Received Date: September 30, 2014; Accepted Date: October 06, 2014; Published Date: October 06, 2014
Citation: Iyudu N, Shkarin S (2014) Constructive Approach to Three Dimensional Sklyanin Algebras. J Generalized Lie Theory Appl 8: e101. doi: 10.4172/1736- 4337.1000e101
Copyright: © 2017 Iyudu N et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
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A three dimensional Sklyanin is the quadratic algebra over a field ? with 3 generators x; y; z given by 3 relations xy - ayx - szz = 0, yz - azy - sxx = 0 and zx - axz - syy = 0, where a,s ∈ ? . A generalized Sklyanin algebra is the algebra given by relations xy - a1yx - s1zz = 0, yz - a2zy - s2xx = 0 and zx - a3xz - s3yy = 0, where ai , si∈ ? . In this paper we announce the following results; the complete proofs will appear elsewhere. We determine explicitly the parameters for which these algebras has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates as well as when these algebras are Koszul and PBW, using constructive combinatorial methods. These provide new direct proofs of results established first by Artin, Tate, and Van Den Bergh.
Quadratic algebras; Koszul algebras; Hilbert series; Grobner bases; PBWalgebras; PHSalgebras
It is well-known that algebras arising in string theory, from the geometry of Calabi-Yau manifolds, i.e. various versions of Calabi- Yau algebras, enjoy the potentiality-like properties. This in essence comes from the symplectic structure on the manifold. The notion of noncommutative potential was first introduced by Kontsevich in . Let , then the quotient vector space Fcyc = F / [F,F]has a simple basis labeled by cyclic words in the alphabet x1… xn. For eachj= 1,…, n in  was introduced a linear map .
So, for any element , which is called potential, one can define a collection of elements an algebra which has a presentation:
is called a potential algebra. This can be generalized to super potential algebras. It is known for 3-dimensional Calabi-Yau that they are always derived from a super potential. But not all super potential algebras are Calabi-Yau. This question was studied in details in [2-5] the conditions on potential which ensure CY has been studied. The most general counterpart of potentiality and its relation to CY (in one of possible definitions) considered in . The simplest example of potential algebras are commutative polynomials. Another important example, which have been studied thoroughly  are Sklyanin algebras. We are aiming here to demonstrate, that such properties of these algebras as PBW, PHS, Kosulity could be obtained by constructive, purely combinatorial and algebraic methods, avoiding the power of geometry demonstrated in [ATV1, ATV2] and later papers continuing this line.
Throughout this paper is an arbitrary field, B is a graded algebra, and the symbol B m stands for the mth graded component of algebra B. If V is an n-dimensional vector space over , then F = F(V) is the tensor algebra of V. For any choice of a basis x1,…, xn in V, F is naturally identified with the free algebra with the generators x1,…, xn. For subsets P1,…, Pk of an algebra B, P1,…, Pk stands for the linear span of all products P1,…, Pk with pj ∈ Pj. We consider a degree grading on the free algebra F: the mth graded component of F is V m.
If R is a subspace of the n2-dimensional space V ⊗V , then the quotient of F by the ideal I generated by R is called a quadratic algebra and denoted A (V, R). For any choice of bases x1 ,…, xn in V and g1 , …, gk in R, A(V,R) is the algebra given by generators x1 ,…, xn and the relations g1 ,… , gk( gj are linear combinations of monomials xi xs for 1 ≤ i, s ≤ n) . Since each quadratic algebra A is degree graded,
We can consider its Hilbert series
Quadratic algebras whose Hilbert series is the same as for the algebra of commutative Polynomials play a particularly important role in physics. We say that A is a PHS (for’ polynomial Hilbert series) if
Following the notation from the Polishchuk, Positselski book , we say that a quadratic algebra A = A(V,R) is a PBW-algebra (Poincare, Birkhoff, Witt) if there are bases x1,…,xn and g1,…,gm in V and R respectively such that with respect to some compatible with multiplication well-ordering on the monomials in x1,…,xn, g1,…,gm is a (non-commutative) Grobner basis of the ideal IA generated by R. In this case, x1,…,xn is called a PBW-basis of A, while g1,…,gm are called the PBW-generators of IA. In order to avoid confusion, we would like to stress from the start that Odesskii  as well as some other authors use the term PBW-algebra for what we have already dubbed PHS. Since we deal with both concepts, we could not possibly call them the same and we opted to follow the notation from .
Another concept playing an important role in this paper is Koszulity. For a quadratic algebra A = A (V,R), the augmentation map equips with the structure of a commutative graded A-bi module. The algebra A is called Koszul if as a graded right A-module has a free resolution with the second last arrow being the augmentation map and with each Mm generated in degree m. The last property is the same as the condition that the matrices of the maps Mm → Mm-1 in the last sequence with respect to some free bases consist of elements of V (=are homogeneous of degree 1).
For Sklyanin algebra Sa,s with 3 generators is the quadratic algebra over with generators x, y, z given by 3 relations
yz − azy − sxx = 0, zx − axz − syy = 0, x − ayx − szz = 0.
Odesskii  proved that in the case a generic Sklyanin algebra is a PHS. That is, for generic(a,s) ∈
By generic he means outside the union of countably many proper algebraic varieties in . In particular, the equality above holds for almost all (a,s) ∈ with respect to the Lebesgue measure. Polishchuk and Positselski  showed in the same setting and with the same understanding of the word generic that for generic (a,s) ∈ , the algebra S is Koszul but is not a PBW-algebra. The same results contained in Artin, Shelter paper . The rather tricky arguments of Odesskii are based upon using a geometric interpretation of S a,s to show the existence of a degree 3 central element in S a,s, which generically happens to be nota zero divisor, the arguments of Polishchuk and Positselski are essentially algebro-geometric. The drawback of this kind of results is that they are of no help, if we wish to determine whether S a,s is a PHS or is Koszul for any specific choice of the parameters. In the present paper we address this issue. Despite the fact that Odesskii [9,11] argues that classical combinatorial techniques are in adequate for determining the Hilbert series of Sklyanin algebras, we use these techniques and they turn out to be quite helpful. Recently Sokolov  asked whether there exist a constructive way to determine, for which paprameters (generalized) Sklyanin algebras are PHS. This motivates us to look for constructive proofs of known results on Koszulity, PBW and PHS properties of 3-dimensional Sklyanin algebras, due to Artin, Tate, Van Den Bergh, which do not use the power of algebraic geometry. We prove the following, and our proof based entirely on Grobner basis computations, relations between Koszul algebras and their Hilbert series, and certain other arguments of combinatorial nature. This approach is substantially different from the proofs in Artin, Tate, Van Den Bergh papers [13,14], for example, they get the fact that Sklyanin algebras are PHS as a byproduct of Koszulity. We do it the other way around, we find the Hilbert series first, and then use it to prove Koszulity .
Theorem 0.1: The algebra S a,s is not a PHS if and only if and only if either a = s = 0 or a3 = s3 = −1. Furthermore, the algebra S a,s is Koszul for any choice of a and s. To complete the picture we determine which of these algebras are PBW.
Theorem 0.2: The algebra S a,s is PBW if and only if either s = 0 or a3 = s3 = −1 or (1 − a)3 = s3 and the equation t2 + t + 1 = 0 has a solution in Note that the condition of solvability of the quadratic equation above is automatically satisfied if is algebraically closed or if has characteristic 3. On the other hand, if the third case is empty.
We also study the case of generalized Sklyanin algebras, namely we show that if instead of keeping coefficients in the relations to be triples of the same numbers p, q, r, we allow them to be all different, the situation changes dramatically. For instance, we show that generically such algebras are finite-dimensional and non-Koszul.
We are grateful to IHES and MPIM for hospitality, support, and excellent research atmosphere. This work is funded by the ERC grant 320974 and the ESC Grant N9038.