A Schur-Weyl type duality for twisted weak modules over a vertex algebra
HTML articles powered by AMS MathViewer
- by Kenichiro Tanabe;
- Proc. Amer. Math. Soc. 152 (2024), 3743-3755
- DOI: https://doi.org/10.1090/proc/16843
- Published electronically: July 17, 2024
- HTML | PDF | Request permission
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
- Toshiyuki Abe, Geoffrey Buhl, and Chongying Dong, Rationality, regularity, and $C_2$-cofiniteness, Trans. Amer. Math. Soc. 356 (2004), no. 8, 3391–3402. MR 2052955, DOI 10.1090/S0002-9947-03-03413-5
- Dražen Adamović, Ching Hung Lam, Veronika Pedić, and Nina Yu, On irreducibility of modules of Whittaker type for cyclic orbifold vertex algebras, J. Algebra 539 (2019), 1–23. MR 3993937, DOI 10.1016/j.jalgebra.2019.08.007
- D. Adamović, C. H. Lam, V. Pedić, and N. Yu, On irreducibility of modules of Whittaker type: twisted modules and nonabelian orbifolds, arXiv:2212.14137, 2022.
- Richard E. Borcherds, Vertex algebras, Kac-Moody algebras, and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 10, 3068–3071. MR 843307, DOI 10.1073/pnas.83.10.3068
- Robbert Dijkgraaf, Cumrun Vafa, Erik Verlinde, and Herman Verlinde, The operator algebra of orbifold models, Comm. Math. Phys. 123 (1989), no. 3, 485–526. MR 1003430, DOI 10.1007/BF01238812
- Chongying Dong, Haisheng Li, and Geoffrey Mason, Compact automorphism groups of vertex operator algebras, Internat. Math. Res. Notices 18 (1996), 913–921. MR 1420556, DOI 10.1155/S1073792896000566
- Chongying Dong, Haisheng Li, and Geoffrey Mason, Regularity of rational vertex operator algebras, Adv. Math. 132 (1997), no. 1, 148–166. MR 1488241, DOI 10.1006/aima.1997.1681
- 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, Twisted representations of vertex operator algebras and associative algebras, Internat. Math. Res. Notices 8 (1998), 389–397. MR 1628239, DOI 10.1155/S1073792898000269
- 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
- Chongying Dong and Geoffrey Mason, On quantum Galois theory, Duke Math. J. 86 (1997), no. 2, 305–321. MR 1430435, DOI 10.1215/S0012-7094-97-08609-9
- Chongying Dong, Li Ren, and Chao Yang, Orbifold theory for vertex algebras and Galois correspondence, J. Algebra 647 (2024), 144–171. MR 4714762, DOI 10.1016/j.jalgebra.2024.02.020
- Chongying Dong and Gaywalee Yamskulna, Vertex operator algebras, generalized doubles and dual pairs, Math. Z. 241 (2002), no. 2, 397–423. MR 1935493, DOI 10.1007/s002090200421
- 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
- Akihide Hanaki, Masahiko Miyamoto, and Daisuke Tambara, Quantum Galois theory for finite groups, Duke Math. J. 97 (1999), no. 3, 541–544. MR 1682988, DOI 10.1215/S0012-7094-99-09720-X
- J. Lepowsky, Calculus of twisted vertex operators, Proc. Nat. Acad. Sci. U.S.A. 82 (1985), no. 24, 8295–8299. MR 820716, DOI 10.1073/pnas.82.24.8295
- James Lepowsky and Haisheng Li, Introduction to vertex operator algebras and their representations, Progress in Mathematics, vol. 227, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2023933, DOI 10.1007/978-0-8176-8186-9
- Hai-Sheng Li, Local systems of twisted vertex operators, vertex operator superalgebras and twisted modules, Moonshine, the Monster, and related topics (South Hadley, MA, 1994) Contemp. Math., vol. 193, Amer. Math. Soc., Providence, RI, 1996, pp. 203–236. MR 1372724, DOI 10.1090/conm/193/02373
- Hai-Sheng Li, Local systems of vertex operators, vertex superalgebras and modules, J. Pure Appl. Algebra 109 (1996), no. 2, 143–195. MR 1387738, DOI 10.1016/0022-4049(95)00079-8
- Masahiko Miyamoto and Kenichiro Tanabe, Uniform product of $A_{g,n}(V)$ for an orbifold model $V$ and $G$-twisted Zhu algebra, J. Algebra 274 (2004), no. 1, 80–96. MR 2040864, DOI 10.1016/j.jalgebra.2003.11.017
- Kenichiro Tanabe, A generalization of twisted modules over vertex algebras, J. Math. Soc. Japan 67 (2015), no. 3, 1109–1146. MR 3376580, DOI 10.2969/jmsj/06731109
- Kenichiro Tanabe, Simple weak modules for the fixed point subalgebra of the Heisenberg vertex operator algebra of rank 1 by an automorphism of order 2 and Whittaker vectors, Proc. Amer. Math. Soc. 145 (2017), no. 10, 4127–4140. MR 3690600, DOI 10.1090/proc/13767
- Kenichiro Tanabe, The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order 2 (Part 1), J. Algebra 575 (2021), 31–66. MR 4222340, DOI 10.1016/j.jalgebra.2021.01.038
- Kenichiro Tanabe, The irreducible weak modules for the fixed point subalgebra of the vertex algebra associated to a non-degenerate even lattice by an automorphism of order $2$ (Part $2$), J. Math. Soc. Japan. (to appear).
- Gaywalee Yamskulna, Vertex operator algebras and dual pairs, ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)–University of California, Santa Cruz. MR 2702470
- 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
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