an:04125631
Zbl 0687.17001
Odesski??, A. V.; Feigin, B. L.
Elliptic Sklyanin algebras
RU
Funkts. Anal. Prilozh. 23, No. 3, 45-54 (1989).
00172645
1989
j
17A45 16T25 16S80
Sklyanin algebra; deformation; polynomial algebra; centers; representations; elliptic solutions of Yang-Baxter equations; quadratic algebras
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.]
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