Hochschild cohomology rings of tame Hecke algebras

Abstract

Let A be a tame Hecke algebra of type A. Based on the minimal projective bimodule resolution \({(\mathbb{P} , \delta)}\) of A constructed by Schroll and Snashall, we first give an explicit description of the so-called “comultiplicative structure” of the generators of each term Pn in \({(\mathbb{P} , \delta)}\) , and then apply it to define a chain map \({\Delta: \mathbb{P} \rightarrow \mathbb{P} \otimes_A \mathbb{P}}\) and thus show that the cup product in the level of cochains for the tame Hecke algebra A is essentially juxtaposition of parallel paths up to sign. As a consequence, we determine the structure of the Hochschild cohomology ring of A under the cup product by giving an explicit presentation by generators and relations.