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

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.