Involve: A Journal of Mathematics

  • Involve
  • Volume 11, Number 4 (2018), 585-608.

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

Daniel J. Reich and Chelsea Walton

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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].

Article information

Source
Involve, Volume 11, Number 4 (2018), 585-608.

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

Permanent link to this document
https://projecteuclid.org/euclid.involve/1522202417

Digital Object Identifier
doi:10.2140/involve.2018.11.585

Mathematical Reviews number (MathSciNet)
MR3778914

Zentralblatt MATH identifier
06864398

Subjects
Primary: 16S38: Rings arising from non-commutative algebraic geometry [See also 14A22] 16G99: None of the above, but in this section 16Z05: Computational aspects of associative rings [See also 68W30]

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

Citation

Reich, Daniel J.; Walton, Chelsea. Explicit representations of 3-dimensional Sklyanin algebras associated to a point of order 2. Involve 11 (2018), no. 4, 585--608. doi:10.2140/involve.2018.11.585. https://projecteuclid.org/euclid.involve/1522202417


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