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

Herscovich, Estanislao ; Li, Ziling

Documenta mathematica, Tome 27 (2022), pp. 1773-1804.

Résumé

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 \textit{M. Suárez-Álvarez} [J. Pure Appl. Algebra 221, No. 8, 1981--1998 (2017; Zbl 1392.16009)] 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.

Détail

Classification : 16E40, 18G10, 16S37
Keywords: Fomin-Kirillov algebra, Hochschild cohomology, Gerstenhaber bracket