Hopf Cyclic Cohomology and Beyond

TL;DR

This paper introduces Hopf cyclic cohomology, highlighting recent advances including coefficient theories and extensions beyond Hopf algebras, with applications to braided monoidal categories.

Contribution

It provides a comprehensive overview of the development of Hopf cyclic cohomology, including new unifications and generalizations beyond traditional Hopf algebra frameworks.

Findings

Introduction of coefficients in Hopf cyclic cohomology

Extension to more general algebraic settings

Discussion of relative Hopf cyclic theory in braided categories

Abstract

This paper is an introduction to Hopf cyclic cohomology with an emphasis on its most recent developments. We cover three major areas: the original definition of Hopf cyclic cohomology by Connes and Moscovici as an outgrowth of their study of transverse index theory on foliated manifolds, the introduction of Hopf cyclic cohomology with coefficients by Hajac-Khalkhali-Rangipour-Sommerhauser, and finally the latest episode on unifying the coefficients as well as extending the notion to more general settings beyond Hopf algebras. In particular, the last section discusses the relative Hopf cyclic theory that arises in the braided monoidal category settings.