Elliptic Sklyanin algebras.

*(Russian)*Zbl 0687.17001A Sklyanin algebra in \(n\) indeterminates is a deformation of the polynomial algebra in the class of \(\mathbb Z\)-graded quadratic (i.e. whose relations are of degree 2) algebras. Such algebras are of interest because they have a describable representation theory relevant to integrable models of statistical physics as can be seen from (scant) references.

In his original work E. K. Sklyanin constructed a deformation (here denoted by \(Q_{4,1}({\mathcal E},\tau))\) of the polynomial algebra in 4 indeterminates parametrized by an elliptic curve \({\mathcal E}\) and its point \(\tau\in {\mathcal E}\). In the paper there are constructed similar algebras \(Q_{n,k}({\mathcal E},\tau)\), where \(k\in (\mathbb Z/n\mathbb Z)^*\), and their centers and representations (in terms of symplectic folias similar to the orbit method) are described. Relations in these algebras are described in terms of elliptic solutions of Yang-Baxter equations. A number of other examples given by V. Cherednik, V. Drinfel’d, A. Vershik in their papers on quadratic algebras are interpreted as particular cases of \(Q_{n,k}({\mathcal E},\tau).\)

The paper is a continuation of an easier to understand but harder to get preprint by the authors [“Sklyanin algebras associated with an elliptic curve”, Inst. Teor. Fiz., Kiev, 1988 (Russian), cf. ref. 8.]

In his original work E. K. Sklyanin constructed a deformation (here denoted by \(Q_{4,1}({\mathcal E},\tau))\) of the polynomial algebra in 4 indeterminates parametrized by an elliptic curve \({\mathcal E}\) and its point \(\tau\in {\mathcal E}\). In the paper there are constructed similar algebras \(Q_{n,k}({\mathcal E},\tau)\), where \(k\in (\mathbb Z/n\mathbb Z)^*\), and their centers and representations (in terms of symplectic folias similar to the orbit method) are described. Relations in these algebras are described in terms of elliptic solutions of Yang-Baxter equations. A number of other examples given by V. Cherednik, V. Drinfel’d, A. Vershik in their papers on quadratic algebras are interpreted as particular cases of \(Q_{n,k}({\mathcal E},\tau).\)

The paper is a continuation of an easier to understand but harder to get preprint by the authors [“Sklyanin algebras associated with an elliptic curve”, Inst. Teor. Fiz., Kiev, 1988 (Russian), cf. ref. 8.]

Reviewer: D.Leites

##### MSC:

17A45 | Quadratic algebras (but not quadratic Jordan algebras) |

16T25 | Yang-Baxter equations |

16S80 | Deformations of associative rings |