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Quantum SL(3,C)'s: the missing case

We study the only missing case in the classification of quantum SL(3,C)'s undertaken in our paper [J. of Algebra 213 (1999), 721--756; see http://arxiv.org/abs/q-alg/9711005], thereby completing this classification.


Introduction
The aim of this paper is to complete the classification of quantum SL(3)'s undertaken in [8].
Roughly speaking, we call a quantum SL(3) any Hopf algebra (over C) whose (finite-dimensional) comodules are "similar" to the modules of the (ordinary) group SL(3, C) (see Definition 1.1). Given such a Hopf algebra A, it will have, among other things, two simple (nonisomorphic) comodules V, W of dimension 3, and the usual decomposition rules for tensor products will imply the existence of Acomodule morphisms (1.1), satisfying certain compatibility conditions (1.2). The tuple L A , consisting of those two comodules and eight morphisms, will be called the basic quantum SL(3) datum (BQD for short) associated to A.
Conversely, starting from an "abstract" BQD L (i.e. where V, W are just vector spaces and (1.1) just linear maps satisfying the aforementioned conditions), we may reconstruct a Hopf algebra A L via the usual Tannakian procedure.
We may now state our main result.
Theorem A. An almost complete version of this theorem was stated and proved in [8]. More precisely, we found a classification of all BQDs, except for one class related to elliptic curves, and we proved the theorem for all BQDs outside this class. In the present paper, we settle the study of this last class, thereby proving Theorem A in full.
Moreover, a geometric analysis of this class of BQDs yields the following contribution to a question raised in the Introduction of [2]. Theorem B. Let B be the quantum three-space associated to an arbitrary quantum SL(3). Then B cannot be a Sklyanin algebra. In other words, the scheme of point modules [3,4] of B (which is a cubic divisor in P 2 ) cannot be an elliptic curve.
The paper is organized as follows. After some recollections from [8] in Section 1 (definition of a quantum SL(3) and of a BQD, and the correspondences between them), Section 2 recalls and studies the form of the only class of BQDs not covered by the results of [8]; this will finish the classification of BQDs. In Section 3, we introduce the shape algebra [8,Section 5] for this class of BQDs and we determine the associated flag variety (in the sense of [9]). These geometric data suggest to view the shape algebra as a twist (in the sense of [10]) of another algebra, which we show in Section 4 to be isomorphic to the shape algebra of another BQD, already covered by the results of [8]. In Section 5, we finish the proof of Theorem A by carrying over the necessary properties from the untwisted to the twisted shape algebra. Theorem B is proved in Section 6 by picking up some leftovers from Section 2.
In the Appendix, we show a result on twists of Koszul algebras that is needed in Section 5, but may also be of independent interest.
Conventions. We denote by Z (resp. N, C) the set of integers (resp. nonnegative integers, complex numbers). All vector spaces, algebras and tensor products are over C. [8] Recall that the group SL(3) is linearly reductive and that its simple modules are parametrized by their highest weights, which are pairs (k, l) ∈ N 2 . Recall further that the dimension of the simple module of highest weight (k, l) is given by d (k,l) := (k + 1)(l + 1)(k + l + 2)/2. For any λ, µ, ν ∈ N 2 , denote by m ν λµ the multiplicity of the simple module of highest weight ν inside the tensor product of those of (respective) highest weights λ and µ. (Recall also that m ν λµ can, in principle, be determined in a purely combinatorial way.)

Recollections from
Our main objects of interest may now be defined as follows (see [8]).
Definition 1.1. We call a quantum SL(3) any (not necessarily commutative) Hopf algebra A (over C) such that (a) there is a family {V λ | λ ∈ N 2 } of simple and pairwise nonisomorphic A-comodules, with dim V λ = d λ , (b) every A-comodule is isomorphic to a direct sum of these, (c) for every λ, µ ∈ N 2 , V λ ⊗ V µ is isomorphic to ν m ν λµ V ν . In particular, if we write V := V (1,0) and W := V (0,1) , then Condition (c) implies the existence of A-comodule morphisms each being unique up to a scalar. We showed in [8, Propositions 3.1 and 3.2] that these maps must, for an appropriate choice of these scalars, satisfy the following compatibility conditions: • ω is a 3rd root of unity, • κ = q −2 + 1 + q 2 and ρ = (q + q −1 ) −2 for some q ∈ C, q = 0, with q 2 either 1 or not a root of unity.
Thus, each quantum SL(3), A, gives rise to a BQD L A (which is really defined only up to equivalence). Conversely, start with a BQD L, and define an algebra A L with (9+9) generators t i j (i, j = 1, 2, 3) and u α β (α, β = 1, 2, 3), and relations (Here, we have chosen bases x 1 , x 2 , x 3 of V and y 1 , y 2 , y 3 of W , and set A(x i ⊗x j ) = A α ij y α , etc.) Then A L possesses a Hopf algebra structure given by The main difficulty is to prove that the Hopf algebra A L is indeed a quantum SL(3) in the sense of Definition 1.1. We were able to do this in [8] for all BQDs, except for one class, which we will describe in the next section.

Classification of Case I.h
In [8, Section 10], we used the possible Jordan normal forms of Q and the value of ω as first criteria for the classification of BQDs. One possibility, called Type I in [8], consists in taking Q = 1 V and ω = 1. The condition Q = 1 V amounts to setting W = V * , with C, c, D, d the obvious canonical maps. (In particular, we now have S 2 = 1 A , reflecting the fact that W is both the left dual and the right dual of V .) In this case, (1.2a) and (1.2e) are automatically satisfied (for κ = 3, so q 2 = 1 and ρ = 1 4 ). Now consider the "quantum determinants," appearing in (1.2c): If L is of Type I, then Conditions (1.2c) imply that with λ, Λ totally antisymmetric and s, S totally symmetric. In particular, choosing dual bases x i in V and y α in W (= V * ), we may view s and S as homogeneous polynomials of degree 3, i.e. as cubic curves in the projective plane P 2 and in its dual plane P 2 * , respectively (unless s = 0 or S = 0). Using the standard classification of cubic curves in P 2 , we were then able in [8] to classify all possible forms of e and E that satisfy Conditions (1.2bfg), except in the case where s is an elliptic curve.
In this particular case, called Case I.h in [8], the bases of V and W may be chosen in such a way that a reads a : with γ = 0 and (α + β) 3 + γ 3 = 0 (so that s is indeed elliptic). It then follows from (1.2b) that A must read Proposition 2.1. Up to base change, we may assume that α = α ′ = 0.
Substituting α = α ′ = 0 back into (2.1) and (2.2) now yields ββ ′ = γγ ′ = 1 2 . Now we still have one degree of freedom to rescale the maps a and A, so we are left with one essential parameter, say This completes the classification of BQDs, undertaken in [8, Section 10].

The shape algebra and its flag variety
Given a BQD L, recall [8, Section 5] that its shape algebra M L is generated by x i (i = 1, 2, 3) and y α (α = 1, 2, 3), with defining relations x j y β This algebra has a natural N 2 -grading M L = (k,l)∈N 2 V (k,l) , with the x i of degree (1, 0) and the y α of degree (0, 1). (Moreover, it carries a natural A L -comodule algebra structure, with each V (k,l) becoming an A L -comodule: these are the natural candidates to show that A L satisfies Definition 1.1.) When L is the BQD of Case I.h (with α = α ′ = 0 and ββ ′ = γγ ′ = 1 2 ; cf. Section 2), the defining relations of M L read (3.1) t y 2 y 3 = y 2 1 t y 3 y 1 = y 2 2 t y 1 y 2 = y 2 3 y 1 x 1 = x 2 y 2 , y 2 x 1 = t x 2 y 3 , t y 3 x 1 = x 2 y 1 t y 1 x 2 = x 3 y 2 , y 2 x 2 = x 3 y 3 , y 3 x 2 = t x 3 y 1 y 1 x 3 = t x 1 y 2 , t y 2 x 3 = x 1 y 3 , y 3 x 3 = x 1 y 1 x 1 y 1 + x 2 y 2 + x 3 y 3 = 0. Now let us determine the flag variety of M L as defined in [9]. To do this, modify Relations (3.1) as follows: in each relation, adorn the right factor of each term with a ′ if the left factor is an x i , and with a ′′ if the left factor is a y α (e.g. the first relation on the fourth line now reads y 1 x ′′ 1 = x 2 y ′ 2 ). View (x 1 : x 2 : x 3 ) and (y 1 : y 2 : y 3 ) as homogeneous coordinates in P 2 and in P 2 * , respectively. Since Relations (3.1) are N 2 -homogeneous, their modified version may now be seen as defining equations for a subscheme Γ ⊂ (P 2 × P 2 * ) × (P 2 × P 2 * ) × (P 2 × P 2 * ).

Proof of Theorem A
In [8], all intermediate results in the (almost complete) proof of Theorem A are valid for an arbitrary BQD L, except for [8,Proposition 5.3] (and the resulting [8,Corollary 5.4]), which relies on a case by case analysis that is not valid in Case I.h. More precisely, the proof of Theorem A will be complete if we show the following facts for this particular case: (a) the shape algebra M L is a Koszul algebra (when viewed as an N-graded algebra via the total grading), Since the BQD L • introduced in Section 4 is already covered by [8], Properties (a) and (b) are true for M L • .
Therefore, M L , being a twist of M L • , also satisfies those two properties: for Property (b), this is automatic, and for Property (a), it follows from Proposition A.1 in the Appendix. Theorem A follows.

Proof of Theorem B
To each BQD L are associated two "quantum three-spaces," namely the quadratic algebras B L := T (V )/(Im a) and C L := T (W )/(Im b). For Case I.h, B L is defined by the following relations: 3 = 0, (The defining relations of C L are similar.) This algebra is one of the regular algebras of dimension 3 studied in [1], where the matrix Q (see Section 2) also played a role in the classification of such algebras. Algebras for which Q = 1 (called of Type A in [1]) give rise to the cubic curve s in P 2 as in Section 2, so let us call this curve the AS-curve. The AS-curve associated to B L is given by Another cubic curve in P 2 has been associated to B L in [3], namely the scheme of its point modules, given by Call this curve the ATV-curve associated to B L .
Remark 6.1. The ATV-curve is defined for an arbitrary regular algebra of dimension 3, whereas the AS-curve is defined only for those of Type A. Note however, as observed in [3], that even when both curves are defined, they do not coincide in general! Recall [3,7] that B L is called a Sklyanin algebra if its ATV-curve is elliptic. But by Proposition 2.1, this is impossible: the ATV-curve of B L must degenerate to a triangle (which may also be viewed as the image of the flag variety X under the projection P 2 × P 2 * → P 2 ).
On the other hand, a case by case analysis shows that none of the other BQDs, classified in [8, Section 10], can give rise to an elliptic ATV-curve. This proves Theorem B.
Remark 6.2. The fact that, for the BQD L of Case I.h, the ATV-curve of B L cannot be elliptic was already visible in (2.3), which was obtained via an elimination procedure from Conditions (1.2fg). In turn, the latter were shown in [8,Section 3] to be related to the existence of an endomorphism of V ⊗ V satisfying the braid relation.
One would of course like to see a more direct and natural link, for an arbitrary BQD L, between the braid relation and the fact that the ATV-curve of B L cannot be elliptic.

Appendix: Multitwists preserve the Koszul property
Let Γ be a monoid and A = γ∈Γ A γ a Γ-graded algebra. Assume that to each γ ∈ Γ, we associate a graded automorphism τ γ of A, in such a way that τ γγ ′ = τ γ τ γ ′ for all γ, γ ′ ∈ Γ. (This is really only a special case of the notion introduced in [10], but it will be sufficient for our purposes: in Section 4, we take τ (k,l) := τ k 1 τ l 2 for each (k, l) ∈ N 2 .) Recall [10] that the twisted algebra A τ is defined to be the vector space γ∈Γ A γ , endowed with the following new multiplication: x * y := xτ γ (y) for all x ∈ A γ , y ∈ A.
Proposition A.1. Consider A and A τ as N-graded algebras via some morphism h : Γ → N, and assume that they are generated by A 1 (= A τ 1 ) = h(γ)=1 A γ . Then A τ is Koszul if and only if A is Koszul.
Remark A.2. If Γ = N, then the categories of N-graded modules of A and of A τ are equivalent thanks to [10,Theorem 3.1], and the Koszul property is a homological property in this category, so the result is immediate.
However, for arbitrary Γ, [10,Theorem 3.1] shows that the categories of Γ-graded modules are equivalent, but this does not imply that those of N-graded modules are. Therefore, a proof is still needed.
By construction, A τ = T (A 1 )/ v −1 (R) , so A τ is again quadratic. Moreover, the sublattice of A ⊗k 1 generated by v −1 (R) is the image under v −1 of that generated by R, so it is still distributive. Applying Backelin's criterion in the reverse direction, we conclude that A τ is Koszul.