The Batalin-Vilkovisky Algebra on Hochschild Cohomology Induced by Infinity Inner Products

Tradler, Thomas

Annales de l'Institut Fourier, Tome 58 (2008) no. 7, pp. 2351-2379.

On définit une structure de BV sur la cohomologie de Hochschild d’une algèbre associative unitaire munie d’une forme bilinéaire symétrique non dégénérée. La structure d’algèbre de Gerstenhaber induite est celle introduite dans l’article originel de Gerstenhaber sur la cohomologie de Hochschild. On étend ce résultat au cas d’une algèbre $A$-infinie unitaire munie d’une forme bilinéaire symétrique $A$-infinie non dégénérée.

We define a BV-structure on the Hochschild cohomology of a unital, associative algebra $A$ with a symmetric, invariant and non-degenerate inner product. The induced Gerstenhaber algebra is the one described in Gerstenhaber’s original paper on Hochschild-cohomology. We also prove the corresponding theorem in the homotopy case, namely we define the BV-structure on the Hochschild-cohomology of a unital $A_{\infty }$-algebra with a symmetric and non-degenerate $\infty$-inner product.

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