Gerstenhaber brackets on Hochschild cohomology of twisted tensor products

TL;DR

This paper develops a method to define Gerstenhaber brackets on Hochschild cohomology for twisted tensor products of algebras and demonstrates its application through explicit computations for quantum complete intersections.

Contribution

It introduces a construction of Gerstenhaber brackets for Hochschild cohomology of twisted tensor products and establishes an isomorphism for certain subalgebras with trivial twisting.

Findings

Gerstenhaber brackets constructed for twisted tensor products

Explicit computations for quantum complete intersections

Subalgebra isomorphism as Gerstenhaber algebras

Abstract

We construct the Gerstenhaber bracket on Hochschild cohomology of a twisted tensor product of algebras, and, as examples, compute Gerstenhaber brackets for some quantum complete intersections arising in work of Buchweitz, Green, Madsen, and Solberg. We prove that a subalgebra of the Hochschild cohomology ring of a twisted tensor product, on which the twisting is trivial, is isomorphic, as Gerstenhaber algebras, to the tensor product of the respective subalgebras of the Hochschild cohomology rings of the factors.