The 4-dimensional Sklyanin algebra
type: Item
instance of: scholarly article
title: The 4-dimensional Sklyanin algebra
zbMATH Open document ID: 0811.17010
author: Joanna M. Staniszkis
published in: Journal of Algebra
publication date: 1994-08-16T00:00:00Z
review text: Let \(A = T(V)/ \langle R\rangle\) be the 4-dimensional Sklyanin algebra attached to an elliptic curve \(E\) and a point \(\tau \in E\) not of order 4; recall that \(V\) is the space of sections of a certain line bundle \(L\) of degree 4 over \(E\) and \(R\subset V\otimes V\). \(A\) has two central elements of degree 2; let \(\Omega \subset V \otimes V\) be the vector space spanned by these elements. Now \(V\) can be identified with the space generated by the 4 Jacobi theta functions, and hence \(V \otimes V\) can be thought of as a space of functions on \(\mathbb{C} \times \mathbb{C}\). The main results of this paper present functional equations satisfied by, and determining, the elements of the spaces \(R\) and \(\Omega\). This allows for new proofs of results from [\textit{E. K. Sklyanin}, Funct. Anal. Appl. 17, 273-284 (1983); translation from Funkts. Anal. Prilozh. 17, No. 4, 34-48 (1983; Zbl 0536.58007)] and [\textit{T. Levasseur} and \textit{S. P. Smith}, Bull. Soc. Math. Fr. 121, 35-90 (1993)], concerning respectively the action of \(A\) on the space of meromorphic functions on one variable, and the annihilators of the point modules.
Mathematics Subject Classification ID: 17B37
Mathematics Subject Classification ID: 16E10
Mathematics Subject Classification ID: 16P40
Mathematics Subject Classification ID: 16W50
Mathematics Subject Classification ID: 14H52
zbMATH DE Number: 614201
zbMATH Keywords: 4-dimensional Sklyanin algebra
zbMATH Keywords: elliptic curve
zbMATH Keywords: Jacobi theta functions
zbMATH Keywords: functional equations
zbMATH Keywords: space of meromorphic functions
zbMATH Keywords: point modules
reviewed by: Nicolás Andruskiewitsch
MaRDI profile type: Publication
full work available at URL: jabr.1994.1178
OpenAlex ID: W2028627394
DOI: 10.1006/JABR.1994.1178
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description: scientific article; zbMATH DE number 614201