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Transactions of the American Mathematical Society

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Associative algebras for (logarithmic) twisted modules for a vertex operator algebra


Authors: Yi-Zhi Huang and Jinwei Yang
Journal: Trans. Amer. Math. Soc. 371 (2019), 3747-3786
MSC (2010): Primary 17B69; Secondary 81T40
DOI: https://doi.org/10.1090/tran/7490
Published electronically: October 1, 2018
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Abstract: 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
Email: yzhuang@math.rutgers.edu

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
Email: jinwei2@ualberta.ca

DOI: https://doi.org/10.1090/tran/7490
Received by editor(s): January 4, 2017
Received by editor(s) in revised form: June 13, 2017
Published electronically: October 1, 2018
Article copyright: © Copyright 2018 American Mathematical Society