On the braided Connes–Moscovici construction

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.