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Finite-dimensional representations of minimal nilpotent W-algebras and zigzag algebras


Author: Alexey Petukhov
Journal: Represent. Theory 22 (2018), 223-245
MSC (2010): Primary 22E47, 16D50, 16D90
DOI: https://doi.org/10.1090/ert/516
Published electronically: November 13, 2018
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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
Email: alex-{}-2@yandex.ru

DOI: https://doi.org/10.1090/ert/516
Keywords: Primitive ideals, self-injective modules, W-algebras, zigzag algebras
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.
Article copyright: © Copyright 2018 American Mathematical Society

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