Cohomology of idempotent braidings, with applications to factorizable monoids

TL;DR

This paper introduces new methods for computing Hochschild (co)homology of monoids derived from idempotent set-theoretic solutions to the Yang-Baxter equation, linking these invariants to the underlying solutions and expanding understanding of their structure.

Contribution

It develops a framework connecting monoid (co)homology with YBE solutions, including a generalization of the Künneth formula and explicit maps for computation.

Findings

Hochschild (co)homology of certain monoids is identified with that of YBE solutions.

A generalized Künneth formula for factorizable monoids is established.

New structural insights into the (co)homology of YBE solutions are obtained.

Abstract

We develop new methods for computing the Hochschild (co)homology of monoids which can be presented as the structure monoids of idempotent set-theoretic solutions to the Yang--Baxter equation. These include free and symmetric monoids; factorizable monoids, for which we find a generalization of the K{\"u}nneth formula for direct products; and plactic monoids. Our key result is an identification of the (co)homologies in question with those of the underlying YBE solutions, via the explicit quantum symmetrizer map. This partially answers questions of Farinati--Garc{\'i}a-Galofre and Dilian Yang. We also obtain new structural results on the (co)homology of general YBE solutions.