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Elliptic Sklyanin algebras. (Russian) Zbl 0687.17001
A 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.]
Reviewer: D.Leites

##### MSC:
 17A45 Quadratic algebras (but not quadratic Jordan algebras) 16T25 Yang-Baxter equations 16S80 Deformations of associative rings