Elliptic Sklyanin algebras. The case of points of finite order (Q1910981)

The elliptic Sklyanin algebras \(Q_{n,k}(E, \tau)\) were introduced in \textit{A. V. Odesskij} and \textit{B. L. Feigin} [Preprint Inst. Theor. Phys., Kiev (1989); see also Funct. Anal. Appl. 23, No. 3, 207-214 (1989); translation from Funkts. Anal. Prilozh. 23, No. 3, 45-54 (1989; Zbl 0687.17001)]. Here \(E\) is an elliptic curve, \(\tau\in E\), \(n\) and \(k\) are positive integers, \(k < n\) is coprime to \(n\). The algebras \(Q_{n,k}(E, \tau)\) are quadratic and have the same Hilbert series as a polynomial ring in \(n\) variables; this is the general notion of ``Sklyanin algebra''. In the paper \textit{A. V. Odesskij} and \textit{B. L. Feigin} [Funct. Anal. Appl. 27, 31-38 (1993; Zbl 0798.16030)], several properties of the algebras \(Q_{n,k}(E, \tau)\) were established in the case when \(\tau\) has infinite order. The purpose of the present paper is to study the case when \(\tau\) has finite order. The center of \(Q_{n,k}(E, \tau)\) and its Poisson structure are determined. For \(n= 4\) and \(\tau\) of order 8, the authors present \(R\)-matrices with spectral parameter associated to the family of 2-dimensional representations. The authors also construct new examples of Sklyanin algebras twisting the multiplication of \(Q_{n,k}(E, \tau)\) by a homogeneous automorphism \(\alpha\). They prove that the twisted algebras \(Q_{n,k}^{(\alpha)}(E, \tau)\) are not isomorphic to each other, under certain mild conditions. For another article concerning the topic of this paper, see \textit{J. Tate} and \textit{M. Van den Bergh} [Invent. Math. 124, 619-647 (1996); see the review Zbl 0876.17012 below ].