Hopf cyclic cohomology and Chern character of equivariant K-theories

Abstract

We extend the Chern character construction of Neshveyev and Tuset to a map whose values lie in Hopf cyclic homology with coefficients, generalizing their definition of K-theory as well. We also introduce the sheaf of equivariant K-theory (with and without coefficients) similar to the equivariant cohomology of Block and Getzler. This construction is much more geometric (it is defined only for the case in which the Hopf algebra and the Hopfmodule algebra are both algebras of functions on some spaces). Thus, we give a geometric definition of the corresponding Chern character, which takes values in a version of Block—Getzler’s sheaf of equivariant cohomology.