Families of Dirac operators and affine quantum groups

Families of Dirac type operators constructed from the supersymmetric Wess-Zumino-Witten model are a useful tool in Fredholm operator realization of twisted K-theory classes on compact Lie groups. They transform in a covariant manner with respect to the action of a central extension of a loop group, the level of the representation giving directly the Dixmier-Douady class of the twisting gerbe. I want to describe a deformation of this system in the language of quantum affine algebras. The loop group covariance property is replaced by a "infinitesimal" Hopf algebra covariance with respect to a quantum enveloping algebra $U_q(\hat g)$ and the Dixmier-Douady class is defined purely algebraically from the action of a central group like element in the Hopf algebra. This is a ongoing project with Antti Harju.
 

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