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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
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Abstract |
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The representation theory of a 3-dimensional Sklyanin algebra depends on its (noncommutative projective algebro-) geometric data: an elliptic curve in , and an automorphism of given by translation by a point. Indeed, by a result of Artin, Tate, and van den Bergh, we have that is module-finite over its center if and only if has finite order. In this case, all irreducible representations of are finite-dimensional and of at most dimension . In this work, we provide an algorithm in Maple to directly compute all irreducible representations of associated to 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 . |
Keywords
Azumaya locus, irreducible representation, Maple algorithm,
3-dimensional Sklyanin algebra
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Mathematical Subject Classification 2010
Primary: 16S38, 16G99, 16Z05
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Milestones
Received: 14 October 2016
Revised: 8 February 2017
Accepted: 22 February 2017
Published: 15 January 2018
Communicated by Michael E. Zieve
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Authors |
| Daniel J. Reich | |
| Department of Mathematics North Carolina State University Raleigh, NC United States |
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| Chelsea Walton | |
| Department of Mathematics Temple University Philadelphia, PA United States |
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