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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Associative algebras for (logarithmic) twisted modules for a vertex operator algebra
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by Yi-Zhi Huang and Jinwei Yang PDF
Trans. Amer. Math. Soc. 371 (2019), 3747-3786 Request permission


We construct two associative algebras from a vertex operator algebra $V$ and a general automorphism $g$ of $V$. The first, called a $g$-twisted zero-mode algebra, is a subquotient of what we call a $g$-twisted universal enveloping algebra of $V$. These algebras are generalizations of the corresponding algebras introduced and studied by Frenkel-Zhu and Nagatomo-Tsuchiya in the (untwisted) case that $g$ is the identity. The other is a generalization of the $g$-twisted version of Zhu’s algebra for suitable $g$-twisted modules constructed by Dong-Li-Mason when the order of $g$ is finite. We are mainly interested in $g$-twisted $V$-modules introduced by the first author in the case that $g$ is of infinite order and does not act on $V$ semisimply. In this case, twisted vertex operators in general involve the logarithm of the variable. We construct functors between categories of suitable modules for these associative algebras and categories of suitable (logarithmic) $g$-twisted $V$-modules. Using these functors, we prove that the $g$-twisted zero-mode algebra and the $g$-twisted generalization of Zhu’s algebra are in fact isomorphic.
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Additional Information
  • Yi-Zhi Huang
  • Affiliation: Department of Mathematics, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019
  • MR Author ID: 239657
  • ORCID: 0000-0002-6121-2539
  • Email:
  • Jinwei Yang
  • Affiliation: Department of Mathematics, University of Notre Dame, 278 Hurley Building, Notre Dame, Indiana 46556
  • Address at time of publication: Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1
  • MR Author ID: 970734
  • Email:
  • Received by editor(s): January 4, 2017
  • Received by editor(s) in revised form: June 13, 2017
  • Published electronically: October 1, 2018
  • © Copyright 2018 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 371 (2019), 3747-3786
  • MSC (2010): Primary 17B69; Secondary 81T40
  • DOI:
  • MathSciNet review: 3917207