#### Vol. 11, No. 4, 2018

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Ethics Statement Editorial Login ISSN: 1944-4184 (e-only) ISSN: 1944-4176 (print) Author Index Coming Soon Other MSP Journals
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 ${ℙ}^{2}$, and an automorphism $\sigma$ 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 $\sigma$ 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 ${ℂ}_{-1}\left[x,y\right]$.

However, your active subscription may be available on Project Euclid at
https://projecteuclid.org/involve

We have not been able to recognize your IP address 35.168.62.171 as that of a subscriber to this journal.
Online access to the content of recent issues is by subscription, or purchase of single articles.

or by using our contact form.

##### Keywords
Azumaya locus, irreducible representation, Maple algorithm, 3-dimensional Sklyanin algebra
##### Mathematical Subject Classification 2010
Primary: 16S38, 16G99, 16Z05
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 12, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.45105111598968506, "perplexity": 877.3672729994496}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-45/segments/1603107877420.17/warc/CC-MAIN-20201021180646-20201021210646-00234.warc.gz"}