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Register a new userThe homotopy Gerstenhaber algebra of Hochschild cochains of a regular algebra is formal
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The solution of Delignes conjecture on Hochschild cochains and the formality of the operad of little disks provide us with a natural homotopy Gerstenhaber algebra structure on the Hochschild cochains of an associative algebra. In this paper we construct a natural chain of quasi-isomorphisms of homotopy Gerstenhaber algebras between the Hochschild cochain complex C(A) of a regular commutative algebra A over a field of characteristic zero and the Gerstenhaber algebra of multiderivations of A. Unlike the original approach of the second author based on the computation of obstructions our method allows us to avoid the bulky Gelfand-Fuchs trick and prove the formality of the homotopy Gerstenhaber algebra structure on the sheaf of polydifferential operators on a smooth algebraic variety, a complex manifold, and a smooth real manifold.
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The Kontsevich-Soibelman solution of the cyclic version of Delignes conjecture and the formality of the operad of little discs on a cylinder provide us with a natural homotopy calculus structure on the pair (C^*(A), C_*(A)) ``Hochschild cochains + Hochschild chains of an associative algebra A. We show that for an arbitrary smooth algebraic variety X with the structure sheaf O_X the sheaf (C^*(O_X), C_*(O_X)) of homotopy calculi is formal. This result was announced in paper [29] by the second and the third author.
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 prove the formality theorem for the differential graded Lie algebra module of Hochschild chains for the algebra of endomorphisms of a smooth vector bundle. We discuss a possible application of this result to a version of the algebraic index theorem for Poisson manifolds.
We introduce a notion of the De Rham complex of a Gerstenhaber algebra which produces a notion of a quasi-BV structure, and allows to classify these structures, generalizing the classical results for polyvector fields.
We discuss some aspects of the representation theory of the deformed Virasoro algebra $virpq$. In particular, we give a proof of the formula for the Kac determinant and then determine the center of $virpq$ for $q$ a primitive N-th root of unity. We derive explicit expressions for the generators of the center in the limit $t=qp^{-1}to infty$ and elucidate the connection to the Hall-Littlewood symmetric functions. Furthermore, we argue that for $q=sqrtN{1}$ the algebra describes `Gentile statistics of order $N-1$, i.e., a situation in which at most $N-1$ particles can occupy the same state.
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