On the braided Connes-Moscovici construction

TL;DR

This paper extends the Connes-Moscovici cyclic cohomology framework to braided Hopf algebras, explicitly computes key operators, introduces twisted modular pairs, and relates these structures through a categorical trace, advancing the understanding of braided cyclic cohomology.

Contribution

It explicitly computes powers of the paracocyclic operator for braided Hopf algebras, introduces twisted modular pairs in involution, and connects the associated (co)cyclic modules via a categorical trace.

Findings

Explicit computation of paracocyclic operator powers

Introduction of twisted modular pairs in involution

Relation between H-module coalgebra and braided structures via categorical trace

Abstract

In $1998$, Connes and Moscovici defined the cyclic cohomology of Hopf algebras. In $2010$, Khalkhali and Pourkia proposed a braided generalization: to any Hopf algebra $H$ in a braided category $\mathcal B$, they associate a paracocyclic object in $\mathcal B$. In this paper we explicitly compute the powers of the paracocyclic operator of this paracocyclic object. Also, we introduce twisted modular pairs in involution for $H$ and derive (co)cyclic modules from them. Finally, we relate the paracocyclic object associated with $H$ to that associated with an $H$-module coalgebra via a categorical version of the Connes-Moscovici trace.