No Arabic abstract
We give BV-structures full description for case of even dimension $k ge 3$ on Hochschild cohomology for exeptional local algebras of quaternion type, defined by parameters $(0,d)$, according to Erdmann classification. This article states as a generalization of similar results about $BV$-structures on quaternion type algebras.
Consider a smooth affine algebraic variety $X$ over an algebraically closed field, and let a finite group $G$ act on it. We assume that the characteristic of the field is greater than the dimension of $X$ and the order of $G$. An explicit formula for multiplication on the Hochschild cohomology of a crossed product of $k[G]$ and $k[X]$ is given in terms of multivector fields on $X$ and $g$-invariant subvarieties of $X$ for $gin G$.
In this paper we investigate the functoriality properties of map-graded Hochschild complexes. We show that the category MAP of map-graded categories is naturally a stack over the category of small categories endowed with a certain Grothendieck topology of 3-covers. For a related topology of infinity-covers on the cartesian morphisms in MAP, we prove that taking map-graded Hochschild complexes defines a sheaf. From the functoriality related to injections between map-graded categories, we obtain Hochschild complexes with support. We revisit Kellers arrow category argument from this perspective, and introduce and investigate a general Grothendieck construction which encompasses both the map-graded categories associated to presheaves of algebras and certain generalized arrow categories, which together constitute a pair of complementary tools for deconstructing Hochschild complexes.
We describe the Gerstenhaber algebra structure on the Hochschild cohomology HH*$(A)$ when $A$ is a quadratic string algebra. First we compute the Hochschild cohomology groups using Barzdells resolution and we describe generators of these groups. Then we construct comparison morphisms between the bar resolution and Bardzells resolution in order to get formulae for the cup product and the Lie bracket. We find conditions on the bound quiver associated to string algebras in order to get non-trivial structures.
We compute the Hochschild cohomology groups $HH^*(A)$ in case $A$ is a triangular string algebra, and show that its ring structure is trivial.
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 Gerstenhabers 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-infinity-algebra with a symmetric and non-degenerate infinity-inner product.