Covariant Dirac Operators on Quantum Groups
arXiv:1009.3913 · doi:10.1063/1.3664284
Abstract
We give a construction of a Dirac operator on a quantum group based on any simple Lie algebra of classical type. The Dirac operator is an element in the vector space $U_q(\g) \otimes \mathrm{cl}_q(\g)$ where the second tensor factor is a -deformation of the classical Clifford algebra. The tensor space $ U_q(\g) \otimes \mathrm{cl}_q(\g)$ is given a structure of the adjoint module of the quantum group and the Dirac operator is invariant under this action. The purpose of this approach is to construct equivariant Fredholm modules and -homology cycles. This work generalizes the operator introduced by Bibikov and Kulish in \cite{BK}.