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

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 ${\mathrm{HH}}^0(A)$ and elements of ${\mathrm{HH}}^n(A)$ for $n\in {\mathbb{N}}_0$, the method by M. Suárez-Álvarez [J. Pure Appl. Algebra 221, No. 8, 1981–1998 (2017; Zbl 1392.16009)] to calculate the Gerstenhaber bracket between elements of ${\mathrm{HH}}^1(A)$ and elements of ${\mathrm{HH}}^n(A)$ 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 ${\mathrm{HH}}^{\bullet }(A)$ is not induced by any Batalin–Vilkovisky generator.