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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

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Finite-dimensional representations of minimal nilpotent W-algebras and zigzag algebras
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by Alexey Petukhov PDF
Represent. Theory 22 (2018), 223-245 Request permission

Abstract:

Let $\frak g$ be a simple finite-dimensional Lie algebra over an algebraically closed field $\mathbb F$ of characteristic 0. We denote by $\mathrm {U}(\frak g)$ the universal enveloping algebra of $\frak g$. To any nilpotent element $e\in \frak g$ one can attach an associative (and noncommutative as a general rule) algebra $\mathrm {U}(\frak g, e)$ which is in a proper sense a “tensor factor” of $\mathrm {U}(\frak g)$. In this article we consider the case in which $e$ belongs to the minimal nonzero nilpotent orbit of $\frak g$. Under these assumptions $\mathrm {U}(\frak g, e)$ was described explicitly in terms of generators and relations. One can expect that the representation theory of $\mathrm {U}(\frak g, e)$ would be very similar to the representation theory of $\mathrm {U}(\frak g)$. For example one can guess that the category of finite-dimensional $\mathrm {U}(\frak g, e)$-modules is semisimple.

The goal of this article is to show that this is the case if $\frak g$ is not simply-laced. We also show that, if $\frak g$ is simply-laced and is not of type $A_n$, then the regular block of finite-dimensional $\operatorname {U}(\frak g, e)$-modules is equivalent to the category of finite-dimensional modules of a zigzag algebra.

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Additional Information
  • Alexey Petukhov
  • Affiliation: The University of Manchester, Oxford Road M13 9PL, Manchester, United Kingdom; and Institute for Information Transmission Problems, Bolshoy Karetniy 19-1, Moscow 127994, Russia
  • MR Author ID: 828039
  • Email: alex-{}-2@yandex.ru
  • Received by editor(s): June 3, 2017
  • Received by editor(s) in revised form: July 3, 2018
  • Published electronically: November 13, 2018
  • Additional Notes: The main result of the research presented in Sections 7, 8 was carried out at the IITP at the expense of Russian Science Foundation (project No. 14-50-00150). Other sections of this work were carried out as a part of a project on W-algebras supported by Leverhulme Trust Grant RPG-2013-293.
  • © Copyright 2018 American Mathematical Society
  • Journal: Represent. Theory 22 (2018), 223-245
  • MSC (2010): Primary 22E47, 16D50, 16D90
  • DOI: https://doi.org/10.1090/ert/516
  • MathSciNet review: 3875450