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

Abstract

In 1982 Sklyanin ( Funct. Anal. Appl. 16 (1982), 27-34) defined a family of graded algebras A( E, τ), depending on an elliptic curve E and a point τ ∈ E which is not 4-torsion. Basic properties of these algebras were established in Smith and Stafford ( Compositio Math. 83 (1992), 259-289) and a study of their representation theory was begun in Levasseur and Smith ( Bull. Soc. Math. France 121 (1993), 35-90). The present paper classifies the finite dimensional simple A-modules when τ is a point of infinite order. Sklyanin ( Funct. Anal. Appl. 17 (1983), 273-284) defines for each k ∈ 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 Levasseur and Smith. In particular, it is proved that every finite 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.