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

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. Suárez-Álvarez 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.