Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-25T05:59:44.987Z Has data issue: false hasContentIssue false

FAMILIES OF DIRAC OPERATORS AND QUANTUM AFFINE GROUPS

Published online by Cambridge University Press:  31 May 2011

JOUKO MICKELSSON*
Affiliation:
Department of Mathematics and Statistics, University of Helsinki, Helsinki, Finland Department of Theoretical Physics, Royal Institute of Technology, Stockholm, Sweden (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Twisted K-theory classes over compact Lie groups can be realized as families of Fredholm operators using the representation theory of loop groups. In this paper we show how to deform the Fredholm family in the sense of quantum groups. The family of Dirac-type operators is parametrized by vectors in the adjoint module for a quantum affine algebra and transforms covariantly under a central extension of the algebra.

Type
Research Article
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Bibikov, P. N. and Kulish, P. P., ‘Dirac operators on quantum SU(2) group and quantum sphere’, J. Math. Sci. (N. Y.) 100 (2000), 20392050.CrossRefGoogle Scholar
[2]Carey, A. L., Mickelsson, J. and Murray, M. K., ‘Bundle gerbes applied to quantum field theory’, Rev. Math. Phys. 12 (2000), 6590.CrossRefGoogle Scholar
[3]Delius, G. W., Gould, M. D., Hüffmann, A. and Zhang, Y.-Z., ‘Quantum Lie algebras associated to U q(gl n) and U q(sl n)’, J. Phys. A 29 (1996), 56115618.CrossRefGoogle Scholar
[4]Freed, D. S., Hopkins, M. J. and Teleman, C., ‘Twisted equivariant K-theory with complex coefficients’, J. Topol. 1 (2008), 1644.CrossRefGoogle Scholar
[5]Harju, A. and Mickelsson, J., in preparation.Google Scholar
[6]Khoroshkin, S. and Tolstoy, V., ‘Twisting of quantized Lie (super)algebras’, in: Quantum Groups, Karpacz, 1994 (Polish Scientific Publishers PWN, Warsaw, 1995), pp. 6384.Google Scholar
[7]Leclerc, B., ‘Fock space representations of ’, Lecture notes, Grenoble 2008, http://cel.archives-ouvertes.fr/docs/00/43/97/41/PDF/LECLERC_IFETE2008.pdf.Google Scholar
[8]Mickelsson, J., ‘Gerbes, (twisted) K-theory, and the supersymmetric WZW model’, in: Infinite Dimensional Groups and Manifolds, IRMA Lectures in Mathematics and Theoretical Physics, 5 (de Gruyter, Berlin, 2004), pp. 93107.CrossRefGoogle Scholar
[9]Neshveyev, S. and Tuset, L., ‘The Dirac operator on compact quantum groups’, J. Reine Angew. Math. 641 (2010), 120.CrossRefGoogle Scholar