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.

16E40, 18G10, 16S37

Fomin-Kirillov algebra, Hochschild cohomology, Gerstenhaber bracket

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Institut Fourier, UMR 5582, Laboratoire de Mathématiques, Université Grenoble Alpes, CS 40700, 38058, Grenoble cedex 9, France

Institut Fourier, UMR 5582, Laboratoire de Mathématiques, Université Grenoble Alpes, CS 40700, 38058, Grenoble cedex 9, France