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

TL;DR

This paper develops an algorithm to explicitly compute all irreducible representations of 3-dimensional Sklyanin algebras associated with a point of order 2, providing concrete classifications and illustrating its application to related algebras.

Contribution

It introduces a Maple algorithm for directly computing irreducible representations of Sklyanin algebras at points of order 2, advancing explicit representation theory methods.

Findings

Successfully computed all irreducible representations for the order 2 case.

Reproduced known results for the skew polynomial ring using the new algorithm.

Provided explicit examples and classifications of representations.

Abstract

The representation theory of a 3-dimensional Sklyanin algebra $S$ depends on its (noncommutative projective algebro-) geometric data: an elliptic curve $E$ in $\mathbb{P}^2$, and an automorphism $\sigma$ of $E$ given by translation by a point. Indeed, by a result of Artin-Tate-van den Bergh, we have that $S$ is module-finite over its center if and only if $\sigma$ has finite order. In this case, all irreducible representations of $S$ are finite-dimensional and of at most dimension $|\sigma|$. In this work, we provide an algorithm in Maple to directly compute all irreducible representations of $S$ associated to $\sigma$ 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 $\mathbb{C}_{-1}[x,y]$.