Mixed vs Stable Anti-Yetter-Drinfeld Contramodule

TL;DR

This paper investigates the differences between mixed and stable anti-Yetter-Drinfeld contramodules in the context of cyclic homology, demonstrating that these two categories are not equivalent through an example involving Sweedler's Hopf algebra.

Contribution

It provides a concrete example showing the non-equivalence of mixed and stable anti-Yetter-Drinfeld contramodules, clarifying their distinct roles in cyclic homology.

Findings

Mixed complexes in stable anti-Yetter-Drinfeld contramodules are not equivalent to mixed anti-Yetter-Drinfeld contramodules.

Sweedler's Hopf algebra serves as a counterexample illustrating the difference.

The categories have distinct differential graded structures.

Abstract

We examine the cyclic homology of the monoidal category of modules over a finite dimensional Hopf algebra, motivated by the need to demonstrate that there is a difference between the recently introduced mixed anti-Yetter-Drinfeld contramodules and the usual stable anti-Yetter-Drinfeld contramodules. Namely, we show that Sweedler's Hopf algebra provides an example where mixed complexes in the category of stable anti-Yetter-Drinfeld contramodules (previously studied) are not equivalent, as differential graded categories to the category of mixed anti-Yetter-Drinfeld contramodules (recently introduced).