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2018 Explicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2
Daniel J. Reich, Chelsea Walton
Involve 11(4): 585-608 (2018). DOI: 10.2140/involve.2018.11.585


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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Daniel J. Reich. Chelsea Walton. "Explicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2." Involve 11 (4) 585 - 608, 2018.


Received: 14 October 2016; Revised: 8 February 2017; Accepted: 22 February 2017; Published: 2018
First available in Project Euclid: 28 March 2018

zbMATH: 06864398
MathSciNet: MR3778914
Digital Object Identifier: 10.2140/involve.2018.11.585

Primary: 16G99 , 16S38 , 16Z05

Keywords: 3-dimensional Sklyanin algebra , Azumaya locus , irreducible representation , Maple algorithm

Rights: Copyright © 2018 Mathematical Sciences Publishers


Vol.11 • No. 4 • 2018
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