Gerstenhaber structure on Hochschild cohomology of the Fomin-Kirillov algebra on 3 generators

TL;DR

This paper computes the Gerstenhaber bracket on the Hochschild cohomology of the Fomin-Kirillov algebra on three generators, introducing a new method for calculating brackets and showing it is not derived from a Batalin-Vilkovisky structure.

Contribution

It introduces a general method for computing Gerstenhaber brackets between Hochschild cohomology elements and applies it to the Fomin-Kirillov algebra, revealing new structural insights.

Findings

Computed the Gerstenhaber bracket for the algebra's Hochschild cohomology.

Developed a method to efficiently calculate brackets between specific cohomology groups.

Showed the bracket is not induced by any Batalin-Vilkovisky generator.

Abstract

The goal of this article is to compute the Gerstenhaber bracket of the Hochschild cohomology of the Fomin-Kirillov algebra on three generators over a field of characteristic different from $2$ and $3$. This is in part based on a general method we introduce to easily compute the Gerstenhaber bracket between elements of $\operatorname{HH}^{0}(A)$ and elements of $\operatorname{HH}^{n}(A)$ for $n \in \mathbb{N}_{0}$, the method by M. Su\'arez-\'Alvarez to calculate the Gerstenhaber bracket between elements of $\operatorname{HH}^{1}(A)$ and elements of $\operatorname{HH}^{n}(A)$ for any $n \in \mathbb{N}_{0}$, as well as an elementary result that allows to compute the remaining brackets from the previous ones. We also show that the Gerstenhaber bracket of $\operatorname{HH}^{\bullet}(A)$ is not induced by any Batalin-Vilkovisky generator.