Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2024 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A Schur-Weyl type duality for twisted weak modules over a vertex algebra
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by Kenichiro Tanabe;
Proc. Amer. Math. Soc. 152 (2024), 3743-3755
DOI: https://doi.org/10.1090/proc/16843
Published electronically: July 17, 2024

Abstract:

Let $V$ be a vertex algebra of countable dimension, $G$ a subgroup of $AutV$ of finite order, $V^{G}$ the fixed point subalgebra of $V$ under the action of $G$, and $\mathscr {S}$ a finite $G$-stable set of inequivalent irreducible twisted weak $V$-modules associated with possibly different automorphisms in $G$. We show a Schur–Weyl type duality for the actions of $\mathscr {A}_{\alpha }(G,\mathscr {S})$ and $V^G$ on the direct sum of twisted weak $V$-modules in $\mathscr {S}$ where $\mathscr {A}_{\alpha }(G,\mathscr {S})$ is a finite dimensional semisimple associative algebra associated with $G,\mathscr {S}$, and a $2$-cocycle $\alpha$ naturally determined by the $G$-action on $\mathscr {S}$. It follows as a natural consequence of the result that for any $g\in G$ every irreducible $g$-twisted weak $V$-module is a completely reducible weak $V^G$-module.
References
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Bibliographic Information
  • Kenichiro Tanabe
  • Affiliation: Faculty of Liberal Arts and Sciences, Tokyo City University, 1-28-1 Tamazutsumi, Setagaya-ku, Tokyo 158-8557, Japan
  • MR Author ID: 616724
  • ORCID: 0000-0002-0936-1596
  • Email: ktanabe@tcu.ac.jp
  • Received by editor(s): April 3, 2023
  • Received by editor(s) in revised form: March 2, 2024
  • Published electronically: July 17, 2024
  • Additional Notes: Research was partially supported by the Grant-in-aid (No. 21K03172) for Scientific Research, JSPS
  • Communicated by: Sarah Witherspoon
  • © Copyright 2024 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 152 (2024), 3743-3755
  • MSC (2020): Primary 17B69
  • DOI: https://doi.org/10.1090/proc/16843
  • MathSciNet review: 4781970