Gerstenhaber structure of the trivial extension’s Hochschild cohomology

Determine explicitly the Gerstenhaber algebra structure on the Hochschild cohomology ring of the trivial extension of the Xu–Snashall algebra.

Background

The paper constructs a symmetric counterexample to the Snashall–Solberg conjecture by taking the trivial extension of the Xu–Snashall algebra. Although the Gerstenhaber algebra structure of the Xu–Snashall algebra’s Hochschild cohomology had been computed in related work, the corresponding structure for its trivial extension was not determined.

The authors explicitly describe this computation as difficult and report that their attempts, including the use of AI assistants, were unsuccessful. Thus, an explicit description of the cup product and Gerstenhaber bracket for the Hochschild cohomology of the trivial extension remains unresolved in the paper.