Vol. 11, No. 4, 2018

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Explicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2

Daniel J. Reich and Chelsea Walton

Vol. 11 (2018), No. 4, 585–608
Abstract

The representation theory of a 3-dimensional Sklyanin algebra S depends on its (noncommutative projective algebro-) geometric data: an elliptic curve E in 2 , and an automorphism σ of E given by translation by a point. Indeed, by a result of Artin, Tate, and van den Bergh, we have that S is module-finite over its center if and only if σ has finite order. In this case, all irreducible representations of S are finite-dimensional and of at most dimension |σ|.

In this work, we provide an algorithm in Maple to directly compute all irreducible representations of S associated to σ of order 2, up to equivalence. Using this algorithm, we compute and list these representations. To illustrate how the algorithm developed in this paper can be applied to other algebras, we use it to recover well-known results about irreducible representations of the skew polynomial ring 1[x,y].

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Keywords
Azumaya locus, irreducible representation, Maple algorithm, 3-dimensional Sklyanin algebra
Mathematical Subject Classification 2010
Primary: 16S38, 16G99, 16Z05
Milestones
Received: 14 October 2016
Revised: 8 February 2017
Accepted: 22 February 2017
Published: 15 January 2018

Communicated by Michael E. Zieve
Authors
Daniel J. Reich
Department of Mathematics
North Carolina State University
Raleigh, NC
United States
Chelsea Walton
Department of Mathematics
Temple University
Philadelphia, PA
United States