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**Natalia Iyudu ^{1*} and Stanislav Shkarin^{2}**

^{1}School of Mathematics, The University of Edinburgh, The King's Buildings, Mayfield Road, Edinburgh, Scotland EH9 3JZ

^{2}Queens's University Belfast, Department of Pure Mathematics, University Road, Belfast, BT7 1NN, UK

- *Corresponding Author:
- Natalia Iyudu

School of Mathematics

The University of Edinburgh

The King's Buildings, Mayfield Road

Edinburgh, Scotland EH9 3JZ

**E-mail:**[email protected]

**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 [1].
Let , then the quotient vector space *Fcyc* = *F* / [*F*,*F*]has a simple basis labeled by cyclic words in the alphabet *x*_{1}… *x _{n}*. For each

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 [6]. The simplest example of potential
algebras are commutative polynomials. Another important example,
which have been studied thoroughly [7] 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 *m*^{th} 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 *x*_{1},…, *x _{n}* in

If *R* is a subspace of the *n*^{2}-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 *x*_{1} ,…, *x _{n}* in

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
[8], we say that a quadratic algebra *A* = *A*(*V*,*R*) is a *PBW-algebra* (Poincare, Birkhoff, Witt) if there are bases *x*_{1},…,*x _{n}* and

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 *M _{m}* generated in degree

For *Sklyanin algebra S ^{a,s}* with 3 generators is the
quadratic algebra over with generators

*yz* − *azy* − *sxx* = 0, *zx* − *axz* − *syy* = 0, *x* − *ayx* − *szz* = 0.

Odesskii [9] 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 [8] 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 [10]. 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

**Theorem 0.1:** *The algebra S ^{a,s} is not a PHS if and only if and only if
either*

**Theorem 0.2:** *The algebra S ^{a,s} is PBW if and only if either s* = 0

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.

- Kontsevich Maxim (1993) Formal (non) commutative symplectic geometry. The Gel’fand Mathematical Seminars, Birkhuser Boston, Boston, MA.
- Dubois Violette M (2007) Multilinear forms and graded algebras.J. Algebra 317: 198-225.
- Bocklandt R, Schedler T, Wemyss M (2010) Super potentials and higher order derivations
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- Estanislao H, Andrea S (2012)
*Hochschild and cyclic homology of Yang-Mills algebras.*J Reine Angew Math 665: 73-156. - Ginzburg V (2007) Calabi-Yau algebras, Cornell University Library 3: 1-79.
- Sklyanin EK (1983) Some algebraic structures connected with the Yang-Baxter equation. Representations of quantum algebra. Functional Analysis and Its Applications 17: 273-284.
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*,*University Lecture Series 37: 596 - Odesskii AV (2002) Elliptic algebras. Russian Math Surv 57: 1127.
- Artin M, Shelter W (1987)
*Graded algebras of global dimension*3*.*Adv in Math 66: 171-216. - Odesskii AV, Feigin BL (1989) Sklyanin's elliptic algebras. Functional Analysis and Its Applications 23: 207-214
- V.Sokolov (2014)
*private communication,*IHES. - Artin M, Tate J, Van den Bergh M (1991) Modules over regular algebras of dimension 3.Invent math 106: 335-388.
- Artin M, Tate J, Vanden Bergh M (1990) Some algebras associated to auto morphism of elliptic curves. The Grothendieck Festschrift, Progr. Math, 86,Birkhuser Boston, Boston, MA, Vol. I, 3385
- Walton C (2012) Representation theory of three-dimensional Sklyanin algebras. Nuclear Phys. B 860: 167-185.

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