Batalin-Vilkovisky structure on Hochschild cohomology of tame Hecke algebras of type \(\mathbf{A}\)
Hecke algebras appear as endomorphism algebras of reduced representations of finite or \(p\)-adic Chevalley groups. The Hochschild cohomology of an associative algebra \(B\), denoted by \(\mathrm{HH}^*(B)\), is an important invariant that is intimately related to simple connectedness, separability and deformation theory. Hochschild cohomology has a rich structure: it is a Gerstenhaber algebra. In particular, it is a graded commutative algebra under the cup product and a graded Lie algebra under the Gerstenhaber bracket. In addition, for some families of algebras, such as symmetric or Calabi-Yau algebras, Hochschild cohomology is a Batalin-Vilkovisky algebra. Let \(A\) be a tame Hecke algebra of type \(\mathbf{A}\). Recently, part of the structure of the Hochschild cohomology of \(A\) have been studied. For example, in [\textit{Y. Xu} and \textit{T. Zhao}, Manuscr. Math. 142, No. 3--4, 491--512 (2013; Zbl 1281.16017)] the graded commutative algebra structure of \(\mathrm{HH}^{\ast}(A)\) has been described. In this paper, the authors calculate the Gerstenhaber bracket and the Batalin-Vilkovisky structure on the Hochschild cohomology of \(A\). In order to compute such structures, the authors construct two comparison maps between the minimal projective bimodule resolution and the reduced bar resolution of \(A\). Using these maps and the method of bilinear form given by \textit{T. Tradler} [Ann. Inst. Fourier 58, No. 7, 2351--2379 (2008; Zbl 1218.16004)], the authors calculate the Batalin-Vilkovisky operator, and then deduce the Gerstenhaber bracket.
scientific article; zbMATH DE number 7595354
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