Invariant Cyclic Homology

TL;DR

This paper introduces a noncommutative version of invariant de Rham cohomology using Hopf algebra structures, defining invariant cyclic homology for algebras and coalgebras, with examples including quantum groups.

Contribution

It develops a new invariant cyclic homology theory for Hopf algebra comodule algebras and modules, extending classical concepts to noncommutative settings.

Findings

Defined invariant cyclic homology for Hopf algebra structures

Established dual theory for coalgebras

Computed examples including quantum group SL(q,2)

Abstract

We define a noncommutative analogue of invariant de Rham cohomology. More precisely, for a triple $(A,\mathcal{H},M)$ consisting of a Hopf algebra $\mathcal{H}$, an $\mathcal{H}$-comodule algebra $A$, an $\mathcal{H}$-module $M$, and a compatible grouplike element $\sigma$ in $\mathcal{H}$, we define the cyclic module of invariant chains on $A$ with coefficients in $M$ and call its cyclic homology the invariant cyclic homology of $A$ with coefficients in $M$. We also develop a dual theory for coalgebras. Examples include cyclic cohomology of Hopf algebras defined by Connes-Moscovici and its dual theory. We establish various results and computations including one for the quantum group $SL(q,2)$.