  A natural extension of the Hopf-cyclic cohomology, with coefficients, is
introduced to encompass topological Hopf algebras. The topological theory
allows to work with infinite dimensional Lie algebras. Furthermore, the
category of coefficients (AYD modules) over a topological Lie algebra and those
over its universal enveloping (Hopf) algebra are isomorphic. For topological
Hopf algebras, the category of coefficients is identified with the
representation category of a topological algebra called the anti-Drinfeld
double. Finally, a topological van Est type isomorphism is detailed, connecting
the Hopf-cyclic cohomology to the relative Lie algebra cohomology with respect
to a maximal compact subalgebra.

K-Theory and Homology