Hopf Algebra Equivariant Cyclic Cohomology, K-theory and a q-Index Formula

For an algebra B coming with an action of a Hopf algebra H and a twist automorphism, we introduce equivariant twisted cyclic cohomology. In the case when the twist is implemented by a modular element in H we establish the pairing between even equivariant cyclic cohomology and equivariant K-theory for B. We then extend this formalism to compact quantum group actions and show that our cyclic cohomology is a target space for the equivariant Chern character of equivariant summable Fredholm modules. We prove an analogue of Julg’s theorem relating equivariant K-theory to ordinary K-theory of the C∗-algebra crossed product, and characterize equivariant vector bundles on quantum homogeneous spaces.