Elliptic Sklyanin algebras. The case of points of finite order
type: Item
instance of: scholarly article
title: Elliptic Sklyanin algebras. The case of points of finite order
zbMATH Open document ID: 0876.17009
DOI: 10.1007/BF01080006
publication date: 1997-11-13T00:00:00Z
review text: 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 ].
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16P40
zbMATH DE Number: 859607
zbMATH Keywords: Sklyanin algebras
zbMATH Keywords: elliptic curve
zbMATH Keywords: finite order
zbMATH Keywords: \(R\)-matrices
zbMATH Keywords: twisted algebras
author: Alexander Odesskii
author: B. Feigin
reviewed by: Nicolás Andruskiewitsch
MaRDI profile type: Publication
cites work: Q4205604
full work available at URL: https://doi.org/10.1007/bf01080006
OpenAlex ID: W2045401496
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 859607