Higher Order Hochschild (Co)homology of Noncommutative Algebras

TL;DR

This paper extends higher order Hochschild (co)homology to noncommutative associative algebras and shows that such a generalization of simplicial sets is limited to one-dimensional cases, broadening the scope of algebraic tools.

Contribution

It introduces a generalization of higher order Hochschild (co)homology to noncommutative algebras and characterizes the simplicial sets suitable for this extension.

Findings

Higher order Hochschild (co)homology can be generalized to noncommutative algebras.

Simplicial sets admit such a generalization only if they are one-dimensional.

Abstract

Hochschild (co)homology and Pirashvili's higher order Hochschild (co)homology are useful tools for a variety of applications including deformations of algebras. When working with higher order Hochschild (co)homology, we can consider the (co)homology of any commutative algebra with symmetric coefficient bimodules, however traditional Hochschild (co)homology is able to be computed for any associative algebra with not necessarily symmetric coefficient bimodules. In a previous paper, the author generalized higher order Hochschild cohomology for multimodule coefficients (which need not be symmetric). In the current paper, we continue to generalize higher order Hochschild (co)homology to work with associative algebras which need not be commutative and in particular, show that simplicial sets admit such a generalization if and only if they are one dimensional.