⊗S ⊗ IdV ⊗j (i+ j +N = n)
A monomial xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xin of length n ≥ N is said to be S-reduced if xi is
Sn,i-reduced for all i, that is, if every connected submonomial of xi of length N is S-reduced.
Now let A = A(V,R) be an N -homogeneous superalgebra, and let S be the X-reduction
operator on V ⊗N such that Ker(S) = R. The algebra A is said to be X-confluent if the pairs
(SN+i,i, SN+i,0) of X-reduction operators on V
⊗N+i are confluent for i = 1, . . . , N − 1. By
(4.7) this amounts to the inequalities
Im(SN+i,i ∨ SN+i,0) ≤ dim
(Im(SN+i,i) + Im(SN+i,0)) (4.8)
being satisfied for i = 1, . . . , N − 1.
Following Berger [5], we denote by Tn the lattice of super subspaces of V
⊗n that is gen-
erated by the subspaces
Rn,i := V
⊗i ⊗R⊗ V ⊗j = Ker(Sn,i) (i+ j +N = n) (4.9)
The superalgebra A is said to be distributive if the lattices Tn are distributive for all n, that is,
C ∩ (D + E) = (C ∩D) + (C ∩ E) holds for all C,D,E ∈ Tn.
The following proposition states the operative facts concerning Koszulity for our purposes.
Part (a) is identical with [5, Thm. 3.11] while (b) is [5, Prop. 3.4].
Proposition 4.4. Let A = A(V,R) be an N -homogeneous superalgebra.
(a) If A is X-confluent for some totally ordered graded basis X of V then A is distributive.
Moreover, letting S denote the X-reduction operator on V ⊗N such that Ker(S) = R,
the classes in A of the S-reduced monomials xi1 ⊗ xi2 ⊗ · · · ⊗ xin with xij ∈ X form
a k-basis of An for all n ≥ N .
(b) Assume that A is distributive and the following “extra condition” is satisfied
Rn+N,0 ∩Rn+N,n ⊆ Rn+N,n−1 (2 ≤ n ≤ N − 1) (4.10)
Then A is N -Koszul.
After these preparations, we are now ready to prove the following result. The quadratic
case N = 2 is due to Gurevich [22]; see also Wambst [44].
Theorem 4.5. Let N ≥ 2 and 0 6= q ∈ k and assume that [n]q 6= 0 for all n ≥ 1. Then, for
every Hecke operator R associated with q, the N -homogeneous superalgebra ΛR,N defined
in (3.8) is N -Koszul.
32 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Proof. Put A = ΛR,N and recall that A = A(V,R) with
R = Im ρN,R(XN ) ⊆ V
The extra condition (4.10) is a consequence of equation (1.19). Indeed, (1.19) implies that the
spaces Rn,i in (4.9) have the form
Rn,i =
i+N−1⋂
s=i+1
Im(ρn,R(Ts) + 1) ⊆ V
⊗n (4.11)
Applying (4.11) with ρ = ρn+N,R we see that the left hand side of (4.10) is identical to
Im(ρ(Ti) + 1) ∩
n+N−1⋂
i=n+1
Im(ρ(Ti) + 1) =
n+N−1⋂
Im(ρ(Ti) + 1)
where the equality holds because n + 1 ≤ N . The last expression is clearly contained in⋂n+N−2
i=n Im(ρ(Ti) + 1), which is identical to the right hand side of (4.10). This establishes
the extra condition (4.10).
In order to prove the distributivity of A, we follow the approach taken in [25]. We first prove
the claim for the standard solution RDJ , i.e., the operator given in Example 1.2 with d = p
and q = 0. As above, fix a basis X = {x1, . . . , xd} of V , ordered by x1 > x2 > · · · > xd,
and consider the basis of V ⊗n consisting of the monomials xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xin for
i = (i1, i2, . . . , in) ∈ {1, 2, . . . , d}
n with the lexicographical ordering. By equation (1.21),
the action of the generators Tj of the Hecke algebra H = Hn,q2 on this basis is given by
Tj(xi) =
q2xi if ij = ij+1
(q2 − 1)xi + qxσj(i) if ij < ij+1
qxσj(i) if ij > ij+1
(4.12)
Here, σj = (j, j + 1) ∈ Sn and σ(i) = (iσ−1(1), iσ−1(2), . . . , iσ−1(n)) for σ ∈ Sn, as in
Example 3.3.
We claim that the H -submodule of V ⊗n that is generated by xi is given by
H (xi) =
i′∈Sn(i)
kxi′ (4.13)
where Sn(i) is the Sn-orbit of i. Indeed, (4.12) implies that each Tσ(xi) with σ ∈ Sn
is a linear combination of basis vectors xi′ with i
′ ∈ Sn(i). Hence, ⊆ certainly holds in
(4.13). For the reverse inclusion, let i∗ denote the unique non-decreasing sequence in Sn(i);
so xi∗ = max{xi′ | i
′ ∈ Sn(i)}. The last formula in (4.12) implies that
T (xi) = q
r(i)xi∗ (4.14)
where T is a suitable finite product of length r(i) ≥ 0 in the generators Tj . Since T is a unit
in H , the inclusion ⊇ holds in (4.13), thereby proving the asserted equality.
Furthermore, (4.14) and (1.18) (with q replaced by q2) give
qr(i)Xn(xi) = Xn(xi∗). (4.15)
N -HOMOGENEOUS SUPERALGEBRAS 33
These elements are nonzero. For, (4.15) implies that the elements Xn(xi∗) span the image of
Xn on V
⊗n, and their number is
d+n−1
which is equal to the rank of Xn (cf. [25, Eq. (5)]).
It follows that Xn(V
⊗n) = Im ρn,RDJ (Xn) has a k-basis consisting of the elements
Xn(xi∗)
∣∣ i∗ = (i1 ≤ i2 ≤ · · · ≤ in) ∈ {1, 2, . . . , d}n
Next, writing
Xn(xi) =
i′∈Sn(i)
λi′xi′ (4.16)
with λi′ ∈ k, we claim that
λσj(i′) =
λi′ if i
′ = σj(i
q±1λi′ otherwise
To prove this, we may assume that i′ 6= σj(i
′). We compute the coefficient of xσj(i′) in
TjXn(xi) in two ways: by (1.18) this coefficient is equal to q
2λσj(i′) while (4.12) yields
the expression qλi′ + (q
2 − q1±1)λσj(i′). The claim follows from this. Writing an arbitrary
σ ∈ Sn as a product of the inversions σj , we see that the coefficients λi′ in (4.16) only differ
by a nonzero scalar, and hence they are all nonzero since Xn(xi) 6= 0.
By Proposition 4.4, it suffices to check the X-confluence conditions (4.8) i = 1, . . . , N−1.
So let S be the X-reduction operator on V ⊗N with Ker(S) = R. It is easy to see from the dis-
cussion above (with n = N ) that S is given by S(xi∗) = (1−XN/λi∗)(xi∗) and S(xi) = xi
for i 6= i∗. According to (4.11) and the discussion above, the dimension of (R⊗V ⊗i)∩(V ⊗i⊗
R) is
d+N+i−1
. Thus, the dimension of the left hand side of (4.8) is dN+i−
d+N+i−1
the other hand the monomials in V ⊗N+i that belong to NRed(SN+i,i) ∩ NRed(SN+i,0) are
exactly those of the form xi∗ with i
∗ ∈ {1, . . . , d}N+i non-decreasing. Their number is pre-
cisely
d+N+i−1
. Therefore, the dimension of Im(SN+i,i) + Im(SN+i,0) = Red(SN+i,i) +
Red(SN+i,0) is at least d
N+i −
d+N+i−1
. This proves the inequality in (4.8), thereby fin-
ishing the proof of the theorem for the case R = RDJ .
In order to deal with an arbitrary Hecke operator R, recall that Hn,q is split semisimple,
having a representative set of simple modules Mλ indexed by the partitions λ ⊢ n; see (1.15).
We denote the representation of Hn,q on Mλ by ρλ; it does not depend on the operator R but
only on the partition λ.
Let us fix a decomposition
V ⊗n =
into simple Hn,q-submodules Mt. Since all Mt are invariant under the operators ρn,R(Tj),
formula (4.11) yields the decomposition
Rn,i =
i+N−1⋂
s=i+1
(ρn,R(Ts) + 1)(Mt) =
Rn,i ∩Mt
for all i. Therefore, by [25, Lemma 1.2], distributivity of the lattice Tn that is generated by
the subspaces Rn,i of V
⊗n is equivalent to distributivity of the lattices Tn ∩Mt (t ∈ T ) that
34 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
are generated by the subspaces
Rn,i ∩Mt =
i+N−1⋂
s=i+1
(ρn,R(Ts) + 1)(Mt)
of Mt. Now, each Mt is isomorphic to Mλ for some λ ⊢ n. Therefore, the lattice Tn ∩Mt is
isomorphic to the lattice of subspaces of Mλ that is generated by the subspaces
i+N−1⋂
s=i+1
(ρλ(Ts) + 1)(Mλ)
with i + N ≤ n. Finally, when d = dimV > n, then all simple Hn,q-modules Mλ appear
in V ⊗n; see [15, Proposition 5.1]. Thus, the distributivity of the lattice associated to RDJ ,
which we have already verified, implies the distributivity of the corresponding lattice for any
Hecke operator R. This completes the proof. �
5. KOSZUL DUALITY AND MASTER THEOREM
In this section, A = A(V,R) denotes an N -homogeneous superalgebra that is assumed to
be N -Koszul (N ≥ 2).
5.1. By Koszulity, the complexes
K(A)n : . . . → An−νN(i) ⊗A
νN (i)
→ An−νN (i−1) ⊗A
νN (i−1)
→ . . .→An → 0
are exact for n > 0. This yields equations in the Grothendieck ring RendA of the category
comodsendA : ∑
(−1)i[An−νN (i)][A
νN (i)
] = 0 (n > 0) (5.1)
In the power series ring RendAJtK over the Grothendieck ring RendA, define the Poincaré
series
PA(t) =
[An]t
n and PA!∗(t) =
[A! ∗n ]t
For any power series P (t) =
n ant
n, we use the notation
PN (t) :=
n≡0,1 mod N
(−1)αN (n)ant
where αN (n) = n − (n mod N) denotes the largest multiple of N less than or equal to n.
Thus, P2(t) = P (t) and in general
PN (−t) =
n≡0,1 mod N
(−1)n mod Nant
(−1)iaνN (i)t
νN (i)
(5.2)
In particular,
PA!∗,N (−t) =
(−1)i[A! ∗νN (i)]t
νN (i)
Equations (5.1) are equivalent to the following Koszul duality formula:
N -HOMOGENEOUS SUPERALGEBRAS 35
Proposition 5.1. For any N -homogeneous Koszul superalgebra A, the identity
PA(t)PA!∗,N (−t) = 1
holds in RendAJtK.
Applying the ring homomorphism χsJtK : RendAJtK → (endA)0̄JtK, where χ
s is the su-
percharacter map as in Corollary 2.3, the formula in Proposition 5.1 takes the following form
in (endA)0̄JtK:
Corollary 5.2.
χsAℓt
m≡0,1 mod N
(−1)m mod Nχs
A! ∗m
 = 1
Analogous formulas hold with the supercharacter χs replaced by the ordinary character χ
or by one of the dimensions dim and sdim.
By (3.32) the coaction of endA on A sends An to An⊗ (endA)n. A similar remark holds
for the endA-coaction on A! ∗; see §4.1. Therefore, both factors in Corollary 5.2 actually
belong to the Rees subring
n≥0 Bnt
n of BJtK, where we have put B = (endA)0̄.
Example 5.3. As an application of the Hilbert series version of Corollary 5.2, we see that the
duals A! of the Yang-Mills algebras A = YMp|q are never 3-Koszul. In fact, by Example 3.7,
we have HA!(t) = 1+dt+d
2t2+dt3+t4 if p = 0 or q = 0 and HA!(t) = 1+dt+d
2t2+dt3
otherwise. In either case, HA!(t)
−1 has a nonzero coefficient at t5, which rules out Koszulity.
5.2. A master theorem modeled on the N -symmetric superalgebra SN (V ). We put A =
SN (V ) and use the notation of Examples 3.3 and 3.8. In particular, we assume that char k = 0
and work with a fixed basis x1, . . . , xd of V = A1 so that î = 0̄ for i ≤ p and î = 1̄ for i > p.
From Example 3.3 (see also Proposition 4.4(a)), we know that a basis of Aℓ is given by the
monomials xi = xi1xi2 . . . xiℓ for sequences i = (i1, . . . , iℓ) ∈ {1, . . . , d}
ℓ such that i has no
connected subsequence j = (j1, . . . , jN ) of length N satisfying
1 ≤ j1 < . . . < jm ≤ p < jm+1 ≤ . . . ≤ jN ≤ d = p+ q
for some m. Adapting notation of Etingof and Pak [16] to our setting, we denote this set of
sequences i by
Λ(p|q,N)ℓ (5.3)
For example, Λ(p|q, 2)ℓ consists of all weakly decreasing sequences i = (i1, . . . , iℓ) with
entries from {1, . . . , d} and such that no repetition occurs in the range {p + 1, . . . , d}.
In order to evaluate the character χsAℓ in Corollary 5.2, recall from (3.34) that the coaction
δA : A → A⊗ endA is given on the generators xi of A by
δA(xi) =
xj ⊗ z
i ∈ A⊗ endA
where zji = x
j ⊗ xi are the canonical generators of the algebra endA. For i = (i1, . . . , iℓ) ∈
Λ(p|q,N)ℓ, we have
δA(xi) = δA(xi1)δA(xi2) . . . δA(xiℓ) ∈ Aℓ ⊗ endA
Since Aℓ ⊗ endA =
i∈Λ(p|q,N)ℓ
xi ⊗ endA, we can define Z(i) ∈ (endA)0̄ by
δA(xi) = xi ⊗ Z(i) + (terms supported on Λ(p|q,N)ℓ \ {i})
36 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Then (2.8) becomes
χsAℓ =
i∈Λ(p|q,N)ℓ
biZ(i) (5.4)
with î = î1 + · · ·+ îℓ.
Now consider the super bialgebra B = O(E(V )) = k[xij | 1 ≤ i, j ≤ d] defined in §2.5.3
and recall that the xij are supercommuting variables of parity î + ĵ over k. Restricting the
comodule Aℓ to B along the map ϕ : endSN (V ) → B, z
j 7→ x
j in (3.35) we must replace
Z(i) in (5.4) by X(i) := ϕ(Z(i)) ∈ B0̄. Thus, writing
xj ⊗ x
i ∈ A⊗ B
and yi = yi1 . . . yiℓ ∈ Aℓ ⊗ B =
j∈Λ(p|q,N)ℓ
xj ⊗ B for i = (i1, . . . , iℓ), we have
yi = xi ⊗X(i) + (terms supported on Λ(p|q,N)ℓ \ {i}) (5.5)
As for the supercharacter of A! ∗m , recall from (4.1) and (3.15) that, for all n ≥ N ,
A! ∗n =
i+j+N=n
V ⊗i ⊗R⊗ V ⊗j = ΛnV
Viewing A! ∗n = Λ
nV as a comodule over B = O(E(V )), the supercharacter of A! ∗n is the
nth elementary supersymmetric function en which we know, by Proposition 2.4, to be iden-
tical to the coefficient at tn of the characteristic function ber(1 + tX) of the generic su-
permatrix X =
1≤i,j≤d
of type p|q; so the diagonal blocks X11 =
1≤i,j≤p
X22 =
p+1≤i,j≤p+q
consist of even entries while all other entries are odd.
To summarize, we obtain the following super-version of [16, Theorem 2].
Theorem 5.4. Let X =
be the generic supermatrix of type p|q. Then
i∈Λ(p|q,N)ℓ
biX(i) tℓ
m≡0,1 mod N
(−1)m mod Nemt
 = 1
holds in the power series ring k[xij | all i, j ]0̄JtK. Here Λ(p|q,N)ℓ and X(i) are defined
by (5.3) and (5.5), respectively, and the em are the coefficients of the characteristic function
ber(1 + tX) =
n≥0 ent
n of X.
5.3. As an application of Theorem 5.4 , we determine the superdimension Hilbert series
HsA(t) =
for the N -symmetric superalgebra A = SN (V ). For the pure even case, this was already done
by Etingof and Pak [16] . The notations of §5.2 remain in effect.
In view of Corollary 2.3, the superdimension Poincaré series follows by applying the counit
ε : B → k to the equation in Theorem 5.4. Indeed, by (2.11), the counit ε sends X 7→ 1d×d,
N -HOMOGENEOUS SUPERALGEBRAS 37
and hence the elements X(i) in (5.5) all map to 1. Therefore, the first factor in Theorem 5.4
becomes
HsA(t) =
i∈Λ(p|q,N)ℓ
For the second factor, note that
ber(1 + t 1d×d) = (1 + t)
by (2.12). Thus,
HsA(t) =
i∈Λ(p|q,N)ℓ


m≡0,1 mod N
(−1)m mod N
if p ≥ q
m≡0,1 mod N
(−1)αN (m)
m+ q − p− 1
q − p− 1
if p < q
(5.6)
where αN (m) = m− (m mod N) denotes the largest multiple of N less than or equal to m
as in §5.1.
5.4. The ordinary Hilbert series HA(t) =
ℓ≥0 dimkAℓ t
ℓ of the N -symmetric superalge-
bra A = SN(V ) is as follows. Recall from §5.2 that
Aℓ = |Λ(p|q,N)ℓ|
and from (3.16) that
A!n =
dn if n < N
r+s=n
q+s−1
if n ≥ N
Therefore, the Hilbert series is
HA(t) =
|Λ(p|q,N)ℓ| t
m≡0,1 mod N
(−1)m mod N
r+s=m
q + s− 1
−1 (5.7)
5.5. Less is known about the Hilbert series of the N -homogeneous superalgebras A = ΛR,N
associated to an arbitrary Hecke operator R : V ⊗2 → V ⊗2 on a vector superspace V ; see
Example 3.4. Recall that A = A(V,R) with R = Im ρR(XN ) ⊆ V
⊗N . For any N -
homogeneous algebra A = A(V,R), we have
A!n = dimk
i+j+N=n
V ⊗j ⊗R⊗ V ⊗i
38 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
by (3.10) and (3.11). For R = Im ρR(XN ) in particular, (1.19) further implies that
i+j+N=n
V ⊗j ⊗R⊗ V ⊗i = ρR(Xn)
holds for n ≥ N . Now [26, Theorem 3.5] implies that
!(t) =
ℓ=1(1 + aℓt)∏s
m=1(1− bmt)
where (r, s) is the birank of R and aℓ and bm are positive real numbers. For example, in the
situation of 5.4, (r, s) = (p, q) and aℓ = bm = 1.
For any complex power series P (t), the power series PN (−t) in (5.2) can be written as
PN (−t) =
(1− ζ−iN )P (ζ
where ζN = e
2πi/N . In particular,
HA!∗,N (−t) =
(1− ζ−iN )
ℓ=1(1 + aℓζ
N t)∏s
m=1(1− bmζ
QN,a,b(t)∏s
m=1(1 + bmt+ . . .+ b
m tN−1)
for some real polynomial QN,a,b(t) with coefficients being polynomial in a = (aℓ) and b =
(bm). Therefore, the Hilbert series of A has the form
HA(t) =
m=1(1 + bmt+ . . .+ b
QN,a,b(t)
(5.8)
Notice that the fraction on the right-hand side is reduced.
In particular, (5.7) has the form
HA(t) =
(1− tN )s
(1− t)sQN,1,1(t)
(5.9)
APPENDIX
For lack of a suitable reference, we include here a proof of Proposition 0.1 that was stated
in the Introduction. Our proof is based on the proof of [8, Proposition 2.1] and on additional
details that were communicated to us by Roland Berger. For the basics concerning graded
algebras, we refer the reader to [9, Chap. II §11] or [6].
As in the Introduction, A =
n≥0 An denotes an arbitrary connected Z≥0-graded k-
algebra and V is a graded subspace of A+ =
n>0An satisfying A+ = V ⊕ A
+. Thus,
T(V )/I
−→ A for some graded ideal I of T(V ). For convenience, we state Proposition 0.1
again:
Proposition. The relation ideal I of A lives in degrees ≥ N if and only if TorAi (k, k) lives in
degrees ≥ νN (i) =
N if i is even
N + 1 if i is odd
N -HOMOGENEOUS SUPERALGEBRAS 39
Proof. Let
P : · · · → Pi
−→ Pi−1
−→ · · ·
−→ P0
−→ k → 0
be a minimal graded-free resolution of the trivial left A-module k. Thus, all Pi have the form
Pi = A⊗ Ei for some graded subspace Ei ⊆ Ker di−1 which is chosen so that
Ker di−1 = Ei ⊕A+Ker di−1 (A.10)
In particular, we may take E0 = k and E1 = V . The differential di : Pi → Pi−1 is the graded
A-module map that is defined by the inclusion Ei →֒ Pi−1. By the graded Nakayama Lemma
(e.g., [9, p. AII.171, Prop. 6]), our choice of Ei implies that
Im di = AEi = Ker di−1 and Ker di ⊆ A+ ⊗ Ei = A+Pi (A.11)
for all i. Consequently, the complex k⊗A P has zero differential, and hence
TorAi (k, k)
∼= k⊗A Pi ∼= Ei
In particular,
TorA0 (k, k)
∼= k and TorA1 (k, k)
∼= V = A+/A
live in degrees 0 = νN (0) and ≥ 1 = νN (1), respectively. Moreover, the kernel of d1 : P1 =
(T(V )/I)⊗ V → P0 = A is exactly I/I ⊗ V , and so
TorA2 (k, k)
∼= Ker d1/A+Ker d1 ∼= I/ (V ⊗ I + I ⊗ V )
Therefore, I lives in degrees ≥ N if and only if TorA2 (k, k) lives in degrees ≥ N = νN (2).
For the remainder of the proof, assume that I lives in degrees ≥ N . We will show by
induction on i that TorAi (k, k) = Ei lives in degrees ≥ νN (i) for all i. The cases i ≤ 2
have been checked above. Assume that Ei lives in degrees ≥ νN (i) and similarly for Ei−1.
By (A.11), we know that Ei+1 ⊆ Ker di ⊆ A+ ⊗ Ei and so Ei+1 certainly lives in degrees
≥ νN (i)+1. Since νN (i)+1 = νN (i+1) when i is even (or when i is arbitrary and N = 2),
we are done in these cases. From now on, we assume that i is odd. We must show that Ei+1
lives in degrees ≥ νN (i + 1) =
N . Since Ei+1 ⊆ Ker di, it suffices to show that di is
injective in degrees < i+1
N , and since Ei lives in degrees ≥ νN (i) =
N+1, our goal is to
show that di is injective on all homogeneous components Pi,n of Pi in degrees n =
N + j
with j = 1, . . . , N − 1. Put m = i−1
N for simplicity and note that
Pi,m+j =
Aj−ℓ ⊗ Ei,m+ℓ (A.12)
Pi−1,m+j =
Aj−k ⊗ Ei−1,m+k (A.13)
since Ei−1 lives in degrees ≥ νN (i − 1) = m. The proposition will be a consequence of the
following claims:
(a) di is injective on all summands Aj−ℓ ⊗ Ei,m+ℓ in (A.12), and
(b) the subspaces di (Aj−ℓ ⊗ Ei,m+ℓ) = Aj−ℓEi,m+ℓ for ℓ = 1, . . . , j form a direct sum
inside Pi−1,m+j .
40 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
In order to prove (a), recall that the restriction of di to Ei,m+ℓ is the inclusion
Ei,m+ℓ →֒ Pi−1,m+ℓ =
Aℓ−k ⊗ Ei−1,m+k
Hence, the effect of di on the ℓ
th summand in (A.12) is the embedding
Aj−ℓ ⊗ Ei,m+ℓ →֒
Aj−ℓ ⊗Aℓ−k ⊗ Ei−1,m+k
followed by the map
Aj−ℓ ⊗Aℓ−k ⊗ Ei−1,m+k −→
Aj−k ⊗ Ei−1,m+k ⊆ Pi−1,m+j
which is given by the multiplication map Aj−ℓ ⊗ Aℓ−k → Aj−k. Since j − k < N , our
hypothesis on I implies that Aj−k ∼= T(V )j−k, and similarly Aj−ℓ ∼= T(V )j−ℓ and Aℓ−k ∼=
T(V )ℓ−k. Therefore, the above multiplication map is identical with the injection T(V )j−ℓ ⊗
T(V )ℓ−k →֒ T(V )j−k in T(V ). This proves (a).
For (b), we proceed by induction on j. The case j = 1 being obvious, let 1 ≤ j ≤ N − 2
and assume that (ii) holds for 1, . . . , j. We wish to show that the subspaces Aj+1−ℓEi,m+ℓ
(ℓ = 1, . . . , j + 1) of Pi−1,m+j+1 form a direct sum. First, by (A.10) we have Ei,m+j+1 ∩
A+Ker di−1 = 0 while
ℓ=1Aj+1−ℓEi,m+ℓ ⊆ A+Ker di−1. Therefore, it suffices to show
that the sum
ℓ=1Aj+1−ℓEi,m+ℓ is direct. To this end, note that Aj+1−ℓ =
d≥1 VdAj+1−d−ℓ
holds for all ℓ ≤ j. Hence,
Aj+1−ℓEi,m+ℓ =
Aj+1−d−ℓEi,m+ℓ
By induction,
Aj+1−d−ℓEi,m+ℓ is a direct sum inside Pi−1,m+j+1−d. Thus, it suffices
to show that the sum
d≥1 VdPi−1,m+j+1−d ⊆ Pi−1,m+j+1 is direct. But (A.13) gives
Pi−1,m+j+1 =
Aj+1−k ⊗ Ei−1,m+k =
T(V )j+1−k ⊗ Ei−1,m+k
where the last equality holds since all j + 1− k < N . Therefore,
VdPi−1,m+j+1−d =
j+1−d⊕
T(V )j+1−d−k ⊗ Ei−1,m+k
as desired. This proves (b), thereby completing the proof of the proposition. �
Acknowledgement. The authors wish to thank Roland Berger for his helpful comments
throughout the completion of this paper. Work on this article was initiated during a visit
of ML to the Université Montpellier 2 in June 2006. ML would like to thank Claude Cibils
for arranging this visit and for his warm hospitality in Montpellier, and James Zhang for help
with some examples in this article.
N -HOMOGENEOUS SUPERALGEBRAS 41
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MATHEMATIK, UNIVERSITY OF DUISBURG-ESSEN, GERMANY AND INSTITUTE OF MATHEMATICS, HANOI,
VIETNAM
E-mail address: hai.phung@uni-duisburg-essen.de
LAMUSE, FACULTÉ DES SCIENCES ET TECHNIQUES, UNIVERSITÉ DE SAINT-ETIENNE, 23 RUE DU DOC-
TEUR PAUL MICHELON, 42023 SAINT-ETIENNE CEDEX 2, FRANCE
E-mail address: benoit.kriegk@univ-st-etienne.fr
DEPARTMENT OF MATHEMATICS, TEMPLE UNIVERSITY, PHILADELPHIA, PA 19122-6094, USA
E-mail address: lorenz@temple.edu
http://arxiv.org/abs/math/0603169
http://arxiv.org/abs/math/0703203
http://arxiv.org/abs/math/0703213
http://arxiv.org/abs/math/0607737
	Introduction
	1. Review of linear superalgebra
	2. The supercharacter
	3. Homogeneous superalgebras
	4. N-Koszul superalgebras
	5. Koszul duality and master theorem
	Appendix
	References

  We develop the theory of N-homogeneous algebras in a super setting, with
particular emphasis on the Koszul property. To any Hecke operator on a vector
superspace, we associate certain superalgebras and generalizing the ordinary
symmetric and Grassmann algebra, respectively. We prove that these algebras are
N-Koszul. For the special case where the Hecke operator is the ordinary
supersymmetry, we derive an $N$-generalized super-version of MacMahon's
classical "master theorem".

Introduction 1
1. Review of linear superalgebra 5
2. The supercharacter 10
3. Homogeneous superalgebras 17
4. N -Koszul superalgebras 27
5. Koszul duality and master theorem 34
Appendix 38
References 41
INTRODUCTION
0.1. The theory of N -homogeneous algebras owes its existence primarily to the concerns of
noncommutative geometry. In fact, as has been expounded by Manin in his landmark pub-
lications [36], [37], quadratic algebras (the case N = 2) provide a convenient framework
for the investigation of quantum group actions on noncommutative spaces. Moreover, certain
Artin-Schelter regular algebras [1], natural noncommutative analogs of ordinary polynomial
algebras, can be presented as associative algebras defined by cubic relations (N = 3). The
latter algebras, as well as many of the quadratic algebras studied by Manin, enjoy the addi-
tional “Koszul property” which will be of central importance in the present article; it will be
reviewed in detail in 0.6 below.
Motivated by these examples and others, Berger [5] initiated the systematic investigation
of N -homogeneous algebras for all N ≥ 2, introducing in particular a natural extension of
the notion of Koszul algebra from the familiar quadratic setting to general N -homogeneous
2000 Mathematics Subject Classification. Primary 16S37, 05A19.
PHH is supported by the DFG through a Heisenberg-Fellowship.
ML’s research is supported in part by NSA Grants H98230-05-1-0025 and H98230-07-1-0008 and by Lever-
hulme Research Interchange Grant F/00158/X.
http://arxiv.org/abs/0704.1888v2
2 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
algebras. Article [5] gives examples of N -Koszul algebras for all N ≥ 2; these are the
so-called N -symmetric algebras, the special case N = 2 being the ordinary symmetric (poly-
nomial) algebra. Following the general outline of Manin’s lecture notes [37] on the case of
quadratic algebras, Berger, Dubois-Violette and Wambst developed the categorical aspects of
N -homogeneous algebras in [7].
0.2. Current interest in N -homogeneous algebras is fueled in part by the fact that they do
occur naturally in mathematical physics and in combinatorics. Indeed, Connes and Dubois-
Violette [10], [11] introduced a class of 3-homogeneous algebras, called Yang-Mills algebras,
which are in fact 3-Koszul. There are two versions of Yang-Mills algebras: in the language of
linear superalgebra, the first kind has even (parity 0̄) algebra generators while the second kind
is generated by odd (parity 1̄) elements.
Combinatorics enters the picture via MacMahon’s celebrated “master theorem” [35], specif-
ically the recent quantum generalization of the master theorem due to Garoufalidis, Lê and
Zeilberger [20]. As has been pointed out by two of the present authors in [28], the yoga of
(quadratic) Koszul algebras leads to a rather effortless and conceptual proof of the quantum
master theorem based on the fact that a certain quadratic algebra, known as quantum affine
space, is Koszul. Further quantum generalizations and super versions of the master theorem
have been obtained by several authors using a variety of approaches; see Foata and Han [17],
[18], [19], Konvalinka and Pak [33], Etingof and Pak [16].
0.3. From an algebraic point of view, MacMahon’s master theorem (MT) in its various incar-
nations finds its most natural explanation by the phenomenon of “Koszul duality”. Indeed, all
versions of MT can be expressed in the form that, for some algebra B, an equation Σ1 ·Σ2 = 1
holds for suitable power series Σ1,Σ2 ∈ BJtK. Here is a brief outline how one can arrive at
such an equation starting with a given N -Koszul algebra A. Associated with A, there is a
graded complex, K(A), which is exact in positive degrees, and a certain endomorphism bial-
gebra, endA, which coacts on all components of K(A). These components therefore define
elements of the representation ring RendA of endA, and exactness of K(A) in positive de-
grees yields an equation in the power series ring RendAJtK. Due to the specific form of K(A),
which is constructed from A together with its so-called dual algebra A!, the equation in ques-
tion does indeed state that ρ1 · ρ2 = 1 holds for suitable ρ1, ρ2 ∈ RendAJtK. The last step
in deriving a MT for A consists in using (super-)characters to transport the abstract duality
equation ρ1 · ρ2 = 1 from RendAJtK to the power series ring over the algebra endA, where it
takes a more explicit and useable form. Here then is the flow chart of our approach:
N -Koszul algebra
// exact Koszul complex
// duality equation
in RendAJtK
// MT for A
The actual labor involved in this process consists in the explicit evaluation of (super-)characters
at the last arrow above. This step is often facilitated by specializing the bialgebra endA, which
is highly noncommutative, to a more familiar algebra B via a homomorphism endA → B.
For example:
• MacMahon’s original MT [35] follows in the manner described above by starting with
A = O(kd) = k[x1, . . . , xd], the ordinary polynomial algebra or “affine space”, and
N -HOMOGENEOUS SUPERALGEBRAS 3
restricting the resulting MT over endO(kd) to the coordinate ring of d × d-matrices,
O(Matd(k)) = k[x
j | 1 ≤ i, j ≤ d].
• As was explained in [28], taking “quantum affine space” Oq(k
d) as the point of de-
parture one arrives at the quantum MT of Garoufalidis, Lê and Zeilberger [20] (and
Konvalinka and Pak [33] in the multi-parameter case). The endomorphism bialgebra
of Oq(k
d) is exactly the algebra of right-quantum matrices as defined in [20].
• Berger’s N -symmetric algebra [5] leads to the N -generalization of the MT proved by
Etingof and Pak [16] using the above approach, again after restricting to O(Matd(k)).
0.4. The present article aims to set forth an extension of the existing theory of N -homogeneous
algebras to the category Vects
of vector superspaces over some base field k. While this does
not give rise to principal obstacles given that [37] and [7] are at hand as guiding references, the
setting of superalgebra requires careful consideration of the order of terms and the so-called
“rule of signs” will be ubiquitous in our formulæ. In view of the potential interdisciplinary
interest of this material, we have opted to keep our presentation reasonably self-contained and
complete.
Therefore, in Sections 1 and 2, we deploy the requisite background material from superal-
gebra in some detail before turning to N -homogeneous superalgebras in Section 3. The latter
section, while following the general outline of [37] and [7] rather closely, also offers explicit
discussions of a number of important examples. We interpolate the pure even and pure odd
Yang-Mills algebras defined by Connes and Dubois-Violette [10], [11] by a family of super-
algebras YMp|q and give a unified treatment of these algebras. (It turns out, however, that the
mixed algebras YMp|q, with p and q both nonzero, are less well-behaved than the pure cases.)
Moreover, we discuss a superized version of the N -symmetric algebras of Berger [5]. Finally,
in Example 3.4, we introduce new N -homogeneous superversions of the symmetric algebra
and the Grassmann algebra of a vector superspace V ; these are associated with any Hecke
operator R : V ⊗2 → V ⊗2 and will be denoted by SR,N and ΛR,N , respectively.
Sections 4 and 5 contain our main results: Theorem 4.5 shows that the superalgebras SR,N
and ΛR,N are in fact N -Koszul, and Theorem 5.4 is superized version of the aforementioned
N -generalized MT of Etingof and Pak [16, Theorem 2]. The special case N = 2 of The-
orem 5.4 is a superization of the original master theorem of MacMahon [35]. The present
article was motivated in part by a comment in Konvalinka and Pak [33, 13.4] asking for a
“real” super-analog of the classical MT.
0.5. A considerable amount of research has been done by mathematical physicists on various
quantum matrix identities. Some of these investigations have been carried out in a super set-
ting; see, e.g., Gurevich, Pyatov and Saponov [23], [24] and the references therein. However,
the techniques employed in these articles appear to be quite different from ours.
After submitting this article, we also learned of recent work of Konvalinka [31], [32] which
not only concerns MacMahon’s MT but also other matrix identities such as the determinantal
identity of Sylvester. These identities are proved in [31], [32] by combinatorial means in
various noncommutative settings including the right-quantum matrix algebra endOq(k
0.6. We conclude this Introduction by reviewing the precise definitions of N -homogeneous
and N -Koszul algebras. Our basic reference is Berger [5]; see also [2], [7], [21].
4 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Let A be a connected Z≥0-graded algebra over a field k; so A =
n≥0An for k-subspaces
An with A0 = k and AnAm ⊆ An+m. Choose a minimal generating set for the algebra A
consisting of homogeneous elements of positive degree; this amounts to choosing a graded
basis for a graded subspace V ⊆ A+ =
n>0An such that A+ = A
+ ⊕ V . The grading
of V imparts a grading to the tensor algebra T(V ) of the space V , and we have a graded
presentation
T(V )/I
for some graded ideal I of T(V ), the ideal of relations of A.
Recall that a graded vector space M =
n∈ZMn is said to live in degrees ≥ n0 if Mn = 0
for all n < n0. Note that the relation ideal I lives in degrees ≥ 2, because T(V )0 ⊕T(V )1 ⊆
k⊕ V and k⊕ V injects into A. Fix an integer N ≥ 2 and define the jump function
νN (i) =
N if i is even
N + 1 if i is odd
(0.1)
The following proposition is identical with [8, Proposition 2.1] except for the fact that we do
not a priori assume A to be generated in degree 1. A proof is given in the Appendix.
Proposition 0.1. The ideal I of relations of A lives in degrees ≥ N if and only if TorAi (k, k)
lives in degrees ≥ νN (i) for all i ≥ 0.
Following Berger [5], the graded algebra A is said to be N -Koszul if TorAi (k, k) is con-
centrated in degree νN (i) for all i ≥ 0. This implies that the space of algebra generators V
is concentrated in degree νN (1) = 1; so the algebra A is 1-generated. Moreover, choosing a
minimal set of homogeneous ideal generators for the relation ideal I amounts to choosing a
graded basis for a graded subspace R ⊆ I such that
I = R⊕ (V ⊗ I + I ⊗ V ) (0.2)
Then TorA2 (k, k)
∼= R and so R must be concentrated in degree νN (2) = N when A is
N -Koszul. To summarize, all N -Koszul algebras are necessarily 1-generated and they have
defining relations in degree N ; so there is a graded isomorphism
A ∼= T(V )/(R) with R ⊆ V ⊗N
Such algebras are called N -homogeneous.
We remark that Green et al. [21] have studied N -Koszul algebras in the more general con-
text where the grading A =
n≥0An is not necessarily connected (A0 = k). In [21, The-
orem 4.1], it is shown that an N -homogeneous algebra A with A0 split semisimple over k is
N -Koszul if and only if the Yoneda Ext-algebra E(A) =
n≥0 Ext
A(A0,A0) is generated
in degrees ≤ 2.
Any N -homogeneous algebra A whose generating space V carries a Z2-grading and whose
defining relations R are Z2-graded is naturally a k-superalgebra, that is, A has a Z2-grading
(“parity”) besides the basic Z≥0-grading (“degree”). As will be reviewed below, this extra
structure provides us with additional functions on Grothendieck rings, namely superdimension
and supercharacters, which lead to natural formulations of the MT in a superized context.
Note, however, that the defining property of N -Koszul algebras makes no reference to the
Z2-grading of A. Thus, an N -homogeneous superalgebra is Koszul precisely if it is Koszul
as an ordinary N -homogeneous algebra (forgetting the Z2-grading).
N -HOMOGENEOUS SUPERALGEBRAS 5
0.7. Throughout k is a commutative field and ⊗ stands for ⊗
. Scalar multiplication in
k-vector spaces will often, but not always, be written on the right while linear maps will
act from the left. We tacitly assume throughout that char k 6= 2; further restrictions on the
characteristic of k will be stated when required.
1. REVIEW OF LINEAR SUPERALGEBRA
1.1. Vector superspaces. A vector superspace over k is a k-vector space V equipped with a
grading by the group Z2 = Z/2Z = {0̄, 1̄}. Thus, we have a decomposition V = V0̄ ⊕ V1̄
with k-subspaces V0̄ and V1̄ whose elements are called even and odd, respectively. In general,
the Z2-degree of a homogeneous element a ∈ V is also called its parity; it will be denoted by
â ∈ Z2. Vector superspaces over k form a category Vect
whose morphisms are given by the
linear maps preserving the Z2-grading; such maps are also called even linear maps.
The dimension of an object V of Vects
is the usual k-linear dimension. We shall use the
notation
d = dim
V , p = dim
V0̄ and q = dimk V1̄
So d = p+ q. The superdimension of a vector superspace V with d < ∞ is defined by
sdimV = p− q ∈ Z
When working with a fixed basis {xi} of a given V in Vect
we shall assume that each xi is
homogeneous; the parity of xi will be denoted by î. The basis x1, x2 . . . is called standard if
î = 0̄ (i ≤ p) and î = 1̄ (i > p).
1.2. Tensors. The tensor product U ⊗V of vector superspaces U and V in Vects
is the usual
tensor product over k of the underlying vector spaces equipped with the natural Z2-grading:
if a, b are homogeneous elements then the parity of a ⊗ b is â + b̂ ∈ Z2. Instead of the usual
symmetry isomorphism U ⊗ V
−→ V ⊗ U for interchanging terms in a tensor product we
shall use the rule of signs, that is, the following functorial supersymmetry isomorphism in
Vects
cU,V : U ⊗ V
−→ V ⊗ U , u⊗ v 7→ (−1)bubvv ⊗ u (1.1)
for u, v homogeneous. (All formulas stated for homogeneous elements only are to be ex-
tended to arbitrary elements by linearity.) The supersymmetry isomorphisms cU,V satisfy
cV,U ◦ cU,V = IdU⊗V , and they are compatible with the usual associativity isomorphims
aU,V,W : (U ⊗V )⊗W ∼= U ⊗ (V ⊗W ) in Vect
, that is, they satisfy the “Hexagon Axiom”;
see [29, Def. XIII.1.1]. Therefore, Vects
is a symmetric tensor category; the unit object is the
field k, with parity 0̄. See [29, Chap. XIII] or [12] for background on tensor categories.
1.3. Homomorphisms. The space Hom
(V,U) of all k-linear maps between vector super-
spaces V and U is again an object of Vects
, with grading Hom
(V,U)0̄ = Homk(V0̄, U0̄) ⊕
(V1̄, U1̄) and Homk(V,U)1̄ = Homk(V0̄, U1̄)⊕Homk(V1̄, U0̄); so
(V,U)0̄ = HomVects
(V,U)
In particular, the linear dual V ∗ = Hom
(V, k) belongs to Vects
. Given homogeneous bases
{xj} of V and {yi} of U we can describe any f ∈ Hom
(V,U) by its matrix F = (F ij ):
f(xj) =
j (1.2)
6 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
When f is an even map then F ij = 0 unless î+ ĵ = 0̄.
For finite-dimensional vector superspaces, we have the following functorial isomorphisms
in Vects
(see, e.g., [43, I.8]):
U ⊗ V ∗ ∼= Hom
(V,U) (1.3)
via (u⊗ f)(v) = u〈f, v〉, and
V ∗1 ⊗ . . . ⊗ V
∼= (Vm ⊗ . . . ⊗ V1)
∗ (1.4)
via 〈f1 ⊗ . . .⊗ fm, vm ⊗ . . .⊗ v1〉 =
i〈fi, vi〉. Here, we use the notation 〈f, v〉 = f(v) for
the evaluation pairing
evV = 〈 . , . 〉 : V
∗ ⊗ V → k
in Vects
. Similarly, we have a pairing
V ⊗ V ∗
cV,V ∗
−→ V ∗ ⊗ V
which yields an isomorphism
−→ V ∗∗ (1.5)
in Vects
The isomorphism (1.3) (which is valid as long as one of U or V is finite-dimensional) has
the following explicit description. Fix homogeneous bases {xj} of V and {yi} of U and let
F = (F ij ) be the matrix of a given f ∈ Homk(V,U) with respect to these bases, as in (1.2).
Let {xj} be the dual basis of V ∗, defined by 〈xj , xℓ〉 = δ
(Kronecker delta). Then the image
of f in U ⊗ V ∗ is given by
i,j yi ⊗ x
jF ij . Note also that xi and x
i have the same parity.
Finally, if U , V and W are vector superspaces, with U finite-dimensional, then the isomor-
phism Id⊗cW,U∗ : V ⊗W ⊗U
∗ ∼−→V ⊗U∗ ⊗W together with (1.3) yields an isomorphism
(U, V ⊗W )
−→ Hom
(U, V )⊗W (1.6)
in Vects
which is explicitly given by (f ⊗ w)(u) = (−1)bwbuf(u) ⊗ w. Similarly, for vector
superspaces U , U ′, V , V ′ with U , U ′ finite-dimensional, there is an isomorphism
(U ⊗ U ′, V ⊗ V ′)
−→ Hom
(U, V )⊗Hom
(U ′, V ′) (1.7)
in Vects
given by (f ⊗ g)(u ⊗ v) = (−1)bgbuf(u)⊗ g(v).
1.4. Supertrace. Let V be a finite-dimensional object of Vects
. The supertrace is the map
str : End
(1.3)
V ⊗ V ∗ −→
(1.3)
k (1.8)
in Vects
. In order to describe the supertrace in terms of matrices, fix a basis {xi} of V
consisting of homogeneous elements and let F = (F ij ) be the matrix of f ∈ Endk(V ) as in
(1.2). Then
str(f) =
biF ii
where î is the parity of xi (and of the dual basis vector x
i ∈ V ∗) as in §1.1. Thus,
str(IdV ) = sdimV.1
N -HOMOGENEOUS SUPERALGEBRAS 7
1.5. Action of the symmetric group. Given vector superspaces V1, . . . , Vn, we can consider
the morphism
ci : V1 ⊗ · · · ⊗ Vi ⊗ Vi+1 ⊗ · · · ⊗ Vn −→ V1 ⊗ · · · ⊗ Vi+1 ⊗ Vi ⊗ · · · ⊗ Vn
in Vects
which interchanges the factors Vi and Vi+1 via cVi,Vi+1 and is the identity on all other
factors. More generally, for any σ ∈ Sn, the symmetric group consisting of all permutations
of {1, 2, . . . , n}, one can define a morphism
cσ : V1 ⊗ · · · ⊗ Vn −→ Vσ−1(1) ⊗ · · · ⊗ Vσ−1(n)
in Vects
as follows. Recall that Sn is generated by the transpositions σ1, . . . , σn−1 where σi
interchanges i and i + 1 and leaves all other elements of {1, 2, . . . , n} fixed. The minimal
length of a product in the σi’s which expresses a given element σ ∈ Sn is called the length of
σ and denoted ℓ(σ); it is given by
ℓ(σ) = #inv(σ) with inv(σ) = {(i, j) | i < j but σ(i) > σ(j)}
Writing σ ∈ Sn as a product of certain σi, the analogous product of the maps ci yields a
morphism cσ as above. This morphism is independent of the way σ is expressed in terms of the
transpositions σi; see [43, I.4.13] or [29, Theorem XIII.1.3]. If all vi ∈ Vi are homogeneous
cσ(v1 ⊗ · · · ⊗ vn) = (−1)
(i,j)∈inv(σ) bvi bvjvσ−1(1) ⊗ · · · ⊗ vσ−1(n) (1.9)
For example, if all vi are even then the ±-sign on the right is +, and if all vi are odd then it is
sgn(σ), the signature of σ.
Taking all Vi = V we obtain a representation c : Sn −→ AutVects
(V ⊗n) where V ⊗n =
V ⊗ · · · ⊗ V (n factors). Letting k[Sn] denote the group algebra of the symmetric group, this
extends uniquely to an algebra map
c : k[Sn] −→ EndVects
(V ⊗n) (1.10)
We will write ca := c(a) for a ∈ k[SN ].
For the dual superspace V ∗, besides the above representation c : k[Sn] −→ EndVects
(V ∗⊗n),
we also have the contragredient representation
c∗ : k[Sn] −→ EndVects
(V ∗⊗n)
for the pairing 〈 . , . 〉 : V ∗⊗n ⊗ V ⊗n → k in (1.4). Explicitly,
〈c∗a(x), y〉 = 〈x, ca∗(y)〉
for all a ∈ k[Sn], x ∈ V
∗⊗n and y ∈ V ⊗n. Here, . ∗ : k[Sn] → k[Sn] is the involution
sending σ ∈ Sn to σ
−1. These two representations are related by
c∗a = cτaτ (1.11)
where τ = (1, n)(2, n−1) · · · ∈ Sn is the order reversal involution. One only needs to check
(1.11) for the transpositions a = σi, which is straightforward.
8 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
1.6. Hecke algebras. We recall some standard facts concerning Hecke algebras; these are
suitable deformations of the group algebra k[Sn] considered above. For details, see [13],
[14].
Fix 0 6= q ∈ k. The Hecke algebra Hn,q is generated as k-algebra by elements T1, . . . , Tn−1
subject to the relations
(Ti + 1)(Ti − q) = 0
TiTi+1Ti = Ti+1TiTi+1
TiTj = TjTi if |i− j| ≥ 2
(1.12)
When q = 1, one has an isomorphism Hn,1
−→ k[Sn], Ti 7→ σi where σi is the transposition
(i, i + 1) as in §1.5. The algebra Hn,q has a k-basis {Tσ | σ ∈ Sn} so that
(i) TId = 1 and Tσi = Ti;
(ii) TσTσi =
Tσσi if ℓ(σσi) = ℓ(σ) + 1;
qTσσi + (q − 1)Tσ otherwise
By k-linear extension of the rule
T ∗σ := Tσ−1 (σ ∈ Sn)
one obtains an involution . ∗ : Hn,q → Hn,q. Moreover, the elements T
i := −qT
q − 1− Ti also satisfy relations (1.12). Therefore,
α(Ti) := −qT
i (1.13)
defines an algebra automorphism α : Hn,q → Hn,q of order 2.
The Hecke algebra Hn,q is always a symmetric algebra, and Hn,q is a split semisimple
k-algebra iff the following condition is satisfied:
[n]q! :=
[i]q 6= 0 where [i]q := 1 + q + · · ·+ q
i−1 (1.14)
More precisely, if (1.14) holds then
Matdλ×dλ(k) (1.15)
where λ runs over all partitions of n and dλ denotes the number of standard λ-tableaux. The
only partitions λ with dλ = 1 are λ = (n) and λ = (1
n). The central primitive idempotents
of Hn,q for these partitions are given by
Xn :=
[n]q!
Tσ (1.16)
Yn :=
[n]q−1 !
(−q)−ℓ(σ)Tσ (1.17)
These idempotents are usually called the q-symmetrizer and the q-antisymmetrizer, respec-
tively. One has
XnTσ = TσXn = q
ℓ(σ)Xn and YnTσ = TσYn = (−1)
ℓ(σ)Yn (1.18)
for σ ∈ Sn. Furthermore, α(Xn) = Yn.
N -HOMOGENEOUS SUPERALGEBRAS 9
For later use, we note the following well-known consequence of (1.18). If M is any Hn,q-
module, with corresponding representation µ : Hn,q → End
(M), then
Im(µ(Xn)) =
Im(µ(Ti) + 1) (1.19)
Indeed, (1.18) implies that Xn = [2]
q (Ti + 1)Xn, which yields the inclusion ⊆. On the
other hand, any m ∈
i=1 Im(µ(Ti) + 1) satisfies (µ(Ti) − q)(m) = 0 for all i, by
(1.12). Therefore, µ(Tσ)(m) = q
ℓ(σ)m holds for all σ ∈ Sn, and hence µ(Xn)(m) =
[n]q!
qℓ(σ)m = m. This proves ⊇.
1.7. Hecke operators. Again, let 0 6= q ∈ k. A Hecke operator (associated to q) on a vector
superspace V is a morphism R : V ⊗2 → V ⊗2 in Vects
satisfying the Hecke equation
(R + 1)(R − q) = 0
and the Yang-Baxter equation
R1R2R1 = R2R1R2
where R1 := R ⊗ IdV : V
⊗3 → V ⊗3 and similarly R2 := IdV ⊗R.
The Hecke equation implies that R is invertible. Moreover, if R is a Hecke operator
associated to q then so is −qR−1.
Defining ρ(Ti) := Id
V ⊗R ⊗ Id
⊗n−i−1
V , one obtains a representation
ρ = ρn,R : Hn,q −→ EndVects
(V ⊗n) (1.20)
The representations ρn,R and ρn,−qR−1 are related by ρn,−qR−1 = ρn,R ◦ α, where α is the
automorphism of Hn,q defined in (1.13).
Example 1.1. The supersymmetry operator cV,V : V ⊗2 → V ⊗2 in (1.1) is a Hecke operator
associated to q = 1, as is its negative, −cV,V . The representation ρcV,V of Hn,1 = k[Sn] in
(1.20) is identical with (1.10).
Example 1.2 (superized Drinfel’d-Jimbo [38], [27]). Let x1, . . . , xd be a standard basis of V
as in §1.1. The super analog R = RDJ of the standard Drinfel’d-Jimbo Hecke operator is
defined as follows. Writing
R(xi ⊗ xj) =
xk ⊗ xlR
the matrix components Rk,li,j ∈ k are given by
i,j =
q2 − q2εi,j
1 + q2εi,j
i,j + (−1)
bibj q
εi,j(q2 + 1)
1 + q2εi,j
Here, εi,j = sgn(i− j). Thus,
ii = q
2 if î = 0̄
ii = −1 if î = 1̄
ij = q
2 − 1 if i < j
ij = (−1)
bibjq if i 6= j
(1.21)
10 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
and Rk,li,j = 0 in all other cases. One checks that R is a Hecke operator that is associated to
2. THE SUPERCHARACTER
2.1. Superalgebras, supercoalgebras etc. An algebra A in Vects
is called a superalgebra
over k; this is just an ordinary k-algebra such that the unit map k → A and the multiplication
µ : A⊗A → A
are morphisms in Vects
. In other words, A is a Z2-graded k-algebra in the usual sense:
A = A0̄ ⊕ A1̄ with k-subspaces A0̄ and A1̄ such that Ar̄As̄ ⊆ Ar+s. Homomorphisms of
superalgebras, by definition, are algebra maps in Vects
, that is, they preserve the Z2-grading.
If V is a vector superspace in Vects
then the tensor algebra T(V ) =
n≥0 V
⊗n is a
superalgebra via the Z2-grading of each V
⊗n as in §1.2. In general, if A is any superalgebra,
then by selecting a Z2-graded subspace V ⊆ A which generates the algebra A, we obtain a
canonical isomorphism of superalgebras
T(V )/(R)
−→ A (2.1)
where (R) is the two-sided ideal of T(V ) that is generated by a Z2-graded linear subspace
R ⊆ T(V ).
Given superalgebras A and B, the tensor product A⊗ B is the superalgebra with the usual
additive structure and grading and with multiplication µA⊗B defined by using the supersym-
metry map (1.1): µA⊗B = (µA ⊗ µB) ◦ (IdA⊗cB,A ⊗ IdB) or, explicitly,
(a⊗ b)(a′ ⊗ b′) = (−1)
ba′bbaa′ ⊗ bb′
for homogeneous a′ ∈ A and b ∈ B. In other words, the canonical images of A and B in
A⊗ B supercommute, in the sense that the supercommutator
[a, b] = ab− (−1)ba
bbba (2.2)
vanishes for any pair of homogeneous elements a ∈ A and b ∈ B.
Supercoalgebras, superbialgebras etc. are defined similarly as suitable objects of Vects
such that all structure maps are maps in Vects
. The compatibility between the comultiplica-
tion ∆ and the multiplication of a superbialgebra B amounts to the following rule:
∆(ab) =
(a),(b)
(−1)ba(2)
bb(1)a(1)b(1) ⊗ a(2)b(2)
for homogeneous elements a, b ∈ B. Here we use the Sweedler notation ∆(a) =
(a) a(1) ⊗
a(2) and a(1), a(2) are chosen homogeneous with â(1) + â(2) = â.
Example 2.1 (Symmetric superalgebra [40, 3.2.5]). The symmetric superalgebra of a given
V in Vects
is defined by
S(V ) = T(V )/ ([v,w]⊗ | v,w ∈ V )
where [v,w]⊗ is the supercommutator (2.2) in T(V ). Ignoring parity, S(V ) is isomorphic to
S(V0̄)⊗Λ(V1̄), where S( . ) and Λ( . ) denote the ordinary symmetric and exterior (Grassmann)
algebras, respectively. The symmetric superalgebra is a Hopf superalgebra: comultiplication
∆: S(V ) → S(V )⊗ S(V ) is given by ∆(v) = v ⊗ 1 + 1⊗ v for v ∈ V and extension to all
N -HOMOGENEOUS SUPERALGEBRAS 11
of S(V ) by multiplicativity. Similarly, the counit ε : S(V ) → k is given by ε(v) = 0 and the
antipode S : S(V ) → S(V ) by S(v) = −v for v ∈ V .
2.2. Comodules. We refer to [29, Chap. III] for background on comodules, comodule alge-
bras etc.
Given a superbialgebra B, we let ComodsB denote the category of all right B-comodules
and B-comodule maps in Vects
. Thus, for any object V in ComodsB, we have a “coaction”
morphism
δV : V → V ⊗B
in Vects
. If x1, . . . , xd is a fixed basis of V consisting of homogeneous elements, with î
denoting the parity of xi as before, then we will write
δV (xj) =
xi ⊗ b
j with b
j ∈ Bbi+bj (2.3)
The tensor product of vector superspaces makes ComodsB into a tensor category: if U and
V are in ComodsB then B coacts on U ⊗ V by
δU⊗V : U ⊗ V
δU⊗δV
−→ U ⊗ B ⊗ V ⊗ B
−→ U ⊗ V ⊗ B ⊗ B
Id⊗µB
−→ U ⊗ V ⊗ B (2.4)
If B is supercommutative as a superalgebra then the supersymmetry cU,V is a B-comodule
morphism, i.e., δV⊗U ◦ cU,V = (cU,V ⊗ IdB) ◦ δU⊗V . Therefore Comod
B is a symmetric
tensor category in this case .
2.3. The supercharacter map. Let B denote a superbialgebra and let V be a finite dimen-
sional object in ComodsB. The coaction δV is an even map in Homk(V, V ⊗B). Consider the
following morphism in Vects
χs : End
δV ◦( . )
−→ Hom
(V, V ⊗ B)
(1.6)
(V )⊗ B
str⊗Id
−→ k⊗ B = B (2.5)
where str is the supertrace as in (1.8). This map will be called the supercharacter map of
V . Forgetting parity and viewing all elements as even, the supertrace becomes the ordinary
trace and the supercharacter becomes the usual character. These will be denoted by tr and χ,
respectively.
In particular, we have the element
χsV := χ
s(IdV ) ∈ B0̄
To obtain explicit formulas, fix a basis x1, . . . , xd of V consisting of homogeneous elements
and let (F ij ) and (b
j) be the matrices of f ∈ Endk(V ) and of δV with respect to this basis as
in (1.2), (2.3). Then
χs(f) =
bibjbijF
i (2.6)
Let ε : B → k denote the counit of B. Then xj =
i xiε(b
j) holds in (2.3). Hence
ε(bij) = δ
j .1k and (2.6) gives
ε(χs(f)) = str(f) (2.7)
12 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
When f is even formula (2.6) becomes χs(f) =
i,j(−1)
bibijF
i , because F
i = 0 unless
î+ ĵ = 0̄. In particular,
χsV =
bibii (2.8)
In the following, we let comodsB denote the full subcategory of Comod
B consisting of all
objects that are finite-dimensional over k. The supercharacter has the following properties
analogous to standard properties of the ordinary character.
Lemma 2.2. Let B denote a superbialgebra and let U , V and W be objects of comodsB .
(a) If f : V → U and g : U → V are B-comodule maps (not necessarily even) then
χs(f ◦ g) = (−1)
bfbgχs(g ◦ f)
(b) For f ∈ End
(V ), g ∈ End
(U) view f ⊗ g ∈ End
(V ⊗ U) as in (1.7). Then
χs(f ⊗ g) = χs(f)χs(g)
(c) Given an exact sequence 0 → U
−→ W → 0 in comodsB, let f ∈ Endk(V )
be such that f(µ(U)) ⊆ µ(U), and let g ∈ End
(U), h ∈ End
(W ) be the maps
induced by f . Then
χs(f) = χs(g) + χs(h)
In particular, χsV = χ
U + χ
W . Moreover, if f ∈ EndcomodsB(V ) is a projection (i.e.,
f2 = f ) then χs(f) = χsIm f .
Proof. (a) Let TV denote the map Hom
(V, V ⊗ B) −→ B in (2.5); so χs(f) = TV (δV ◦ f).
Since f and g are comodule maps, we have δU ◦ f = (f ⊗ IdB) ◦ δV and similarly for g.
Putting h = δU ◦ f ∈ Hom
(V,U ⊗ B) we obtain χs(f ◦ g) = TU (δU ◦ f ◦ g) = TU (h ◦ g)
and χs(g ◦ f) = TV (δV ◦ g ◦ f) = TV ((g ⊗ IdB) ◦ h). Therefore, we must show that
TU (h ◦ g) = (−1)
bfbgTV ((g ⊗ IdB) ◦ h)
Using the identification Hom
(V,U⊗B) ∼= Hom
(V,U)⊗B as in (1.6), write h =
i fi⊗bi
with fi ∈ Hom
(V,U), bi ∈ B, and f̂i + b̂i = ĥ = f̂ . Then h ◦ g ∈ Hom
(U,U ⊗ B)
becomes the element (
i fi ⊗ bi) ◦ g =
i(−1)
bbibg(fi ◦ g) ⊗ bi ∈ End
(U) ⊗ B, and
(g ⊗ IdB) ◦ hHom
(V, V ⊗B) becomes
i(g ◦ fi)⊗ bi. The standard identity str(fi ◦ g) =
bfibgstr(g ◦ fi) (cf., e.g., [40, p. 165 §3(b)]) now yields
TU (h ◦ g) =
bbibgstr(fi ◦ g)⊗ bi
bbibg+bfibgstr(g ◦ fi)⊗ bi
= (−1)
bfbgTV ((g ⊗ IdB) ◦ h)
as desired.
(b) Fix homogeneous k-bases {xi} and {yℓ} of V and U , respectively, and write x̂i = î,
ŷℓ = ℓ̂ as usual. Moreover, let (F
j ) and (G
m) be the matrices of f and g for these bases, as
N -HOMOGENEOUS SUPERALGEBRAS 13
in (1.2). Then {xi ⊗ yℓ} is a basis of V ⊗ U , with xi ⊗ yℓ having parity î+ ℓ̂. Moreover,
(f ⊗ g)(xj ⊗ ym) = (−1)
bgbjf(xj)⊗ g(ym)
= (−1)bg
xi ⊗ yℓΦ
j,m with Φ
j,m = (−1)
(bℓ+ bm)bjF ijG
because Gmℓ = 0 unless ℓ̂+ m̂ = ĝ. Similarly, writing δV (xj) =
i xi ⊗ b
j with b
j ∈ Bbi+bj
and δU (ym) =
ℓ yℓ ⊗ c
m with c
m ∈ Bbℓ+bm, one obtains using (2.4)
δV⊗U (xj ⊗ ym) =
xi ⊗ yℓ ⊗Ψ
j,m with Ψ
j,m = (−1)
(bi+bj)bℓbijc
Therefore, formula (2.6) becomes
χs(f ⊗ g) =
i,ℓ,j,m
(−1)(
bi+bℓ)(bj+ bm)Ψ
i,ℓ,j,m
bibj+bℓbmbijF
= χs(f)χs(g)
(c) Choose a basis {xi} of V consisting of homogeneous elements so that xi = µ(yi) for
i ≤ dimU and let (F ij ) be the matrix of f for this basis. Then F
j = 0 for i > dimU ,
j ≤ dimU . Moreover, the yi form a basis of U and the zi = π(xi) form a basis of W ,
and the matrices of g and h for these bases are (F ij )i,j≤dimU and (F
j )i,j>dimU , respectively.
Similarly, if (bij) is the matrix of δV with respect to the basis basis {xi} as in (2.3) then b
j = 0
for i > dimU , j ≤ dimU , and the matrices of δU and δW for the given bases are (b
j)i,j≤dimU
and (bij)i,j>dimU , respectively. Therefore,
χs(f) =
bibjbijF
i,j≤dimU
bibjbijF
i,j>dimU
bibjbijF
= χs(g) + χs(h)
The remaining assertions are clear. �
2.4. The Grothendieck ring. Let B be a superbialgebra and let
RB = K0(comod
denote the Grothendieck group of the category comodsB. Thus, for each V in comod
B, there
is an element [V ] ∈ RB and each short exact sequence 0 → U → V → W → 0 in comod
gives rise to an equation [V ] = [U ] + [W ] in RB. The group RB is in fact a ring with
multiplication given by the tensor product of B-comodules. If B is supercommutative as a
superalgebra then the ring RB is commutative; see §2.2.
14 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Both the ordinary dimension and the superdimension are additive on short exact sequences
and multiplicative on tensor products. Hence they yield ring homomorphisms
dim, sdim: RB → Z
Parts (b) and (c) of Lemma 2.2 and formula (2.7) have the following immediate consequence:
Corollary 2.3. The map [V ] 7→ χsV yields a well-defined ring homomorphism χ
s : RB → B0̄.
Furthermore, the following diagram commutes
Z can.
Forgetting the Z2-grading, the corollary also gives the more familiar version with χ and
dim in place of χs and sdim, respectively.
2.5. General linear supergroup and Berezinian. Let V in Vects
be finite-dimensional and
fix a standard basis x1, . . . , xd with î = 0̄ (i ≤ p) and î = 1̄ (i > p).
2.5.1. For each supercommutative k-superalgebra R we denote by E(V )(R) the set of all
R-linear maps V ⊗ R → V ⊗ R in Vects
. Using the identification EndR(V ⊗ R) ∼=
(V, V ⊗R) ∼= End
(V )⊗R (see (1.6)), we may view E(V )(R) as the even subspace
of End
(V )⊗R:
E(V )(R) = (End
(V )⊗R)0̄
This defines a functor E(V ) from the category of supercommutative k-superalgebras to the
category of semigroups.
2.5.2. Tensoring the supertrace str : End
(V ) → k of (1.8) with IdR, we obtain an R-linear
supertrace map str : End
(V )⊗R → R in Vects
which restricts to a map E(V )(R) → R0̄.
The given standard basis x1, . . . , xd of V is an R-basis of V ⊗ R. In terms of this basis, an
element φ ∈ E(V )(R) is given by
φ(xj) =
j with Φ
j ∈ Rbi+bj (2.9)
Thus φ is described by a supermatrix Φ =
in standard form over R:
(2.10)
where A =
i,j≤p
and D =
i,j>p
are square matrices with entries in R0̄ while C,D
are matrices over R1̄. The supertrace of φ is given by
str(φ) =
biΦii = tr(A)− tr(D) =: str(Φ)
N -HOMOGENEOUS SUPERALGEBRAS 15
2.5.3. The functor E(V ) is represented by a supercommutative k-superbialgebra which coacts
on V ; this algebra will be denoted by
B = O(E(V ))
Thus, there is a natural isomorphism of E(V ) with the functor Hom(B, ?) of parity preserving
algebra homorphisms. In particular, the identity map on B corresponds to an element ξ ∈
E(V )(B). Let X = (xij)d×d be the matrix of ξ, as in (2.9). The elements x
j have parity
î + ĵ and they form a set of supercommuting algebraically independent generators of B. In
fact, B is isomorphic to the symmetric superalgebra S(V ∗ ⊗ V ), with xij 7→ x
i ⊗ xj , where
{xi} ⊆ V ∗ is the dual basis for the given basis of V .
We can think of X as the generic supermatrix with respect to the given basis of V : any
supermatrix Φ =
as in (2.9) comes from an algebra map B → R via xij 7→ Φ
j . The
canonical coaction δ : V → V ⊗B, the comultiplication ∆ and the counit ε of B are given by
δ(xj) =
xi ⊗ x
∆(xij) =
xik ⊗ x
ε(xij) = δ
(2.11)
These formulas can also be written as δ(x1, . . . , xd) = (x1, . . . , xd) ⊗X, ∆(X) = X ⊗X
and ε(X) = 1.
2.5.4. Similarly, GL(V )(R) is defined, for any supercommutative k-superalgebra R, as the
set of all invertible R-linear endomorphism of V ⊗R in Vects
. The condition for a superma-
trix Φ in standard form (as in (2.10)) to be invertible is that A and D are invertible as ordinary
matrices over R0̄. In this case, the inverse of Φ is given by
Φ−1 =
(A−BD−1C)−1 −A−1B(D − CA−1B)−1
−D−1C(A−BD−1C)−1 (D − CA−1B)−1
See Berezin [3, Theorem 3.1 and Lemma 3.2]. The element
ber(Φ) := det(A) det(D − CA−1B)−1 = det(D)−1 det(A−BD−1C) (2.12)
is called the superdeterminant or Berezinian of Φ; it is an invertible element of R0̄.
The functor GL(V ) is represented by a supercommutative Hopf superalgebra O(GL(V ))
which is generated over B = O(E(V )) by det(X11)−1 and det(X22)−1, where X11 =(
i,j≤p
and X22 =
i,j>p
are the even blocks of the generic supermatrix X. By [3,
Theorem 3.3], the Berezinian ber(X) is a group-like element in O(GL(V )).
2.6. Supersymmetric functions and exterior powers. Throughout this section, V will de-
note a finite-dimensional vector superspace over k. We assume that the characteristic of k is
zero.
16 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
2.6.1. Let
sgn(σ)σ ∈ k[Sn]
be the antisymmetrizer idempotent of the group algebra k[Sn] and define
nV := Im cYn ⊆ V
⊗n (2.13)
where c : k[Sn] → EndVects
(V ⊗n) is as in (1.10). Thus, ΛnV is the space of antisymmetric
n-tensors,
nV = {y ∈ V ⊗n | cσ(y) = sgn(σ)y for all σ ∈ Sn}
For later use, we describe an explicit basis of ΛnV . To this end, fix a standard basis
x1, . . . , xd of V , with î = 0̄ for i ≤ p and î = 1̄ for i > p. Then the products xi =
xi1 ⊗ xi2 ⊗ · · · ⊗ xin for sequences i = (i1, i2, . . . , in) ∈ {1, 2, . . . , d}
n form a graded basis
of V ⊗n that is permuted up to a ±-sign by the action of Sn on V
⊗n; see formula (1.9):
cσ(xi) = sgni(σ)xσ(i) (2.14)
sgni(σ) = (−1)
(p,q)∈inv(σ)
bip biq and σ(i) = (iσ−1(1), iσ−1(2), . . . , iσ−1(n))
Therefore, by elementary properties of monomial group representations, a k-basis of ΛnV
is given by the nonzero elements cYn(xi) where i ranges over a transversal for the Sn-
action on {1, 2, . . . , d}n. Such a transversal is provided by the weakly increasing sequences
i ∈ {1, 2, . . . , d}n. Moreover, for a weakly increasing i, it is easily seen from (2.14) that
cYn(xi) = 0 holds precisely if iℓ = iℓ+1 ≤ p for some ℓ. Therefore, a basis of Λ
nV is given
by the elements cYn(xi) with i = (i1 < i2 < · · · < im < im+1 ≤ · · · ≤ in) ∈ {1, 2, . . . , d}
and im ≤ p < im+1.
In particular,
m+m′=n
q +m′ − 1
(2.15)
where p = dim
V0̄ and q = dimk V1̄. Equivalently, the generating power series in ZJtK for
the sequence dim
nV is given by
nV tn =
(1 + t)p
(1− t)q
(2.16)
When q > 0 then all ΛnV are nonzero. For additional details on exterior powers, see, e.g.,
[43, Sections I.5 and I.7].
2.6.2. Consider the super bialgebra B = O(E(V )) as defined in §2.5.3 and recall that V is
in comodsB. The representation c : k[Sn] → EndVects
(V ⊗n) of (1.10) actually has image in
EndcomodsB(V
⊗n), since B is supercommutative. Therefore, ΛnV also belongs to comodsB
and we can define the nth elementary supersymmetric function by
en := χ
ΛnV = χ
s(cYn) ∈ B0̄
Here, the equality χs
ΛnV = χ
s(cYn) holds by Lemma 2.2(c).
Similarly, one defines the nth super power sum by
pn := χ
s(c(1,2,...,n)) ∈ B0̄
N -HOMOGENEOUS SUPERALGEBRAS 17
where (1, 2, . . . , n) ∈ Sn the cyclic permutation mapping 1 7→ 2 7→ 3 7→ . . . 7→ n 7→ 1. In
terms of the generic supermatrix X from §2.5.3, one has
pn = str(X
Modulo the space spanned by the Lie commutators fg− gf with f, g ∈ k[Sn], the follow-
ing relation is easily seen to hold in k[Sn]:
nYn ≡
(−1)i−1(1, 2, . . . , i)Yn−i
(with Y0 = 1). Applying the function χ
s ◦ c : k[Sn] → B0̄ to this relation and using
Lemma 2.2(a),(b), one obtains the Newton relations:
nen =
(−1)i−1pien−i
Let t be a formal parameter (of parity 0̄) and consider the generating functions P (t) =∑
n≥1 pnt
n−1 and E(t) =
n≥0 ent
n in B0̄JtK. The Newton relations can be written in
the form P (−t) = d
logE(t); see, e.g., [34, p. 23]. Combining this with the identity
ber(exp(tX)) = exp(str(tX))
due to Berezin ([3, Chapter 3] or [40, p. 167]), one obtains the following expansion for the
characteristic function ber(1 + tX) of generic supermatrix X:
Proposition 2.4. ber(1 + tX) =
n≥0 ent
This proposition is known; see, e.g., Khudaverdian and Voronov [30, Prop. 1].
3. HOMOGENEOUS SUPERALGEBRAS
3.1. N -homogeneous superalgebras. Let N be an integer with N ≥ 2. A homogeneous
superalgebra of degree N or N -homogeneous superalgebra is an algebra A of the form (2.1)
with V finite-dimensional and R ⊆ V ⊗N :
A = A(V,R) ∼= T(V )/(R)
The assumption R ⊆ V ⊗N implies that, besides the usual Z2-grading (“parity”), A also has a
connected Z+-grading (“degree”),
The algebra A is generated by A1 = V and all homogeneous components An are finite-
dimensional objects of Vects
. In fact,
An ∼= V
⊗n/Rn with Rn := (R) ∩ V
i+j+N=n
V ⊗i ⊗R⊗ V ⊗j (3.1)
Note that Rn = 0 for n < N ; so An ∼= V
⊗n if n < N .
Morphisms of N -homogeneous superalgebras f : A = A(V,R) → A′ = A(V ′, R′) are
morphism of superalgebras which also respect the Z+-grading. Equivalently, by restricting to
degree 1, we have a morphism f1 : A1 = V → A
1 = V
′ in Vects
whose N th tensor power sat-
isfies f⊗N1 (R) ⊆ R
′. Thus, one has a category HNAlg
of N -homogeneous k-superalgebras.
18 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Finally, N -homogeneous superalgebras with N = 2 are called quadratic superalgebras; for
N = 3, they are called cubic, etc..
3.2. Some examples. In order to explicitly describe a certain N -homogeneous superalgebra
A = A(V,R), we will usually fix a Z2-graded k-basis x1, . . . , xd of V = A1 and denote the
the parity of xi by î, as in §1.1. The xi form a set of algebra generators for A. Following
Manin [38],[39], the d-tuple f = (1̂, . . . , d̂) ∈ Zd2 is called the format of the basis {xi}.
Example 3.1 (Quantum superspace [39]). For a fixed family q of scalars 0 6= qij ∈ k (1 ≤
i < j ≤ d) and a given format f = (1̂, . . . , d̂) ∈ Zd2 of the basis x1, . . . , xd, the quadratic
superalgebra A = Sfq is defined as the factor of T(V ) modulo the ideal generated by the
elements
ri := xi ⊗ xi ∈ (V
⊗2)0̄ (̂i = 1̄) (3.2)
rij := xj ⊗ xi − qij(−1)
bibjxi ⊗ xj ∈ (V
bi+bj
(i < j) (3.3)
Thus, the algebra Sfq is generated by x1, . . . , xd subject to the defining relations
xixi = 0 (̂i = 1̄)
xjxi = qij(−1)
bibjxixj (i < j).
In the special case where all qij = 1, the algebra S
q is the symmetric superalgebra S(V ) of V
as in Example 2.1.
The ordered monomials of the form xm11 x
2 . . . x
, with
i mi = n, mi ≥ 0 for all i
and mi ≤ 1 if î = 1̄, form a k-basis of the n
th homogeneous component of Sfq. Therefore,
(Sfq)n =
r+s=n
r + p− 1
(3.4)
where dimV0̄ = p and dimV1̄ = q as usual.Thus, the generating series of the dimensions is
(Sfq)nt
(1 + t)q
(1− t)p
Example 3.2 (Yang-Mills algebras [11],[10]). Fix a collection of elements x1, . . . , xd (d ≥
2), numbered so as to have parity î = 0̄ for i ≤ p and î = 1̄ for i > p. Let G = (gij) ∈ GLd(k)
be an invertible symmetric d × d-matrix satisfying gij = 0 if î 6= ĵ and consider the cubic
superalgebra A that is generated by elements x1, . . . , xd subject to the relations
gij [xi, [xj , xk]] = 0 (k = 1, . . . , d) (3.5)
Here [ . , . ] is the supercommutator (2.2). The algebra A will be denoted by YMp|q (q = d−p).
In particular, the pure even algebra YMd|0 is the ordinary Yang-Mills algebra introduced in
[10] while YM0|d is the super Yang-Mills algebra as in [11].
As usual, put V =
i kxi and let [ . , . ]⊗ denote the supercommutator in T(V ). Further-
more, put rk =
i,j gij [xi, [xj , xk]⊗]⊗ and R =
k krk ⊆ V
⊗3; so YMp|q = T(V )/(R).
Using the symmetry of G, we may replace the rk by simpler relations as follows. Choose an
N -HOMOGENEOUS SUPERALGEBRAS 19
invertible d × d-matrix C = (cij) with cij = 0 if î 6= ĵ and such that C
trGC is diagonal,
i,j cirgijcjs = gsδ
s . Replace the bases {xi} of V and {rk} of R by the new bases
ijxj and sk =
kℓrℓ where C
−1 = (cij). Note that yi has parity î and sk has
parity k̂, the parity of rk. A simple calculation shows that sk =
i 6=k gi[yi, [yi, yk]⊗]⊗. Thus
we obtain the following defining relations for the generators y1, . . . yd of YM
i 6=k
gi[yi, [yi, yk]] = 0 (k = 1, . . . , d) (3.6)
The resulting algebras for d = 2 are as follows. Putting x = y1 and y = y2 we have two
defining relations: [x, [x, y]] = 0 and [y, [y, x]] = 0. In the pure even case (x̂ = ŷ = 0̄),
the supercommutators are the ordinary Lie commutators. So YM2|0 is the enveloping algebra
of the Heisenberg Lie algebra; see [1, (0.4)]. In the pure odd case (x̂ = ŷ = 1̄), the two
relations can be written as x2y = yx2 and yx2 = x2y. The resulting algebra YM0|2 is a cubic
Artin-Schelter algebra of type S1 [1, (8.6)]. Thus, both unmixed algebras are Artin-Schelter
regular of global dimension 3. In the mixed case, however (x̂ = 0̄, ŷ = 1̄), the relations say
that x commutes with the Lie commutator [x, y] while y anticommutes: y[x, y] = −[x, y]y.
Thus, [x, y] is a normal element of YM1|1 and YM1|1/([x, y]) is a polynomial algebra in two
variables over k. Moreover, the calculation
[x, y]2 = [x, [x, y]y] = −[x, y[x, y]] = −[x, y]2
shows that [x, y]2 = 0. Thus, the algebra YM1|1 is noetherian with Gelfand-Kirillov dimen-
sion 2 and infinite global dimension.
Returning to the case of general d ≥ 2, we now concentrate on the unmixed algebras intro-
duced by Connes and Dubois-Violette. We will denote these algebras by YM+ = YMd|0 and
YM− = YM0|d. In all formulas below, + applies to YM+ and − to YM−. The generators
i 6=k gi[yi, [yi, yk]⊗]⊗ of the space of relations R can be written as sk =
ℓ yℓ⊗mℓk =
ℓmkℓ ⊗ yℓ with
mℓk =
gℓ (yℓ ⊗ yk − (1± 1)yk ⊗ yℓ) for ℓ 6= k
i 6=k giyi ⊗ yi for ℓ = k
Thus, putting Y = (y1, . . . , yd) and letting M denote the d × d-matrix over YM
± whose
(ℓ, k)-entry is the image of mℓk, the defining relations (3.6) can be written as
YM = 0 or MY tr = 0 (3.7)
The defining relations (3.6) for A = YM− amount to the even element
i giy
i ∈ A2 being
central in A.
Example 3.3 (N -symmetric superalgebra; cf. [5]). Let N ≥ 2 be given and let V be a vector
superspace V over a field k with char k = 0 or char k > N . Define
SN (V ) = A(V,R) with R = Λ
NV = cYN
⊆ V ⊗N
where YN is the antisymmetrizer idempotent of the group algebra k[SN ]; see (2.13). This
defines a functor SN ( . ) : Vect
→ HNAlg
. Since 2cY2 is the supercommutator in T(V ), the
algebra S2(V ) is just the symmetric superalgebra S(V ) of V ; see Example 3.1. The algebra
SN (V ), for a pure even space V = V0̄ and general N ≥ 2, has been introduced in [5].
20 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
If 2 ≤ M ≤ N then, viewing k[SM ] as a subalgebra of k[SN ] as usual, the antisymmetriz-
ers of k[SN ] and k[SM ] satisfy YN = YMa for some a ∈ k[SN ]. Therefore,
R = cYN
⊆ cYM
= cYM
⊗ V ⊗(N−M)
This shows that the identity map on V extends to an epimorphism of superalgebras SN(V ) ։
SM (V ).
Now assume that dimk V = d and fix a standard basis x1, . . . , xd of V , with î = 0̄ for
i ≤ p and î = 1̄ for i > p. From the basis for ΛNV exhibited in §2.6.1 we obtain that the
algebra SN (V ) is generated by x1, . . . , xd subject to the relations
(p,q)∈inv(σ) 1+
bip biqxi
σ−1(1)
σ−1(2)
. . . xi
σ−1(N)
with 1 ≤ i1 < i2 < · · · < im ≤ p = dim
V0̄ < im+1 ≤ · · · ≤ iN ≤ d = dimk V ; see
formula (2.14).
Example 3.4. The following construction generalizes Example 3.3. Fix N ≥ 2 and 0 6= q ∈ k
and assume that condition (1.14) is satisfied. Given a Hecke operator R : V ⊗2 → V ⊗2 on a
vector superspace V we define the N -homogeneous superalgebra
ΛR,N := A(V,R) with R = Im ρR(XN ) ⊆ V
⊗N (3.8)
where XN ∈ HN,q is the q-symmetrizer (1.16) and ρR is the representation (1.20) of HN,q.
We also put
SR,N := Λ−qR−1,N = A(V,R) with R = Im ρR(YN ) ⊆ V
⊗N (3.9)
where YN ∈ HN,q is the antisymmetrizer (1.17). The algebra SN(V ) in Example 3.3 is
identical with ScV,V ,N (q = 1).
3.3. The dual of a homogeneous superalgebra. Let A = A(V,R) be an N -homogeneous
superalgebra. The dual A! of A is defined by
A! = A(V ∗, R⊥)
where, R⊥ ⊆ V ∗⊗N is the (homogeneous) subspace consisting of all elements that vanish on
R ⊆ V ⊗N , using (1.4) in order to evaluate elements of V ∗⊗N on V ⊗N . Thus, (3.1) takes the
A!n = V
∗⊗n/R⊥n with R
i+j+N=n
V ∗⊗i ⊗R⊥ ⊗ V ∗⊗j (3.10)
Identifying V ∗⊗n with the linear dual of V ⊗n via (1.4), we have V ∗⊗i ⊗ R⊥ ⊗ V ∗⊗j =(
V ⊗j ⊗R⊗ V ⊗i
. Hence,
R⊥n =
i+j+N=n
V ⊗j ⊗R⊗ V ⊗i
(3.11)
The canonical isomorphism V
−→ V ∗∗ in (1.5) leads to an isomorphism V ⊗N
V ∗∗⊗N which maps R onto R⊥⊥. Hence,
A! ! ∼= A (3.12)
N -HOMOGENEOUS SUPERALGEBRAS 21
Moreover, if f : A = A(V,R) → A′ = A(V ′, R′) is any morphism in HNAlg
then the
transpose of f1 : V → V
′ induces a morphism f ! : (A′)! → A! in HNAlg
. Thus, we have a
contravariant quasi-involutive dualization functor A 7→ A!, f 7→ f ! on HNAlg
Example 3.5. The dual of A(V, 0) = T(V ) is A(V ∗, V ∗⊗N ); so
T(V )! = T(V ∗)/
V ∗⊗N
In particular, letting V = k be the unit object of Vects
, we have A(k, 0) = k[t] (polynomial
algebra) and A(k, 0)! = k[d]/(dN ), with t and d both having degree 1 and parity 0̄.
Example 3.6 (Dual of quantum superspace). We will describe the dual A! of quantum su-
perspace A = Sfq; see Example 3.1. Fix a homogeneous k-basis x1, . . . , xd with format f
for V , and let x1, . . . , xd denote the dual basis of V ∗; this basis also has format f . Evaluat-
ing an arbitrary element f =
ℓ,m fℓmx
ℓ ⊗ xm ∈ V ∗⊗2 on one of the generating relations
ri, rij ∈ R in (3.2), (3.3) we obtain 〈f, ri〉 = fii and 〈f, rij〉 = fij − qij(−1)
bibjfji. Therefore,
the space R⊥ ⊆ V ∗⊗2 has a basis consisting of the elements sℓ := xℓ ⊗ xℓ (ℓ̂ = 0̄) and
sℓ,k := xℓ ⊗ xk + qkℓ(−1)
bkbℓxk ⊗ xℓ (k < ℓ). In summary, A! is generated by x1, . . . , xd
subject to the defining relations
xℓxℓ = 0 (ℓ̂ = 0̄)
xℓxk = −qkℓ(−1)
bkbℓxkxℓ (k < ℓ).
Thus, A! is isomorphic to quantum superspace Sf
with q′ij = (−1)
bi+bjqij and f
′ = f +
(1̄, . . . , 1̄) the format obtained from f by parity reversal in all components.
Example 3.7 (Duals of the Yang-Mills algebras). Continuing with the notation of Exam-
ple 3.2, we now desribe the algebra A! for A = YMp|q. We assume that char k = 0 and work
with generators y1, . . . , yd of A satisfying (3.6).
Let y1, . . . , yd denote the basis of V ∗ given by 〈yi, yj〉 = δ
j and put γ =
yi ∈ V ∗⊗2. Then, for the generators sk =
i 6=k gi[yi, [yi, yk]⊗]⊗ of R as in Example 3.2,
one computes
〈ya ⊗ yb ⊗ yc, sk〉 = gcδ
k + (−1)
bbgbδ
k − (−1)
babk(1 + (−1)ba)gaδ
〈yi ⊗ γ, sk〉 = δ
(3.13)
Therefore, the map ϕ 7→ ϕ−
k〈ϕ, sk〉y
k ⊗ γ is an epimorphism V ∗⊗3 ։ R⊥ ⊂ V ∗⊗3. We
obtain that the algebra A! is generated by y1, . . . , yd subject to the relations
yaybyc = (gcδ
a + (−1)
bbgbδ
c − (−1)ba
bb(1 + (−1)ba)gaδ
b)g (3.14)
where g = 1
iyi is the image of γ in A.
Since A! is 3-homogeneous, we clearly have A!0 = k, A
i = V ∗ and A!2 =⊕
i,j ky
iyj ∼= V ∗⊗2. By (3.13), the elements yag form a k-basis of A!3 = V
∗⊗3/R⊥ ∼= R∗.
Using the defining relations (3.14) it is not hard to see that A!4 = kg
2 and A!n = 0 for n ≥ 5.
If A = YMp|q is of mixed type (i.e., p 6= 0 and q 6= 0) then g2 = 0.
22 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Example 3.8 (Dual of the N -symmetric superalgebra). Recall from Example 3.3 that SN (V ) =
A(V,R) with R = cYN
. Since YN is central in k[SN ] and stable under the inversion
involution ∗ of k[SN ], it follows from (1.11) that
〈x, cYN (y)〉 = 〈cYN (x), y〉
holds for all x ∈ V ∗⊗N and y ∈ V ⊗N . Therefore,
R⊥ = KerV ∗⊗N (cYN ) = (1− cYN )
V ∗⊗N
and so
SN (V )
! = A
V ∗, (1 − cYN )(V
∗⊗N )
Note that ⋂
i+j+N=n
V ⊗i ⊗R⊗ V ⊗j = cYn
(3.15)
holds for all n ≥ N . This follows from (1.19). Alternatively, as has been noted in Exam-
ple 3.3, we have cYn (V
⊗n) ⊆ R ⊗ V ⊗(n−N). In the same way, one sees that cYn (V
⊗n) ⊆
V ⊗i ⊗ R ⊗ V ⊗j whenever i + j + N = n. For the reverse inclusion, note that each
x ∈ V ⊗i ⊗ R ⊗ V ⊗j satisfies cσℓ(x) = −x for all transpositions σℓ = (ℓ, ℓ + 1) ∈ Sn
with i < ℓ < i+N . Hence, the left hand side of (3.15) is contained in the space of antisym-
metric n-tensors, ΛnV = cYn (V
⊗n), thereby proving (3.15). We deduce from (3.10), (3.11)
and (2.15) that
SN (V )
dn if n < N
r+s=n
q+s−1
if n ≥ N
(3.16)
where d = dim
V , p = dim
V0̄ and q = dimk V1̄.
3.4. The operations ◦ and • on HNAlg
. Let A = A(V,R) and A′ = A(V ′, R′) be N -
homogeneous superalgebras. Following [37] and [7] we define the white and black products
A ◦ A′ and A • A′ by
A ◦A′ = A
V ⊗ V ′, cπN
R⊗ V ′⊗N + V ⊗N ⊗R′
A •A′ = A
V ⊗ V ′, cπN
where πN ∈ S2N is the inverse of the permutation
(1, 2, . . . , 2N) 7→ (1, N + 1, 2, N + 2, . . . , k,N + k, . . . ,N, 2N)
Explicitly, cπN : V
⊗N ⊗ V ′⊗N −→ (V ⊗ V ′)⊗N is the morphism in Vects
that is given by
v1 ⊗ . . . vN ⊗ v
1 ⊗ . . . v
= (−1)
bvj (v1 ⊗ v
1)⊗ . . . (vN ⊗ v
N ) (3.17)
Hence, cπN (R⊗R
′) and cπN
R⊗ V ′⊗N + V ⊗N ⊗R′
are homogeneous subspaces of (V⊗
V ′)⊗N and so A ◦ A′ and A • A′ belong to HNAlg
Under the isomorphism (V ′∗⊗V ∗)⊗N
−→ (V ⊗V ′)∗⊗N which comes from (1.4), the rela-
tions cπN
R′⊥ ⊗R⊥
of A′! • A! map onto the relations
R⊗ V ′⊗N + V ⊗N ⊗R′
of (A ◦ A′)!. In fact, by (1.11) we have c∗πN = cπN , because πN τ = τπN , and so 〈x, y〉 =
〈cπN (x), cπN (y)〉 holds for all x ∈ V
′∗ ⊗N ⊗ V ∗⊗N and y ∈ V ⊗N ⊗ V ′⊗N . Therefore,
canonically,
(A ◦A′)! ∼= A
′! • A! and (A • A′)! ∼= A′! ◦ A! (3.18)
N -HOMOGENEOUS SUPERALGEBRAS 23
the two identities being equivalent by (3.12).
By definition of ◦, the canonical isomorphisms k⊗V ∼= V ∼= V ⊗ k in Vects
give isomor-
phisms A(k, 0) ◦ A ∼= A ∼= A ◦ A(k, 0) in HNAlg
, and (3.18) yields similar isomorphisms
for •, with A(k, 0)! = k[d]/(dN ) replacing A(k, 0) = k[t]; see Example 3.5.
The supersymmetry isomorphism cV,V ′ : V ⊗ V
′ ∼−→ V ′ ⊗ V in Vects
(see (1.1)) yields
isomorphisms
A ◦ A′ ∼= A
′ ◦ A and A • A′ ∼= A′ • A (3.19)
in HNAlg
. To see this, note that the following diagram of isomorphisms in Vects
commutes:
V ⊗N ⊗ V ′⊗N
cπN //
V⊗N,V ′⊗N
(V ⊗ V ′)
V,V ′
V ′⊗N ⊗ V ⊗N cπN
// (V ′ ⊗ V )
with v1⊗. . . vN⊗v
1⊗. . . v
N 7→ (−1)
bvj (v′1⊗v1)⊗. . . (v
N⊗vN ) in both composites.
Therefore, putting RA◦A′ = cπN
R⊗ V ′⊗N + V ⊗N ⊗R′
and similarly for RA′◦A etc., we
V,V ′
(RA◦A′) =
cπN ◦ cV ⊗N ,V ′⊗N
R⊗ V ′⊗N + V ⊗N ⊗R′
= cπN
R′ ⊗ V ⊗N + V ′⊗N ⊗R
= RA′◦A
In the same way, one sees that c⊗N
V,V ′
(RA•A′) = RA′•A. This proves (3.19).
Similarly, the associativity isomorphism aV,V ′,V ′′ : (V ⊗ V
′) ⊗ V ′′ ∼= V ⊗ (V ′ ⊗ V ′′) in
Vects
leads to isomorphisms
(A ◦ A′) ◦ A′′ ∼= A ◦ (A
′ ◦ A′′) and (A •A′) • A′′ ∼= A • (A′ • A′′) (3.20)
in HNAlg
. This is a consequence of the following commutative diagram of isomorphisms in
Vects
V ⊗N ⊗ V ′⊗N
⊗ V ′′⊗N
cπN⊗Id
V ⊗N,V ′⊗N,V ′′⊗N
(V ⊗ V ′)
⊗ V ′′⊗N cπN
// ((V ⊗ V ′)⊗ V ′′)
V,V ′,V ′′
V ⊗N ⊗
V ′⊗N ⊗ V ′′⊗N
Id⊗cπN
// V ⊗N ⊗ (V ′ ⊗ V ′′)⊗N cπN
// (V ⊗ (V ′ ⊗ V ′′))
Finally, the compatibility between the isomorphisms cV,V ′ and aV,V ′,V ′′ (see §1.2) is inher-
ited by the isomorphisms (3.19) and (3.20) in HNAlg
. To summarize:
Proposition 3.9. The operations ◦ and • both make the category HNAlg
of N -homogeneous
k-superalgebras into a symmetric tensor category, with unit objects A(k, 0) = k[t] for ◦ and
A(k, 0)! = k[d]/(dN ) for •.
24 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
3.5. The superalgebra map i : A◦A′ → A⊗A′. Let A = A(V,R) and A′ = A(V ′, R′) be
objects of HNAlg
. The superalgebra A⊗A′ is generated by V ⊕ V ′ subject to the relations
R+R′ ⊆ (V ⊕ V ′)⊗N and [V, V ′]⊗ ⊆ (V ⊕ V
where [ . , . ]⊗ is the supercommutator (2.2) in the tensor algebra, as usual. Thus, A ⊗ A
not N -homogeneous when N ≥ 3. Nonetheless, there always is an injective superalgebra
homomorphism i : A ◦ A′ → A ⊗ A′ which is defined as follows. The linear embedding
V ⊗ V ′ →֒ T(V )⊗ T(V ′) extends uniquely to a superalgebra map
ι̃ : T(V ⊗ V ′) → T(V )⊗ T(V ′) (3.21)
which doubles degrees: the restriction of ι̃ to degree n is the embedding
T(V ⊗ V ′)n = (V ⊗ V
−→ V ⊗n ⊗ V ′⊗n ⊆ (T(V )⊗ T(V ′))2n
in Vects
, where cπn is as in (3.17). Thus, ι̃ identifies the superalgebra T(V ⊗ V
′) with the
(super) Segre product
n≥0 V
⊗n ⊗ V ′⊗n of T(V ) and T(V ′).
The map ι̃ sends RA◦A′ = cπN
R⊗ V ′⊗N + V ⊗N ⊗R′
⊆ (V ⊗V ′)⊗N to R⊗V ′⊗N +
V ⊗N ⊗R′, the kernel of the canonical epimorphism V ⊗N ⊗ V ′⊗N ։ AN ⊗A
N . Thus:
Proposition 3.10. The algebra map ι̃ in (3.21) passes down to yield an injective homomor-
phism k-superalgebras i : A ◦ A′ ֌ A ⊗ A′ which doubles degree. The image of i is the
super Segre product
n≥0An ⊗A
n of A and A
3.6. Internal Hom. The isomorphisms (1.3) and (1.4) together with associativity lead to a
functorial isomorphism
(U ⊗ V,W ∗) ∼= Hom
(U, (V ⊗W )∗)
in Vects
. Explicitly, if g ∈ Hom
(U ⊗ V,W ∗) and g′ ∈ Hom
(U, (V ⊗W )∗) correspond to
each other under the above isomorphism then
〈g(u ⊗ v), w〉 = 〈g′(u), v ⊗ w〉 (3.22)
holds for all u ∈ U , v ∈ V and w ∈ W .
In particular, by restricting to 0̄-components, we have a k-linear isomorphism
HomVects
(U ⊗ V,W ∗) ∼= HomVects
(U, (V ⊗W )∗) (3.23)
This isomorphism leads to
Proposition 3.11. There is a functorial isomorphism
HomHNAlgs
(A • B, C) ∼= HomHNAlgs
(A, C ◦ B!)
Proof. We follow Manin [37, 4.2]. Let A = A(U,R), B = A(V, S) and C = A(W,T ) be
N -homogeneous superalgebras. We will prove the proposition in the following equivalent
form; see (3.12) and (3.18):
HomHNAlgs
(A • B, C!) ∼= HomHNAlgs
(A, (B • C)!)
Recall that C! = A(W ∗, T⊥) and (B•C)! = A((V ⊗W )∗, (cπN (S⊗T ))
⊥). Let g : U⊗V →
W ∗ be a morphism in Vects
and let g′ : U → (V ⊗ W )∗ be the morphism in Vects
N -HOMOGENEOUS SUPERALGEBRAS 25
corresponds to g under (3.23). We must show that, for homogeneous subspaces R ⊆ U⊗N ,
S ⊆ V ⊗N and T ⊆ W⊗N ,
g⊗N (cπN (R ⊗ S)) ⊆ T
⊥ ⇔ g′⊗N (R) (cπN (S ⊗ T ))
Identifying T⊥⊥ with T as in §3.3, the first inclusion is equivalent to
〈g⊗N (cπN (R ⊗ S)) , T 〉 = 0 (3.24)
while the second inclusion states that
〈g′⊗N (R), cπN (S ⊗ T )〉 = 0 (3.25)
But (3.22) shows that (3.24) and (3.25) are equivalent, which proves the proposition. �
Proposition 3.11 says that the tensor category (HNAlg
, •) has an internal Hom which is
given by
Hom(A,B) = B ◦ A!
Explicitly, Hom(A,B) is an object of HNAlg
which represents the functor (HNAlg
)op →
Sets, X 7→ HomHNAlgs
(X • A,B); so there is an isomorphism of functors
HomHNAlgs
(? • A,B) ∼= HomHNAlgs
(?,Hom(A,B))
By general properties of Hom (see [12, Def. 1.6]), the morphism IdHom(A,B) corresponds to
a morphism
µ : Hom(A,B) • A → B (3.26)
in HNAlg
satisfying the following universal property: for any morphism f : X • A → B in
HNAlg
there exists a unique morphism g : X → Hom(A,B) such that the following diagram
commutes:
X • A
g•IdA
Hom(A,B) • A
In degree 1, the map µ is simply IdV ⊗evU : V ⊗ U
∗ ⊗ U −→ V ⊗ k = V .
From Hom(B, C) • Hom(A,B) • A
Id •µ
−→ Hom(B, C) • B
−→ C one obtains in this way a
composition morphism
m : Hom(B, C) •Hom(A,B) → Hom(A, C) (3.27)
in HNAlg
. The morphisms µ and m satisfy the obvious associativity properties.
3.7. The superbialgebra endA. Following Manin [37, 4.2] we define
hom(A,B) = Hom(A!,B!)! = A! • B
for A, B in HNAlg
. Applying the dualization functor to (3.26), (3.27) and recalling (3.18),
we obtain morphisms
δ◦ : A → B ◦ hom(B,A)
∆◦ : hom(A, C) → hom(A,B) ◦ hom(B, C)
26 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
in HNAlg
. The associativity properties of µ and m translate into corresponding coassociativ-
ity properties for δ◦ and ∆◦. Following δ◦ and ∆◦ by the algebra map i of Proposition 3.10,
we obtain superalgebra maps
δ : A → B ⊗ hom(B,A) (3.28)
∆: hom(A, C) → hom(A,B)⊗ hom(B, C) (3.29)
Now take A = B = C = A(V,R) and put endA = hom(A,A); so
endA = A! • A = A(V ∗ ⊗ V, cπN (R
⊥ ⊗R)) (3.30)
Then (3.29) yields a coassociative superalgebra map
∆: endA → endA⊗ endA
Moreover, by Proposition 3.11, the morphism A!
−→ A! ∼= k[t] ◦ A! corresponds to a mor-
phism endA = A! • A → k[t] in HNAlg
. Following this morphism by the map t 7→ 1 we
obtain a superalgebra map
ε : endA → k
which in degree 1 is the usual evaluation pairing evV : V
∗ ⊗ V → k in Vects
. Finally, (3.28)
provides us with a superalgebra map
δA : A → A⊗ endA (3.31)
Note that δA maps the degree n-component of A according to
−→ (A ◦ endA)n
−→ An ⊗ (endA)n →֒ An ⊗ endA (3.32)
Fixing a graded k-basis x1, . . . , xd of V and denoting the dual basis of V
∗ by x1, . . . , xd as
before, endA has algebra generators
zij := x
i ⊗ xj (3.33)
of degree-1 and parity î+ ĵ. In terms of these generators, the maps ε, δA and ∆ are given by
i ) = δ
i or ε(Z) = 1
δA(xj) =
xi ⊗ z
j or δA(x1, . . . , xd) = (x1, . . . , xd)⊗ Z
∆(zij) =
zik ⊗ z
j or ∆(Z) = Z ⊗ Z
(3.34)
where Z = (zij)d×d.
Proposition 3.12. Let A = A(V,R) be an N -homogeneous k-superalgebra.
(a) With ∆ as comultiplication and ε as counit, the superalgebra endA becomes a super-
bialgebra. Moreover, δA makes A into a graded right endA-comodule superalgebra.
N -HOMOGENEOUS SUPERALGEBRAS 27
(b) Given any k-superalgebra B and a morphism of superalgebras δ : A → A⊗ B satis-
fying δ(V ) ⊆ V ⊗ B, there is a unique morphism of superalgebras ϕ : endA → B
such that the following diagram commutes:
δA $$J
A⊗ endA
IdA ⊗ϕ
The proposition is proved as in [37, §5] or [7, Theorem 3].
Example 3.13. When A = A(V, 0) = T(V ), we have endA = A(V ∗⊗V, 0) = T(V ∗⊗V );
endT(V ) = T(V ∗ ⊗ V )
the free superalgebra generated by the elements zij in (3.33).
Example 3.14. By Examples 3.3 and 3.8, we have
endSN (V ) = A
V ∗ ⊗ V, cπN
(1− cYN )(V
∗⊗N )⊗ cYN (V
For example, the algebra end S2(V ) is generated by the elements z
j with parity î+ ĵ subject
to the relations
] + (−1)
bi1 bi2+(bi1+bi2)bj1 [z
] = 0
where [ . , . ] is the supercommutator (2.2). This algebra is highly noncommutative, even for a
pure even space V .
Let O(E(V )) = S(V ∗ ⊗ V ) be the supercommutative superbialgebra as in §2.5.3, with
generators xij . There is a map of superbialgebras
ϕ : endSN (V ) → O(E(V )) , z
j 7→ x
j (3.35)
Indeed, write B = O(E(V )) for brevity and recall the coaction δ : V → V ⊗ B, xj 7→∑
i xi ⊗ x
j from (2.11). Since cYN ∈ EndcomodsB(V
⊗N ) (see §2.6.2), the map δ extends to a
map of superalgebras
δ : SN (V ) → SN (V )⊗ B
Therefore, Proposition 3.12(b) yields the desired ϕ. Note that the coaction of end SN (V ) on
V , when restricted along ϕ, becomes the canonical coaction of O(E(V )) on V ; see (2.11) and
(3.34).
4. N -KOSZUL SUPERALGEBRAS
Throughout this section, we fix an N -homogeneous superalgebra A = A(V,R).
28 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
4.1. The graded dual A!∗. The graded dual
A!∗ =
A! ∗n
of A! has a natural structure of a graded right endA-comodule. Indeed, the linear dual A! ∗n
of the degree n-component of A! embeds into V ⊗n as follows. Recall from (3.11) that
A! ∗n =
V ⊗n if n < N
i+j+N=n V
⊗i ⊗R⊗ V ⊗j if n ≥ N
(4.1)
This identification makes the graded dual A!∗ into a graded right endA-comodule. For, by
(3.32) the coaction δA restricts in degree 1 to a map V → V ⊗endA which makes T(V ) into a
graded right endA-comodule superalgebra. The structure map T(V ) → T(V )⊗endA sends
R → R ⊗ endA. Therefore, each V ⊗i ⊗ R ⊗ V ⊗j is a endA-subcomodule of V ⊗(i+j+N),
and hence A! ∗n is a endA-subcomodule of V
⊗n. Finally, for all n ≥ 0,
A! ∗n+1 ⊆ V ⊗A
n and A
n+N ⊆ V
⊗N ⊗A! ∗n ∩R⊗ V
⊗n = R⊗A! ∗n (4.2)
4.2. The Koszul complex. The map
A⊗ V ⊗(i+1) → A⊗ V ⊗i
a⊗ (v1 ⊗ · · · ⊗ vi+1) 7→ av1 ⊗ (v2 ⊗ · · · ⊗ vi+1)
is a morphism in the category ComodsendA of right endA-comodules, because the endA-
coaction δA in (3.31) is a superalgebra map. Furthermore, this map is a left A-module map
which preserves total degree, and it restricts to a map of endA-subcomodules
d : A⊗A! ∗i+1 → AV ⊗A
i →֒ A ⊗A
which is the A-linear extension of the embedding (4.2). The map dN sends A! ∗i+N to AR ⊗
A! ∗i = 0; so d
N = 0. In other words, we have an N -complex
K(A) : . . .
−→ A⊗A! ∗i+1
−→ A⊗A! ∗i
−→ . . .
−→ A −→ 0 (4.3)
in ComodsendA consisting of graded-free left A-modules and A-module maps which pre-
serve total degree. Therefore, K(A) splits into a direct sum of N -complexes K(A)n =⊕
i+j=nAi ⊗A
j in comod
endA.
Following [7], the Koszul complex K(A) defined by Berger in [5] can be described as the
following contraction of K(A):
K(A) : . . .
−→ A⊗A! ∗N+1
−→ A⊗A! ∗N
−→ A⊗A! ∗1
−→ A −→ 0 (4.4)
This is an ordinary complex in ComodsendA which splits into a direct sum of complexes
K(A)n in comodsendA. The i
th components of K(A) and of K(A)n are given by
K(A)i = A⊗A
ν(i) and K(A)
i = An−ν(i) ⊗A
with ν(i) = νN (i) as in (0.1). The differential on K(A) is
δi : K(A)i → K(A)i−1 where δi =
dN−1 for i even
d for i odd
N -HOMOGENEOUS SUPERALGEBRAS 29
Writing A+ =
n>0 An = AV as usual, we have
Ker δi ⊆ A+K(A)i
for all i. Indeed, this is clear for odd i, since δi = d is injective on A
. For even i, the
restriction of δi = d
N−1 to A! ∗
is given by dN−1 : A! ∗
= A! ∗
ν(i−1)+N−1
→֒ V ⊗(N−1) ⊗
ν(i−1)
−→ AN−1 ⊗A
ν(i−1)
→֒ A ⊗A! ∗
ν(i−1)
where the first embedding comes from (4.2).
Since A! ∗
= A! ∗1 = V and A
= A! ∗N = R by (4.1), the start of the Koszul complex,
augmented by the canonical map
A ։ k = A/A+
is as follows:
−→ A⊗ V
δ1=mult
−→ A −→ k −→ 0 (4.5)
This piece is easily seen to be exact: writing A = T(V )/I with I = (R) = I⊗V +T(V )⊗R
as in (0.2), the map T(V )+ = T(V ) ⊗ V ։ A ⊗ V
։ A+ has kernel I . Thus, Ker δ1 =
I/I ⊗ V = Im δ2. Hence (4.5) is the start of the minimal graded-free resolution of the left
A-module k.
4.3. N -homogeneous Koszul superalgebras. Recall from the Introduction that an N -homoge-
neous superalgebra A is called N -Koszul if TorAi (k, k) is concentrated in degree νN (i) for
all i ≥ 0. By [5, Proposition 2.12] or [8, Theorem 2.4], this happens exactly if the Koszul
complex K(A) is exact in degrees i > 0 and in view of (4.5), this amounts to exactness of
K(A) in degrees i ≥ 2. In this case,
K(A) −→ k −→ 0
is the minimal graded-free resolution of the trivial left A-module k.
The Yoneda Ext-algebra E(A) =
i≥0 Ext
A(k, k) of an N -Koszul superalgebra A has
the following description in terms of the dual algebra A!:
ExtiA(k, k)
ν(i) (i ≥ 0)
Moreover, identifying ExtiA(k, k) and A
, the Yoneda product f · g and the A!-product fg
for f ∈ ExtiA(k, k) = A
and g ∈ ExtjA(k, k) = A
are related by f · g = (−1)ijfg
when N = 2, and
f · g =
fg if i or j is even
0 if i and j are both odd
for N > 2; see [21, Theorem 9.1], [8, Proposition 3.1].
Example 4.1. Quadratic algebras having a PBW-basis are 2-Koszul; see, e.g., [41, Chap. 4,
Theorem 3.1]. This applies in particular to quantum superspace A = Afq; see Example 3.1. A
PBW-basis in this case is given by the collection of ordered monomials xm11 x
2 . . . x
d with
mi ≥ 0 for all i and mi ≤ 1 if î = 1̄, as in Example 3.1. For a more general result, see [41,
Chap. 4, Theorem 8.1].
30 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Example 4.2. The unmixed Yang-Mills algebras A = YM± (see Example 3.2) were shown
to be 3-Koszul in [10], [11]. Indeed, letting A[ℓ] denotes the shift of A that is defined by
A[ℓ]n = Aℓ+n, the defining relations for A in the form (3.7) imply that the following complex
of graded-free left A-modules is exact:
0 −→ A[−4]
−→ A[−3]d
−→ A[−1]d
·Y tr
−→ A −→ k −→ 0 (4.6)
The piece A[−3]d
−→ A[−1]d
·Y tr
−→ A −→ k −→ 0 is identical with (4.5). Therefore, (4.6)
is the minimal graded-free resolution of k. The resolution shows that each TorAi (k, k) is con-
centrated in degree ν3(i), and hence A is 3-Koszul. It also follows that (4.6) is isomorphic to
K(A) → k → 0. In particular, (4.6) confirms the dimensions of the corresponding compo-
nents A!n in Example 3.7. As has been pointed out in [10], [11], it follows from (4.6) that the
Hilbert series HA(t) =
n≥0 dimkAn t
n of A = YM± has the form
HA(t) =
1− dt+ dt3 − t4
(1− t2)(1− dt+ t2)
If d > 2 then the series has a pole in the interval (0, 1), and hence dim
An grows exponen-
tially with n. Therefore, A is not noetherian in this case; see Stephenson and Zhang [42].
The mixed Yang-Mills algebras A = YMp|q with p 6= 0 and q 6= 0, on the other hand, are
never 3-Koszul. For YM1|1 this follows from the description given in Example 3.2: this alge-
bra has infinite global dimension. In general, one can check that the so-called extra condition
(see (4.10) below) fails for A, and so A cannot be Koszul, by [5, Prop. 2.7].
Example 4.3. It has been shown in [5, Theorem 3.13] that the N -symmetric algebra SN(V )
of a pure even space V over a field of characteristic 0 is N -Koszul. An extension of this result
will be offered in Theorem 4.5 below.
4.4. Confluence and Koszulity. For the convenience of the reader, we recall the notions of
reduction operators and confluence and their relation to the Koszul property. Complete details
can be found in Berger [4], [5].
Let V in Vects
be given along with a graded basis X = {x1, . . . xd} that is ordered by
x1 > x2 > · · · > xd. The tensors (“monomials”) xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xiN for i =
(i1, i2, . . . , iN ) ∈ {1, 2, . . . , d}
N form a basis of V ⊗N which will be given the lexicographical
ordering. An X-reduction operator on V ⊗N is a projection S ∈ EndVects
(V ⊗N ) such that
either S(xi) = xi or S(xi) < xi holds for each i, where the latter inequality means that
S(xi) is a linear combination (possibly 0) of monomials < xi. The monomials xi satisfying
S(xi) = xi are called S-reduced, all other monomials are S-nonreduced. We denote by
Red(S) and NRed(S)) the (super) subspaces of V ⊗N that are generated by the S-reduced
monomials and the S-nonreduced monomials, respectively; so V ⊗N = Red(S) ⊕ NRed(S)
and Im(S) = Red(S).
Let LX(V
⊗N ) denote the collection of all X-reduction operators on V ⊗N . The proof of
[4, Theorem 2.3] shows that the application S 7→ Ker(S) is a bijection between LX(V
and the set of all super subspaces of V ⊗N . Hence LX(V
⊗N ) inherits a lattice structure: for
S, S′ ∈ LX(V
⊗N ) one has X-reduction operators S ∧ S′ and S ∨ S′ on V ⊗N which are
N -HOMOGENEOUS SUPERALGEBRAS 31
defined by
Ker(S ∧ S′) = Ker(S) + Ker(S′)
Ker(S ∨ S′) = Ker(S) ∩Ker(S′)
A pair (S, S′) of X-reduction operators is said to be confluent if
Red(S ∨ S′) = Red(S) + Red(S′)
Since the inclusion ⊇ is always true, confluence of (S, S′) is equivalent to the inequality
Im(S ∨ S′) ≤ dim
(Im(S) + Im(S′)) (4.7)
Let n ≥ N . Any X-reduction operator S on V ⊗N gives rise to X-reduction operators Sn,i
on V ⊗n which are defined by
Sn,i := IdV ⊗i ⊗S ⊗ IdV ⊗j (i+ j +N = n)
A monomial xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xin of length n ≥ N is said to be S-reduced if xi is
Sn,i-reduced for all i, that is, if every connected submonomial of xi of length N is S-reduced.
Now let A = A(V,R) be an N -homogeneous superalgebra, and let S be the X-reduction
operator on V ⊗N such that Ker(S) = R. The algebra A is said to be X-confluent if the pairs
(SN+i,i, SN+i,0) of X-reduction operators on V
⊗N+i are confluent for i = 1, . . . , N − 1. By
(4.7) this amounts to the inequalities
Im(SN+i,i ∨ SN+i,0) ≤ dim
(Im(SN+i,i) + Im(SN+i,0)) (4.8)
being satisfied for i = 1, . . . , N − 1.
Following Berger [5], we denote by Tn the lattice of super subspaces of V
⊗n that is gen-
erated by the subspaces
Rn,i := V
⊗i ⊗R⊗ V ⊗j = Ker(Sn,i) (i+ j +N = n) (4.9)
The superalgebra A is said to be distributive if the lattices Tn are distributive for all n, that is,
C ∩ (D + E) = (C ∩D) + (C ∩ E) holds for all C,D,E ∈ Tn.
The following proposition states the operative facts concerning Koszulity for our purposes.
Part (a) is identical with [5, Thm. 3.11] while (b) is [5, Prop. 3.4].
Proposition 4.4. Let A = A(V,R) be an N -homogeneous superalgebra.
(a) If A is X-confluent for some totally ordered graded basis X of V then A is distributive.
Moreover, letting S denote the X-reduction operator on V ⊗N such that Ker(S) = R,
the classes in A of the S-reduced monomials xi1 ⊗ xi2 ⊗ · · · ⊗ xin with xij ∈ X form
a k-basis of An for all n ≥ N .
(b) Assume that A is distributive and the following “extra condition” is satisfied
Rn+N,0 ∩Rn+N,n ⊆ Rn+N,n−1 (2 ≤ n ≤ N − 1) (4.10)
Then A is N -Koszul.
After these preparations, we are now ready to prove the following result. The quadratic
case N = 2 is due to Gurevich [22]; see also Wambst [44].
Theorem 4.5. Let N ≥ 2 and 0 6= q ∈ k and assume that [n]q 6= 0 for all n ≥ 1. Then, for
every Hecke operator R associated with q, the N -homogeneous superalgebra ΛR,N defined
in (3.8) is N -Koszul.
32 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Proof. Put A = ΛR,N and recall that A = A(V,R) with
R = Im ρN,R(XN ) ⊆ V
The extra condition (4.10) is a consequence of equation (1.19). Indeed, (1.19) implies that the
spaces Rn,i in (4.9) have the form
Rn,i =
i+N−1⋂
s=i+1
Im(ρn,R(Ts) + 1) ⊆ V
⊗n (4.11)
Applying (4.11) with ρ = ρn+N,R we see that the left hand side of (4.10) is identical to
Im(ρ(Ti) + 1) ∩
n+N−1⋂
i=n+1
Im(ρ(Ti) + 1) =
n+N−1⋂
Im(ρ(Ti) + 1)
where the equality holds because n + 1 ≤ N . The last expression is clearly contained in⋂n+N−2
i=n Im(ρ(Ti) + 1), which is identical to the right hand side of (4.10). This establishes
the extra condition (4.10).
In order to prove the distributivity of A, we follow the approach taken in [25]. We first prove
the claim for the standard solution RDJ , i.e., the operator given in Example 1.2 with d = p
and q = 0. As above, fix a basis X = {x1, . . . , xd} of V , ordered by x1 > x2 > · · · > xd,
and consider the basis of V ⊗n consisting of the monomials xi = xi1 ⊗ xi2 ⊗ · · · ⊗ xin for
i = (i1, i2, . . . , in) ∈ {1, 2, . . . , d}
n with the lexicographical ordering. By equation (1.21),
the action of the generators Tj of the Hecke algebra H = Hn,q2 on this basis is given by
Tj(xi) =
q2xi if ij = ij+1
(q2 − 1)xi + qxσj(i) if ij < ij+1
qxσj(i) if ij > ij+1
(4.12)
Here, σj = (j, j + 1) ∈ Sn and σ(i) = (iσ−1(1), iσ−1(2), . . . , iσ−1(n)) for σ ∈ Sn, as in
Example 3.3.
We claim that the H -submodule of V ⊗n that is generated by xi is given by
H (xi) =
i′∈Sn(i)
kxi′ (4.13)
where Sn(i) is the Sn-orbit of i. Indeed, (4.12) implies that each Tσ(xi) with σ ∈ Sn
is a linear combination of basis vectors xi′ with i
′ ∈ Sn(i). Hence, ⊆ certainly holds in
(4.13). For the reverse inclusion, let i∗ denote the unique non-decreasing sequence in Sn(i);
so xi∗ = max{xi′ | i
′ ∈ Sn(i)}. The last formula in (4.12) implies that
T (xi) = q
r(i)xi∗ (4.14)
where T is a suitable finite product of length r(i) ≥ 0 in the generators Tj . Since T is a unit
in H , the inclusion ⊇ holds in (4.13), thereby proving the asserted equality.
Furthermore, (4.14) and (1.18) (with q replaced by q2) give
qr(i)Xn(xi) = Xn(xi∗). (4.15)
N -HOMOGENEOUS SUPERALGEBRAS 33
These elements are nonzero. For, (4.15) implies that the elements Xn(xi∗) span the image of
Xn on V
⊗n, and their number is
d+n−1
which is equal to the rank of Xn (cf. [25, Eq. (5)]).
It follows that Xn(V
⊗n) = Im ρn,RDJ (Xn) has a k-basis consisting of the elements
Xn(xi∗)
∣∣ i∗ = (i1 ≤ i2 ≤ · · · ≤ in) ∈ {1, 2, . . . , d}n
Next, writing
Xn(xi) =
i′∈Sn(i)
λi′xi′ (4.16)
with λi′ ∈ k, we claim that
λσj(i′) =
λi′ if i
′ = σj(i
q±1λi′ otherwise
To prove this, we may assume that i′ 6= σj(i
′). We compute the coefficient of xσj(i′) in
TjXn(xi) in two ways: by (1.18) this coefficient is equal to q
2λσj(i′) while (4.12) yields
the expression qλi′ + (q
2 − q1±1)λσj(i′). The claim follows from this. Writing an arbitrary
σ ∈ Sn as a product of the inversions σj , we see that the coefficients λi′ in (4.16) only differ
by a nonzero scalar, and hence they are all nonzero since Xn(xi) 6= 0.
By Proposition 4.4, it suffices to check the X-confluence conditions (4.8) i = 1, . . . , N−1.
So let S be the X-reduction operator on V ⊗N with Ker(S) = R. It is easy to see from the dis-
cussion above (with n = N ) that S is given by S(xi∗) = (1−XN/λi∗)(xi∗) and S(xi) = xi
for i 6= i∗. According to (4.11) and the discussion above, the dimension of (R⊗V ⊗i)∩(V ⊗i⊗
R) is
d+N+i−1
. Thus, the dimension of the left hand side of (4.8) is dN+i−
d+N+i−1
the other hand the monomials in V ⊗N+i that belong to NRed(SN+i,i) ∩ NRed(SN+i,0) are
exactly those of the form xi∗ with i
∗ ∈ {1, . . . , d}N+i non-decreasing. Their number is pre-
cisely
d+N+i−1
. Therefore, the dimension of Im(SN+i,i) + Im(SN+i,0) = Red(SN+i,i) +
Red(SN+i,0) is at least d
N+i −
d+N+i−1
. This proves the inequality in (4.8), thereby fin-
ishing the proof of the theorem for the case R = RDJ .
In order to deal with an arbitrary Hecke operator R, recall that Hn,q is split semisimple,
having a representative set of simple modules Mλ indexed by the partitions λ ⊢ n; see (1.15).
We denote the representation of Hn,q on Mλ by ρλ; it does not depend on the operator R but
only on the partition λ.
Let us fix a decomposition
V ⊗n =
into simple Hn,q-submodules Mt. Since all Mt are invariant under the operators ρn,R(Tj),
formula (4.11) yields the decomposition
Rn,i =
i+N−1⋂
s=i+1
(ρn,R(Ts) + 1)(Mt) =
Rn,i ∩Mt
for all i. Therefore, by [25, Lemma 1.2], distributivity of the lattice Tn that is generated by
the subspaces Rn,i of V
⊗n is equivalent to distributivity of the lattices Tn ∩Mt (t ∈ T ) that
34 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
are generated by the subspaces
Rn,i ∩Mt =
i+N−1⋂
s=i+1
(ρn,R(Ts) + 1)(Mt)
of Mt. Now, each Mt is isomorphic to Mλ for some λ ⊢ n. Therefore, the lattice Tn ∩Mt is
isomorphic to the lattice of subspaces of Mλ that is generated by the subspaces
i+N−1⋂
s=i+1
(ρλ(Ts) + 1)(Mλ)
with i + N ≤ n. Finally, when d = dimV > n, then all simple Hn,q-modules Mλ appear
in V ⊗n; see [15, Proposition 5.1]. Thus, the distributivity of the lattice associated to RDJ ,
which we have already verified, implies the distributivity of the corresponding lattice for any
Hecke operator R. This completes the proof. �
5. KOSZUL DUALITY AND MASTER THEOREM
In this section, A = A(V,R) denotes an N -homogeneous superalgebra that is assumed to
be N -Koszul (N ≥ 2).
5.1. By Koszulity, the complexes
K(A)n : . . . → An−νN(i) ⊗A
νN (i)
→ An−νN (i−1) ⊗A
νN (i−1)
→ . . .→An → 0
are exact for n > 0. This yields equations in the Grothendieck ring RendA of the category
comodsendA : ∑
(−1)i[An−νN (i)][A
νN (i)
] = 0 (n > 0) (5.1)
In the power series ring RendAJtK over the Grothendieck ring RendA, define the Poincaré
series
PA(t) =
[An]t
n and PA!∗(t) =
[A! ∗n ]t
For any power series P (t) =
n ant
n, we use the notation
PN (t) :=
n≡0,1 mod N
(−1)αN (n)ant
where αN (n) = n − (n mod N) denotes the largest multiple of N less than or equal to n.
Thus, P2(t) = P (t) and in general
PN (−t) =
n≡0,1 mod N
(−1)n mod Nant
(−1)iaνN (i)t
νN (i)
(5.2)
In particular,
PA!∗,N (−t) =
(−1)i[A! ∗νN (i)]t
νN (i)
Equations (5.1) are equivalent to the following Koszul duality formula:
N -HOMOGENEOUS SUPERALGEBRAS 35
Proposition 5.1. For any N -homogeneous Koszul superalgebra A, the identity
PA(t)PA!∗,N (−t) = 1
holds in RendAJtK.
Applying the ring homomorphism χsJtK : RendAJtK → (endA)0̄JtK, where χ
s is the su-
percharacter map as in Corollary 2.3, the formula in Proposition 5.1 takes the following form
in (endA)0̄JtK:
Corollary 5.2.
χsAℓt
m≡0,1 mod N
(−1)m mod Nχs
A! ∗m
 = 1
Analogous formulas hold with the supercharacter χs replaced by the ordinary character χ
or by one of the dimensions dim and sdim.
By (3.32) the coaction of endA on A sends An to An⊗ (endA)n. A similar remark holds
for the endA-coaction on A! ∗; see §4.1. Therefore, both factors in Corollary 5.2 actually
belong to the Rees subring
n≥0 Bnt
n of BJtK, where we have put B = (endA)0̄.
Example 5.3. As an application of the Hilbert series version of Corollary 5.2, we see that the
duals A! of the Yang-Mills algebras A = YMp|q are never 3-Koszul. In fact, by Example 3.7,
we have HA!(t) = 1+dt+d
2t2+dt3+t4 if p = 0 or q = 0 and HA!(t) = 1+dt+d
2t2+dt3
otherwise. In either case, HA!(t)
−1 has a nonzero coefficient at t5, which rules out Koszulity.
5.2. A master theorem modeled on the N -symmetric superalgebra SN (V ). We put A =
SN (V ) and use the notation of Examples 3.3 and 3.8. In particular, we assume that char k = 0
and work with a fixed basis x1, . . . , xd of V = A1 so that î = 0̄ for i ≤ p and î = 1̄ for i > p.
From Example 3.3 (see also Proposition 4.4(a)), we know that a basis of Aℓ is given by the
monomials xi = xi1xi2 . . . xiℓ for sequences i = (i1, . . . , iℓ) ∈ {1, . . . , d}
ℓ such that i has no
connected subsequence j = (j1, . . . , jN ) of length N satisfying
1 ≤ j1 < . . . < jm ≤ p < jm+1 ≤ . . . ≤ jN ≤ d = p+ q
for some m. Adapting notation of Etingof and Pak [16] to our setting, we denote this set of
sequences i by
Λ(p|q,N)ℓ (5.3)
For example, Λ(p|q, 2)ℓ consists of all weakly decreasing sequences i = (i1, . . . , iℓ) with
entries from {1, . . . , d} and such that no repetition occurs in the range {p + 1, . . . , d}.
In order to evaluate the character χsAℓ in Corollary 5.2, recall from (3.34) that the coaction
δA : A → A⊗ endA is given on the generators xi of A by
δA(xi) =
xj ⊗ z
i ∈ A⊗ endA
where zji = x
j ⊗ xi are the canonical generators of the algebra endA. For i = (i1, . . . , iℓ) ∈
Λ(p|q,N)ℓ, we have
δA(xi) = δA(xi1)δA(xi2) . . . δA(xiℓ) ∈ Aℓ ⊗ endA
Since Aℓ ⊗ endA =
i∈Λ(p|q,N)ℓ
xi ⊗ endA, we can define Z(i) ∈ (endA)0̄ by
δA(xi) = xi ⊗ Z(i) + (terms supported on Λ(p|q,N)ℓ \ {i})
36 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
Then (2.8) becomes
χsAℓ =
i∈Λ(p|q,N)ℓ
biZ(i) (5.4)
with î = î1 + · · ·+ îℓ.
Now consider the super bialgebra B = O(E(V )) = k[xij | 1 ≤ i, j ≤ d] defined in §2.5.3
and recall that the xij are supercommuting variables of parity î + ĵ over k. Restricting the
comodule Aℓ to B along the map ϕ : endSN (V ) → B, z
j 7→ x
j in (3.35) we must replace
Z(i) in (5.4) by X(i) := ϕ(Z(i)) ∈ B0̄. Thus, writing
xj ⊗ x
i ∈ A⊗ B
and yi = yi1 . . . yiℓ ∈ Aℓ ⊗ B =
j∈Λ(p|q,N)ℓ
xj ⊗ B for i = (i1, . . . , iℓ), we have
yi = xi ⊗X(i) + (terms supported on Λ(p|q,N)ℓ \ {i}) (5.5)
As for the supercharacter of A! ∗m , recall from (4.1) and (3.15) that, for all n ≥ N ,
A! ∗n =
i+j+N=n
V ⊗i ⊗R⊗ V ⊗j = ΛnV
Viewing A! ∗n = Λ
nV as a comodule over B = O(E(V )), the supercharacter of A! ∗n is the
nth elementary supersymmetric function en which we know, by Proposition 2.4, to be iden-
tical to the coefficient at tn of the characteristic function ber(1 + tX) of the generic su-
permatrix X =
1≤i,j≤d
of type p|q; so the diagonal blocks X11 =
1≤i,j≤p
X22 =
p+1≤i,j≤p+q
consist of even entries while all other entries are odd.
To summarize, we obtain the following super-version of [16, Theorem 2].
Theorem 5.4. Let X =
be the generic supermatrix of type p|q. Then
i∈Λ(p|q,N)ℓ
biX(i) tℓ
m≡0,1 mod N
(−1)m mod Nemt
 = 1
holds in the power series ring k[xij | all i, j ]0̄JtK. Here Λ(p|q,N)ℓ and X(i) are defined
by (5.3) and (5.5), respectively, and the em are the coefficients of the characteristic function
ber(1 + tX) =
n≥0 ent
n of X.
5.3. As an application of Theorem 5.4 , we determine the superdimension Hilbert series
HsA(t) =
for the N -symmetric superalgebra A = SN (V ). For the pure even case, this was already done
by Etingof and Pak [16] . The notations of §5.2 remain in effect.
In view of Corollary 2.3, the superdimension Poincaré series follows by applying the counit
ε : B → k to the equation in Theorem 5.4. Indeed, by (2.11), the counit ε sends X 7→ 1d×d,
N -HOMOGENEOUS SUPERALGEBRAS 37
and hence the elements X(i) in (5.5) all map to 1. Therefore, the first factor in Theorem 5.4
becomes
HsA(t) =
i∈Λ(p|q,N)ℓ
For the second factor, note that
ber(1 + t 1d×d) = (1 + t)
by (2.12). Thus,
HsA(t) =
i∈Λ(p|q,N)ℓ


m≡0,1 mod N
(−1)m mod N
if p ≥ q
m≡0,1 mod N
(−1)αN (m)
m+ q − p− 1
q − p− 1
if p < q
(5.6)
where αN (m) = m− (m mod N) denotes the largest multiple of N less than or equal to m
as in §5.1.
5.4. The ordinary Hilbert series HA(t) =
ℓ≥0 dimkAℓ t
ℓ of the N -symmetric superalge-
bra A = SN(V ) is as follows. Recall from §5.2 that
Aℓ = |Λ(p|q,N)ℓ|
and from (3.16) that
A!n =
dn if n < N
r+s=n
q+s−1
if n ≥ N
Therefore, the Hilbert series is
HA(t) =
|Λ(p|q,N)ℓ| t
m≡0,1 mod N
(−1)m mod N
r+s=m
q + s− 1
−1 (5.7)
5.5. Less is known about the Hilbert series of the N -homogeneous superalgebras A = ΛR,N
associated to an arbitrary Hecke operator R : V ⊗2 → V ⊗2 on a vector superspace V ; see
Example 3.4. Recall that A = A(V,R) with R = Im ρR(XN ) ⊆ V
⊗N . For any N -
homogeneous algebra A = A(V,R), we have
A!n = dimk
i+j+N=n
V ⊗j ⊗R⊗ V ⊗i
38 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
by (3.10) and (3.11). For R = Im ρR(XN ) in particular, (1.19) further implies that
i+j+N=n
V ⊗j ⊗R⊗ V ⊗i = ρR(Xn)
holds for n ≥ N . Now [26, Theorem 3.5] implies that
!(t) =
ℓ=1(1 + aℓt)∏s
m=1(1− bmt)
where (r, s) is the birank of R and aℓ and bm are positive real numbers. For example, in the
situation of 5.4, (r, s) = (p, q) and aℓ = bm = 1.
For any complex power series P (t), the power series PN (−t) in (5.2) can be written as
PN (−t) =
(1− ζ−iN )P (ζ
where ζN = e
2πi/N . In particular,
HA!∗,N (−t) =
(1− ζ−iN )
ℓ=1(1 + aℓζ
N t)∏s
m=1(1− bmζ
QN,a,b(t)∏s
m=1(1 + bmt+ . . .+ b
m tN−1)
for some real polynomial QN,a,b(t) with coefficients being polynomial in a = (aℓ) and b =
(bm). Therefore, the Hilbert series of A has the form
HA(t) =
m=1(1 + bmt+ . . .+ b
QN,a,b(t)
(5.8)
Notice that the fraction on the right-hand side is reduced.
In particular, (5.7) has the form
HA(t) =
(1− tN )s
(1− t)sQN,1,1(t)
(5.9)
APPENDIX
For lack of a suitable reference, we include here a proof of Proposition 0.1 that was stated
in the Introduction. Our proof is based on the proof of [8, Proposition 2.1] and on additional
details that were communicated to us by Roland Berger. For the basics concerning graded
algebras, we refer the reader to [9, Chap. II §11] or [6].
As in the Introduction, A =
n≥0 An denotes an arbitrary connected Z≥0-graded k-
algebra and V is a graded subspace of A+ =
n>0An satisfying A+ = V ⊕ A
+. Thus,
T(V )/I
−→ A for some graded ideal I of T(V ). For convenience, we state Proposition 0.1
again:
Proposition. The relation ideal I of A lives in degrees ≥ N if and only if TorAi (k, k) lives in
degrees ≥ νN (i) =
N if i is even
N + 1 if i is odd
N -HOMOGENEOUS SUPERALGEBRAS 39
Proof. Let
P : · · · → Pi
−→ Pi−1
−→ · · ·
−→ P0
−→ k → 0
be a minimal graded-free resolution of the trivial left A-module k. Thus, all Pi have the form
Pi = A⊗ Ei for some graded subspace Ei ⊆ Ker di−1 which is chosen so that
Ker di−1 = Ei ⊕A+Ker di−1 (A.10)
In particular, we may take E0 = k and E1 = V . The differential di : Pi → Pi−1 is the graded
A-module map that is defined by the inclusion Ei →֒ Pi−1. By the graded Nakayama Lemma
(e.g., [9, p. AII.171, Prop. 6]), our choice of Ei implies that
Im di = AEi = Ker di−1 and Ker di ⊆ A+ ⊗ Ei = A+Pi (A.11)
for all i. Consequently, the complex k⊗A P has zero differential, and hence
TorAi (k, k)
∼= k⊗A Pi ∼= Ei
In particular,
TorA0 (k, k)
∼= k and TorA1 (k, k)
∼= V = A+/A
live in degrees 0 = νN (0) and ≥ 1 = νN (1), respectively. Moreover, the kernel of d1 : P1 =
(T(V )/I)⊗ V → P0 = A is exactly I/I ⊗ V , and so
TorA2 (k, k)
∼= Ker d1/A+Ker d1 ∼= I/ (V ⊗ I + I ⊗ V )
Therefore, I lives in degrees ≥ N if and only if TorA2 (k, k) lives in degrees ≥ N = νN (2).
For the remainder of the proof, assume that I lives in degrees ≥ N . We will show by
induction on i that TorAi (k, k) = Ei lives in degrees ≥ νN (i) for all i. The cases i ≤ 2
have been checked above. Assume that Ei lives in degrees ≥ νN (i) and similarly for Ei−1.
By (A.11), we know that Ei+1 ⊆ Ker di ⊆ A+ ⊗ Ei and so Ei+1 certainly lives in degrees
≥ νN (i)+1. Since νN (i)+1 = νN (i+1) when i is even (or when i is arbitrary and N = 2),
we are done in these cases. From now on, we assume that i is odd. We must show that Ei+1
lives in degrees ≥ νN (i + 1) =
N . Since Ei+1 ⊆ Ker di, it suffices to show that di is
injective in degrees < i+1
N , and since Ei lives in degrees ≥ νN (i) =
N+1, our goal is to
show that di is injective on all homogeneous components Pi,n of Pi in degrees n =
N + j
with j = 1, . . . , N − 1. Put m = i−1
N for simplicity and note that
Pi,m+j =
Aj−ℓ ⊗ Ei,m+ℓ (A.12)
Pi−1,m+j =
Aj−k ⊗ Ei−1,m+k (A.13)
since Ei−1 lives in degrees ≥ νN (i − 1) = m. The proposition will be a consequence of the
following claims:
(a) di is injective on all summands Aj−ℓ ⊗ Ei,m+ℓ in (A.12), and
(b) the subspaces di (Aj−ℓ ⊗ Ei,m+ℓ) = Aj−ℓEi,m+ℓ for ℓ = 1, . . . , j form a direct sum
inside Pi−1,m+j .
40 PHÙNG HÔ HAI, BENOIT KRIEGK, AND MARTIN LORENZ
In order to prove (a), recall that the restriction of di to Ei,m+ℓ is the inclusion
Ei,m+ℓ →֒ Pi−1,m+ℓ =
Aℓ−k ⊗ Ei−1,m+k
Hence, the effect of di on the ℓ
th summand in (A.12) is the embedding
Aj−ℓ ⊗ Ei,m+ℓ →֒
Aj−ℓ ⊗Aℓ−k ⊗ Ei−1,m+k
followed by the map
Aj−ℓ ⊗Aℓ−k ⊗ Ei−1,m+k −→
Aj−k ⊗ Ei−1,m+k ⊆ Pi−1,m+j
which is given by the multiplication map Aj−ℓ ⊗ Aℓ−k → Aj−k. Since j − k < N , our
hypothesis on I implies that Aj−k ∼= T(V )j−k, and similarly Aj−ℓ ∼= T(V )j−ℓ and Aℓ−k ∼=
T(V )ℓ−k. Therefore, the above multiplication map is identical with the injection T(V )j−ℓ ⊗
T(V )ℓ−k →֒ T(V )j−k in T(V ). This proves (a).
For (b), we proceed by induction on j. The case j = 1 being obvious, let 1 ≤ j ≤ N − 2
and assume that (ii) holds for 1, . . . , j. We wish to show that the subspaces Aj+1−ℓEi,m+ℓ
(ℓ = 1, . . . , j + 1) of Pi−1,m+j+1 form a direct sum. First, by (A.10) we have Ei,m+j+1 ∩
A+Ker di−1 = 0 while
ℓ=1Aj+1−ℓEi,m+ℓ ⊆ A+Ker di−1. Therefore, it suffices to show
that the sum
ℓ=1Aj+1−ℓEi,m+ℓ is direct. To this end, note that Aj+1−ℓ =
d≥1 VdAj+1−d−ℓ
holds for all ℓ ≤ j. Hence,
Aj+1−ℓEi,m+ℓ =
Aj+1−d−ℓEi,m+ℓ
By induction,
Aj+1−d−ℓEi,m+ℓ is a direct sum inside Pi−1,m+j+1−d. Thus, it suffices
to show that the sum
d≥1 VdPi−1,m+j+1−d ⊆ Pi−1,m+j+1 is direct. But (A.13) gives
Pi−1,m+j+1 =
Aj+1−k ⊗ Ei−1,m+k =
T(V )j+1−k ⊗ Ei−1,m+k
where the last equality holds since all j + 1− k < N . Therefore,
VdPi−1,m+j+1−d =
j+1−d⊕
T(V )j+1−d−k ⊗ Ei−1,m+k
as desired. This proves (b), thereby completing the proof of the proposition. �
Acknowledgement. The authors wish to thank Roland Berger for his helpful comments
throughout the completion of this paper. Work on this article was initiated during a visit
of ML to the Université Montpellier 2 in June 2006. ML would like to thank Claude Cibils
for arranging this visit and for his warm hospitality in Montpellier, and James Zhang for help
with some examples in this article.
N -HOMOGENEOUS SUPERALGEBRAS 41
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MATHEMATIK, UNIVERSITY OF DUISBURG-ESSEN, GERMANY AND INSTITUTE OF MATHEMATICS, HANOI,
VIETNAM
E-mail address: hai.phung@uni-duisburg-essen.de
LAMUSE, FACULTÉ DES SCIENCES ET TECHNIQUES, UNIVERSITÉ DE SAINT-ETIENNE, 23 RUE DU DOC-
TEUR PAUL MICHELON, 42023 SAINT-ETIENNE CEDEX 2, FRANCE
E-mail address: benoit.kriegk@univ-st-etienne.fr
DEPARTMENT OF MATHEMATICS, TEMPLE UNIVERSITY, PHILADELPHIA, PA 19122-6094, USA
E-mail address: lorenz@temple.edu
http://arxiv.org/abs/math/0603169
http://arxiv.org/abs/math/0703203
http://arxiv.org/abs/math/0703213
http://arxiv.org/abs/math/0607737
	Introduction
	1. Review of linear superalgebra
	2. The supercharacter
	3. Homogeneous superalgebras
	4. N-Koszul superalgebras
	5. Koszul duality and master theorem
	Appendix
	References
