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FAMILIES OF DIRAC OPERATORS AND QUANTUM AFFINE GROUPS

  • JOUKO MICKELSSON (a1) (a2)

Abstract

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.

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References

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[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.
[2]Carey, A. L., Mickelsson, J. and Murray, M. K., ‘Bundle gerbes applied to quantum field theory’, Rev. Math. Phys. 12 (2000), 6590.
[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.
[4]Freed, D. S., Hopkins, M. J. and Teleman, C., ‘Twisted equivariant K-theory with complex coefficients’, J. Topol. 1 (2008), 1644.
[5]Harju, A. and Mickelsson, J., in preparation.
[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.
[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.
[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.
[9]Neshveyev, S. and Tuset, L., ‘The Dirac operator on compact quantum groups’, J. Reine Angew. Math. 641 (2010), 120.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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