Constructing tensor products of modules for <i>C</i><sub>2</sub>-cofinite vertex operator superalgebras

doi:10.1007/s11464-014-0369-3

Front. Math. China

2014, Vol. 9

Issue (3) : 477-494 https://doi.org/10.1007/s11464-014-0369-3

RESEARCH ARTICLE

Constructing tensor products of modules for C₂-cofinite vertex operator superalgebras

Jianzhi HAN(

)

Department of Mathematics, Sichuan University, Chengdu 610064, China

Download: PDF(183 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

For any C₂-cofinite vertex operator superalgebra V, the tensor product and the P(z)-tensor product of any two admissible V-modules of finite length are proved to exist, which are shown to be isomorphic, and their constructions are given explicitly in this paper.

Keywords vertex operator superalgebra tensor product C₂-cofiniteness

Corresponding Author(s): Jianzhi HAN

Issue Date: 24 June 2014

Cite this article:

Jianzhi HAN. Constructing tensor products of modules for C₂-cofinite vertex operator superalgebras[J]. Front. Math. China, 2014, 9(3): 477-494.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-014-0369-3
https://academic.hep.com.cn/fmc/EN/Y2014/V9/I3/477

1	AbeT, BuhlG, DongC Y. Rationality, regularity, and C2-cofiniteness. Trans Amer Math Soc, 2004, 356: 3391-3402 doi: 10.1090/S0002-9947-03-03413-5
2	BuhlG. A spanning set for VOA modules. J Algebra, 2002, 254: 125-151 doi: 10.1016/S0021-8693(02)00056-X
3	BuhlG, KaraaliG. Spanning sets for Möbius vertex algebras satisfying arbitrary difference conditions. J Algebra, 2008, 320: 3345-3364 doi: 10.1016/j.jalgebra.2008.06.038
4	DongC Y, HanJ Z. Some finite properties for vertex operator superalgebras. Pacific J Math, 2012, 258: 269-290 doi: 10.2140/pjm.2012.258.269
5	DongC Y, HanJ Z. On rationality of vertex operator superalgebras. Int Math Res Not doi: 10.1093/imrn/rnt077
6	DongC Y, LiH S, MasonG. Twisted representations of vertex operator algebras. Math Ann, 1998, 310: 571-600 doi: 10.1007/s002080050161
7	DongC Y, LiH S, MasonG. Vertex operator algebras and associative algebras. J Algebra, 1998, 206: 67-96 doi: 10.1006/jabr.1998.7425
8	DongC Y, RenL. Representations of vertex operator algebras and bimodules. J Algebra, 2013, 384: 212-226 doi: 10.1016/j.jalgebra.2013.03.010
9	DongC Y, ZhaoZ P. Modularity in orbifold theory for vertex operator superalgebras. Comm Math Phys, 2005, 260: 227-256 doi: 10.1007/s00220-005-1418-2
10	FrenkelI B, HuangY-Z, LepowskyJ. On Axiomatic Approaches to Vertex Operator Algebras and Modules. Mem Amer Math Soc, Vol 104, No 494. Providence: Amer Math Soc, 1993
11	FrenkelI B, ZhuY C. Vertex operator algebras associated to representations of affine and Virasoro algebras. Duke Math J, 1992, 66: 123-168 doi: 10.1215/S0012-7094-92-06604-X
12	GaberdielM, NeitzkeA. Rationality, quasirationality and finite W-algebras. Comm Math Phys, 2003, 238: 305-331
13	HuangY Z. Cofiniteness conditions, projective covers and the logarithmic tensor product theory. J Pure Appl Algebra, 2009, 213: 458-475 doi: 10.1016/j.jpaa.2008.07.016
14	HuangY Z, LepowskyJ. A theory of tensor products for module categories for a vertex operator algebra, I. Selecta Math, 1995, 1: 699-756 doi: 10.1007/BF01587908
15	HuangY Z, LepowskyJ. A theory of tensor products for module categories for a vertex operator algebra, III. J Pure Appl Algebra, 1995, 100: 141-171 doi: 10.1016/0022-4049(95)00049-3
16	HuangY Z, LepowskyJ, ZhangL. Logarithmic tensor product theory for generalized modules for a conformal vertex algebra. arXiv: 0710.2687
17	HuangY Z, YangJ W. Logarithmic intertwining operators and associative algebras. J Pure Appl Algebra, 2012, 216: 1467-1492 doi: 10.1016/j.jpaa.2011.12.006
18	LiH S. Representation Theory and Tensor Product Theory for Vertex Operator Algebras. Ph D Thesis, Rutgers University, 1994
19	LiH S. Local systems of vertex operators, vertex superalgebras and modules. J Pure Appl Algebra, 1996, 109: 143-195 doi: 10.1016/0022-4049(95)00079-8
20	LiH S. Some finiteness properties of regular vertex operator algebras. J Algebra, 1999, 212: 495-514 doi: 10.1006/jabr.1998.7654
21	MilasA. Weak modules and logarithmic intertwining operators for vertex operator algebras. In: BermanS, FendleyP, HuangY-Z, et al, eds. Recent Developments in Infinite-Dimensional Lie Algebras and Conformal Field Theory. Contemp Math, Vol 297. Providence: Amer Math Soc, 2002, 201-225 doi: 10.1090/conm/297/05099
22	MiyamotoM. Modular invariance of vertex operator algebras satisfying C2-cofiniteness. Duke Math J, 2004, 122: 51-91 doi: 10.1215/S0012-7094-04-12212-2
23	YamauchiH. Modularity on vertex operator algebras arising from semisimple primary vectors. Internat J Math, 2004, 15: 87-109 doi: 10.1142/S0129167X04002193
24	ZhuY C. Modular invariance of characters of vertex operator algebras. J Amer Math Soc, 1996, 9: 237-302 doi: 10.1090/S0894-0347-96-00182-8

[1]	Dong LIU, Xiufu ZHANG. Tensor product weight modules of Schrödinger-Virasoro algebras[J]. Front. Math. China, 2019, 14(2): 381-393.
[2]	Xin WANG, Yuan SHEN. Artin-Schelter regularity of twisted tensor products[J]. Front. Math. China, 2018, 13(5): 1141-1167.
[3]	Meixiang CHEN,Qinghua CHEN. Tensor products of tilting modules[J]. Front. Math. China, 2017, 12(1): 51-62.
[4]	Yuan CHEN. Module-relative-Hochschild (co)homology of tensor products[J]. Front Math Chin, 2012, 7(3): 415-426.

Viewed

Full text

Abstract

Cited

Shared

Discussed