Explicit Realization of Hopf Cyclic Cohomology Classes of Bicrossed Product Hopf Algebras

TL;DR

This paper constructs explicit Hopf cyclic cohomology classes for a bicrossed product Hopf algebra derived from Lie groups and realizes these classes through explicit cocycles in the cyclic cohomology of a related convolution algebra.

Contribution

It provides a novel explicit construction of Hopf cyclic cohomology classes for bicrossed product Hopf algebras from Lie groups, with concrete cocycle representatives.

Findings

Explicit Hopf action with invariant trace on convolution algebra

Construction of Hopf cyclic cohomology classes

Realization of classes as explicit cocycles in cyclic cohomology

Abstract

We construct a Hopf action, with an invariant trace, of a bicrossed product Hopf algebra $\cH=\big( \cU(\Fg_1) \acr \cR(G_2) \big)^{\cop}$ constructed from a matched pair of Lie groups $G_1$ and $G_2$, on a convolution algebra $\cA=C_c^{\ify}(G_1)\rtimes G_2^{\delta}$. We give an explicit way to construct Hopf cyclic cohomology classes of our Hopf algebra $\cH$ and then realize these classes in terms of explicit representative cocycles in the cyclic cohomology of the convolution algebra $\cA$.