On Cyclic Cohomology of x-Hopf algebras

TL;DR

This paper investigates the cyclic cohomology of various x-Hopf algebras, introduces new modules and cocyclic modules, and generalizes the Connes-Moscovici characteristic map for these structures.

Contribution

It computes cyclic cohomology for specific x-Hopf algebras and introduces a pairing that extends the Connes-Moscovici characteristic map to x-Hopf algebras.

Findings

Computed cyclic cohomology for universal enveloping algebras, quantum algebraic tori, and others.

Introduced stable anti Yetter-Drinfeld modules and cocyclic modules for x-Hopf algebras.

Established a pairing that transfers cyclic cocycles between x-Hopf algebras and algebras.

Abstract

In this paper we study the cyclic cohomology of certain x-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici x-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of x-Hopf algebras which generalizes the Connes-Moscovici characteristic map to x-Hopf algebras. This enables us to transfer the x-Hopf algebra cyclic cocycles to algebra cyclic cocycles.