On the Hochschild homology of involutive algebras

TL;DR

This paper explores the homological properties of involutive associative algebras, establishing that involutive Hochschild cohomology computes Z/2-invariants intersected with the center, and introduces a related Hochschild homology theory.

Contribution

It proves that Braun's involutive Hochschild cohomology is a derived functor and defines a new involutive Hochschild homology theory as a derived functor involving Z/2-coinvariants.

Findings

Braun's involutive Hochschild cohomology equals Z/2-invariants intersected with the center.

Introduces a new involutive Hochschild homology as a derived functor.

Provides a description of the homology via pushout of Z/2-coinvariants and abelianization.

Abstract

We study the homological algebra of bimodules over involutive associative algebras. We show that Braun's definition of involutive Hochschild cohomology in terms of the complex of involution-preserving derivations is indeed computing a derived functor: the Z/2-invariants intersected with the center. We then introduce the corresponding involutive Hochschild homology theory and describe it as the derived functor of the pushout of Z/2-coinvariants and abelianization.