## 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

## 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.## References

- Dražen Adamović and Antun Milas,
*Lattice construction of logarithmic modules for certain vertex algebras*, Selecta Math. (N.S.)**15**(2009), no. 4, 535–561. MR**2565050**, DOI 10.1007/s00029-009-0009-z - Bojko Bakalov,
*Twisted logarithmic modules of vertex algebras*, Comm. Math. Phys.**345**(2016), no. 1, 355–383. MR**3509017**, DOI 10.1007/s00220-015-2503-9 - Bojko Bakalov and McKay Sullivan,
*Twisted logarithmic modules of free field algebras*, J. Math. Phys.**57**(2016), no. 6, 061701, 18. MR**3510308**, DOI 10.1063/1.4953249 - David Brungs and Werner Nahm,
*The associative algebras of conformal field theory*, Lett. Math. Phys.**47**(1999), no. 4, 379–383. MR**1693759**, DOI 10.1023/A:1007525300192 - Chongying Dong, Haisheng Li, and Geoffrey Mason,
*Twisted representations of vertex operator algebras*, Math. Ann.**310**(1998), no. 3, 571–600. MR**1615132**, DOI 10.1007/s002080050161 - Chongying Dong, Haisheng Li, and Geoffrey Mason,
*Vertex operator algebras and associative algebras*, J. Algebra**206**(1998), no. 1, 67–96. MR**1637252**, DOI 10.1006/jabr.1998.7425 - Igor Frenkel, James Lepowsky, and Arne Meurman,
*Vertex operator algebras and the Monster*, Pure and Applied Mathematics, vol. 134, Academic Press, Inc., Boston, MA, 1988. MR**996026** - Igor B. Frenkel and Yongchang Zhu,
*Vertex operator algebras associated to representations of affine and Virasoro algebras*, Duke Math. J.**66**(1992), no. 1, 123–168. MR**1159433**, DOI 10.1215/S0012-7094-92-06604-X - Yi-Zhi Huang,
*Generalized rationality and a “Jacobi identity” for intertwining operator algebras*, Selecta Math. (N.S.)**6**(2000), no. 3, 225–267. MR**1817614**, DOI 10.1007/PL00001389 - Yi-Zhi Huang,
*Differential equations, duality and modular invariance*, Commun. Contemp. Math.**7**(2005), no. 5, 649–706. MR**2175093**, DOI 10.1142/S021919970500191X - Yi-Zhi Huang,
*Generalized twisted modules associated to general automorphisms of a vertex operator algebra*, Comm. Math. Phys.**298**(2010), no. 1, 265–292. MR**2657819**, DOI 10.1007/s00220-010-0999-6 - Y.-Z. Huang, J. Lepowsky, and L. Zhang,
*Logarithmic tensor category theory, V: Convergence condition for intertwining maps and the corresponding compatibility condition*, to appear; arXiv:1012.4199. - Yi-Zhi Huang and Jinwei Yang,
*On functors between module categories for associative algebras and for $\Bbb {N}$-graded vertex algebras*, J. Algebra**409**(2014), 344–361. MR**3198845**, DOI 10.1016/j.jalgebra.2014.04.004 - Kiyokazu Nagatomo and Akihiro Tsuchiya,
*Conformal field theories associated to regular chiral vertex operator algebras. I. Theories over the projective line*, Duke Math. J.**128**(2005), no. 3, 393–471. MR**2145740**, DOI 10.1215/S0012-7094-04-12831-3 - V. Turaev,
*Homotopy field theory in dimension 3 and crossed group-categories*, arxiv:math.GT/0005291. 2000. - Jinwei Yang,
*Twisted representations of vertex operator algebras associated to affine Lie algebras*, J. Algebra**484**(2017), 88–108. MR**3656714**, DOI 10.1016/j.jalgebra.2017.03.041 - Yongchang Zhu,
*Modular invariance of characters of vertex operator algebras*, J. Amer. Math. Soc.**9**(1996), no. 1, 237–302. MR**1317233**, DOI 10.1090/S0894-0347-96-00182-8

## 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: 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
- MR Author ID: 970734
- Email: jinwei2@ualberta.ca
- 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: https://doi.org/10.1090/tran/7490
- MathSciNet review: 3917207