Covariant Dirac operators on quantum groups

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 Uq(g)⊗ cl q(g), where the second tensor factor is a q-deformation of the classical Clifford algebra. The tensor space Uq(g)⊗ cl q(g) is given by 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 K-homology cycles. This work generalizes the operator introduced by P. N. Bibikov and P. P. Kulish [J. Math. Sci. (N.Y.) 100, 2039–2050 (2000)].