A detailed look on actions on Hochschild complexes especially the degree $1$ co-product and actions on loop spaces

TL;DR

This paper revisits Hochschild actions on complexes, clarifies the coproduct's properties, and extends the understanding of loop space operations, especially in the context of graded Gorenstein Frobenius algebras.

Contribution

It provides a new perspective on Hochschild actions, removing the commutativity assumption, and extends the coproduct's well-definedness to broader algebraic contexts.

Findings

Coproduct well-defined modulo constant loops

Extension to graded Gorenstein Frobenius algebras

Homotopy of anti-symmetrized coproduct

Abstract

We explain our previous results about Hochschild actions [Kau07a, Kau08a] pertaining in particular to the coproduct which appeared in a different form in [GH09] and provide a fresh look at the results. We recall the general action, specialize to the aforementioned coproduct and prove that the assumption of commutativity, made for convenience in [Kau08a], is not needed. We give detailed background material on loop spaces, Hochschild complexes and dualizations, and discuss details and extensions of these techniques which work for all operations of [Kau07a, Kau08a]. With respect to loop spaces, we show that the co--product is well defined modulo constant loops and going one step further that in the case of a graded Gorenstein Frobenius algebra, the co--product is well defined on the reduced normalized Hochschild complex. We discuss several other aspects such as ``time reversal'' duality and several homotopies of operations induced by it. This provides a cohomology operation which is a homotopy of the anti--symmetrization of the coproduct. The obstruction again vanishes on the reduced normalized Hochschild complex if the Frobenius algebra is graded Gorenstein. Further structures such as ``animation'' and the BV structure and a coloring for operations on chains and cochains and a Gerstenhaber double bracket are briefly treated.