Available in electronic format
Available in print format		 Transacrions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)
Previous issue | Table of contents | Next issue
Articles in press | Recently posted articles | Most recent issue | All issues
Previous Article | Next Article
     

Group cohomology and gauge equivalence of some twisted quantum doubles

Author(s): Geoffrey Mason; Siu-Hung Ng
Journal: Trans. Amer. Math. Soc. 353 (2001), 3465-3509.
MSC (2000): Primary 57T05, 16S40, 16W30
Posted: April 24, 2001
Retrieve article in: PDF DVI PostScript
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: We study the module category associated to the quantum double of a finite abelian group $G$ twisted by a 3-cocycle, which is known to be a braided monoidal category, and investigate the question of when two such categories are equivalent. We base our discussion on an exact sequence which interweaves the ordinary and Eilenberg-Mac Lane cohomology of $G$. Roughly speaking, this reveals that the data provided by such module categories is equivalent to (among other things) a finite quadratic space equipped with a metabolizer, and also a pair of rational lattices $L\subseteq M$ with $L$ self-dual and integral.

References:

[AF92]
Frank W. Anderson and Kent R. Fuller, Rings and categories of modules, Second ed., Springer-Verlag, New York, 1992. MR 94i:16001

[Bor86]
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 87m:17033

[Bro82]
Kenneth S. Brown, Cohomology of groups, Springer-Verlag, New York-Berlin, 1982; Corrected reprint, Springer-Verlag, 1994. MR 83k:20002; MR 96a:20072

[CP95]
Vyjayanthi Chari and Andrew Pressley, A guide to quantum groups, Cambridge University Press, Cambridge, 1995, Corrected reprint of the 1994 original. MR 96h:17014

[DL93]
Chongying Dong and James Lepowsky, Generalized vertex algebras and relative vertex operators, Birkhäuser Boston, Inc., Boston, MA, 1993. MR 95b:17032

[DPR92]
R. Dijkgraaf, V. Pasquier, and P. Roche, Quasi-Hopf algebras, group cohomology and orbifold models, Integrable systems and quantum groups (Pavia, 1990), World Sci. Publishing, River Edge, NJ, 1992, pp. 75-98. MR 92m:81238

[Dri87]
V. G. Drinfel'd, Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 798-820. MR 89f:17017

[Dri90]
V. G. Drinfel'd, Quasi-Hopf algebras, Leningrad Math. J. 1 (1990), 1419-1457. MR 91b:17016

[DW90]
Robbert Dijkgraaf and Edward Witten, Topological gauge theories and group cohomology, Comm. Math. Phys. 129 (1990), no. 2, 393-429. MR 91g:81133

[EM50a]
Samuel Eilenberg and Saunders Mac Lane, Cohomology theory of Abelian groups and homotopy theory. I, Proc. Nat. Acad. Sci. U. S. A. 36 (1950), 443-447. MR 12:350c

[EM50b]
Samuel Eilenberg and Saunders Mac Lane, Cohomology theory of Abelian groups and homotopy theory. II, Proc. Nat. Acad. Sci. U. S. A. 36 (1950), 657-663. MR 12:520a

[FHL93]
Igor B. Frenkel, Yi Zhi Huang, and James Lepowsky, On axiomatic approaches to vertex operator algebras and modules, Mem. Amer. Math. Soc. 104 (1993), no. 494, viii+64. MR 94a:17007

[FLM88]
Igor Frenkel, James Lepowsky, and Arne Meurman, Vertex operator algebras and the Monster, Academic Press Inc., Boston, MA, 1988. MR 90h:17026

[HL92]
Yi Zhi Huang and James Lepowsky, Toward a theory of tensor products for representations of a vertex operator algebra, Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol. 1, 2 (New York, 1991) (River Edge, NJ), World Sci. Publishing, 1992, pp. 344-354. MR 94k:17045

[HL94a]
Yi-Zhi Huang and James Lepowsky, Tensor products of modules for a vertex operator algebra and vertex tensor categories, Lie theory and geometry, Birkhäuser Boston, Boston, MA, 1994, pp. 349-383. MR 96e:17061

[HL94b]
Yi-Zhi Huang and James Lepowsky, A theory of tensor products for module categories for a vertex operator algebra. I, Surikaisekikenkyusho Kokyuroku (1994), no. 883, 148-203, Geometric aspects of infinite integrable systems (Japanese) (Kyoto, 1993). MR 98a:17048

[HL95a]
Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra. I, II, Selecta Math. (N.S.) 1 (1995), no. 4, 699-756, 757-786. MR 98a:17047

[HL95b]
Yi-Zhi Huang and James Lepowsky, A theory of tensor products for module categories for a vertex operator algebra. III, J. Pure Appl. Algebra 100 (1995), no. 1-3, 141-171. MR 98a:17049

[HO97]
Reinhard Häring-Oldenburg, Reconstruction of weak quasi Hopf algebras, J. Algebra 194 (1997), no. 1, 14-35. MR 98h:16062

[JS93]
André Joyal and Ross Street, Braided tensor categories, Adv. Math. 102 (1993), no. 1, 20-78. MR 94m:18008

[Kas95]
Christian Kassel, Quantum groups, Springer-Verlag, New York, 1995. MR 96e:17041

[Kne70]
Manfred Knebusch, Grothendieck- und Wittringe von nichtausgearteten symmetrischen Bilinearformen, S.-B. Heidelberger Akad. Wiss. Math.-Natur. Kl. 1969/70 (1969/1970), 93-157. MR 42:6001

[Lam99]
T. Y. Lam, Lectures on modules and rings, Springer-Verlag, New York, 1999. MR 99i:16001

[Mac52]
Saunders Mac Lane, Cohomology theory of Abelian groups, Proceedings of the International Congress of Mathematicians, Cambridge, Mass., 1950, vol. 2, Amer. Math. Soc., Providence, R. I., 1952, pp. 8-14. MR 13:529d

[Maj92]
Shahn Majid, Tannaka-Kre{\u{\i}}\kern.15emn theorem for quasi-Hopf algebras and other results, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990), Amer. Math. Soc., Providence, RI, 1992, pp. 219-232. MR 93k:16073

[Mas95]
Geoffrey Mason, The quantum double of a finite group and its role in conformal field theory, Groups '93 Galway/St. Andrews, Vol. 2, Cambridge Univ. Press, Cambridge, 1995, pp. 405-417. MR 97a:11067

[MH73]
John Milnor and Dale Husemoller, Symmetric bilinear forms, Springer-Verlag, New York, 1973, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 73. MR 58:22129

[MS89]
Gregory Moore and Nathan Seiberg, Classical and quantum conformal field theory, Comm. Math. Phys. 123 (1989), no. 2, 177-254. MR 90e:81216

[Swe69]
Moss E. Sweedler, Hopf algebras, W. A. Benjamin, Inc., New York, 1969, Mathematics Lecture Note Series. MR 40:5705

[tD79]
Tammo tom Dieck, Transformation groups and representation theory, Springer, Berlin, 1979. MR 82c:57025

[TY98]
Daisuke Tambara and Shigeru Yamagami, Tensor categories with fusion rules of self-duality for finite abelian groups, J. Algebra 209 (1998), no. 2, 692-707. MR 2000b:18013

[Wal63]
C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281-298. MR 28:133

Similar Articles:

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 57T05, 16S40, 16W30

Retrieve articles in all Journals with MSC (2000): 57T05, 16S40, 16W30

Additional Information:

Geoffrey Mason
Affiliation: Department of Mathematics, University of California, Santa Cruz, California 95064
Email: gem@cats.ucsc.edu

Siu-Hung Ng
Affiliation: Department of Mathematics, University of California, Santa Cruz, California 95064
Address at time of publication: Department of Mathematics, Towson University, Baltimore, Maryland 21252
Email: rng@towson.edu

DOI: 10.1090/S0002-9947-01-02771-4
PII: S 0002-9947(01)02771-4
Received by editor(s): December 8, 1999
Received by editor(s) in revised form: July 24, 2000
Posted: April 24, 2001
Additional Notes: Research of the first author was supported by the National Science Foundation and the Regents of the University of California.
Copyright of article: Copyright 2001, American Mathematical Society

Forward Citation(s):

Information for authors on submitting citations

The following works have cited this article

Höhn, Gerald, Genera of vertex operator algebras and three-dimensional, Vertex operator algebras in mathematics and physics (Toronto, ON 2000), Fields Inst. Commun., vol. 39, Amer. Math. Soc., Providence, RI, 2003, pp. 89--107. MR MR2029792 (2005m:11077)

  AMS Website Logo Small Comments: webmaster@ams.org
© Copyright 2008, American Mathematical Society
Privacy Statement 		Search the AMSPowered by Google