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Register a new userSimple modules over the 4-dimensional Sklyanin Algebras at points of finite order
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S. Paul Smith
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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 $0 e a in A$, then there exists a simple module $S$ such that $a.S e 0.$ 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 $n e 0,2,4$ 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.
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The 4-dimensional Sklyanin algebras are a well-studied 2-parameter family of non-commutative graded algebras, often denoted A(E,tau), that depend on a quartic elliptic curve E in P^3 and a translation automorphism tau of E. They are graded algebras generated by four degree-one elements subject to six quadratic relations and in many important ways they behave like the polynomial ring on four indeterminates apart from the minor difference that they are not commutative. They are elliptic analogues of the enveloping algebra of sl(2,C) and the quantized enveloping algebras U_q(gl_2). Recently, Cho, Hong, and Lau, conjectured that a certain 2-parameter family of algebras arising in their work on homological mirror symmetry consists of 4-dimensional Sklyanin algebras. This paper shows their conjecture is false in the generality they make it. On the positive side, we show their algebras exhibit features that are similar to, and differ from, analogous features of the 4-dimensional Sklyanin algebras in interesting ways. We show that most of the Cho-Hong-Lau algebras determine, and are determined by the graph of a bijection between two 20-point subsets of the projective space P^3. The paper also examines a 3-parameter family of 4-generator 6-relator algebras admitting presentations analogous to those of the 4-dimensional Sklyanin algebras. This class includes the 4-dimensional Sklyanin algebras and most of the Cho-Hong-Lau algebras.
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