Irreducible Representations of the 4-Dimensional Sklyanin Algebra at Points of Infinite Order

Abstract

In 1982 E.K. Sklyanin 13] de ned a family of graded algebras A(E; ), depending on an elliptic curve E and a point 2 E which is not 4-torsion. Basic properties of these algebras were established in 16], and a study of their representation theory was begun in 7]. The present paper classi es the nite dimensional simple A-modules when is a point of in nite order. Sklyanin 14] de nes for each k 2 N a representation of A in a certain k-dimensional subspace of theta functions of order 2(k ? 1): We prove that these are irreducible representations, and that any other simple module is obtained by twisting one of these by an automorphism of A. The automorphism group of A is explicitly computed. The method of proof relies on results in 7]. In particular, it is proved that every nite dimensional simple module is a quotient of a line module. An important part of the analysis is a determination of the 1-critical A-modules, and the fact that such a module is (equivalent to) a quotient of a line module by a shifted line module.

Cocycle twists of algebras

“…Such algebras were studied in [60,61,35], not to mention [67], where they are viewed as a special case of a more general construction.…”

“…While we will discuss Zhang twists fully in §2.2.2, we define a special class of them here. (ii) fat points of all multiplicities were classified by Smith and Staniszkis in [61].…”

“…Fat point modules over A(α, β, γ) were realised in [61] as factor modules of line modules (see Definition 2.1.30 with n = 1).…”

“…We will use results of Staniszkis and Smith from [61]. Although the primary focus of their paper is to classify the finite-dimensional simple A-modules, they also describe the N-graded automorphism group of A, denoted Aut N-alg (A), which we describe below.…”

Homogenized 𝔰𝔩(2)

“…erated in degree 0, having constant Hilbert series, and a fat point is a module that is an ,4-module that is equivalent to a fat point module in the sense that they give isomorphic objects of Proj(^4). This is reminiscent of some of the results on the Sklyanin algebra in [9]. Homogenized sl(2, C) shares some other common features with the Sklyanin algebra; for example annihilators of line modules behave in a similar way.…”