Simple modules over the 4-dimensional Sklyanin Algebras at points of finite order

TL;DR

This paper studies simple modules over 4-dimensional Sklyanin algebras at points of finite order, revealing their dimensions, polynomial identities, and associated division algebras, with implications for their structure and center.

Contribution

It classifies simple modules over these algebras at finite order points, showing their dimensions, polynomial identities, and describing their associated division algebras and Azumaya loci.

Findings

Simple modules have dimension ≤ n, with most having dimension exactly n.

The algebra satisfies a polynomial identity of degree 2n.

The associated division algebra has a rational center.

Abstract

In 1982 E.K. Sklyanin defined a family of graded algebras $A(E,\tau)$, depending on an elliptic curve $E$ and a point $\tau \in E$ that is not 4-torsion. The present paper is concerned with the structure of $A$ when $\tau$ is a point of finite order, $n$ say. It is proved that every simple $A$-module has dimension $\le n$ and that "almost all" have dimension precisely $n$. There are enough finite dimensional simple modules to separate elements of $A$; that is, if $0\ne a \in A$, then there exists a simple module $S$ such that $a.S \ne 0.$ Consequently $A$ satisfies a polynomial identity of degree $2n$ (and none of lower degree). Combined with results of Levasseur and Stafford it follows that $A$ is a finite module over its center. Therefore one may associate to $A$ a coherent sheaf, ${\mathcal A}$ say, of finite ${\mathcal O}_S$ algebras where $S$ is the projective 3-fold determined by the center of $A$. We determine where ${\mathcal A}$ is Azumaya, and prove that the division algebra ${\rm Fract}({\mathcal A})$ has rational center. Thus, for each $E$ and each $\tau \in E$ of order $n \ne 0,2,4$ one obtains a division algebra of degree $s$ over the rational function field of ${\mathbb P}^3$, where $s=n$ if $n$ is odd, and $s={{1} \over {2}} n$ if $n$ is even. The main technical tool in the paper is the notion of a "fat point" introduced by M. Artin. A key preliminary result is the classification of the fat points: these are parametrized by a rational 3-fold.