Algebra/Topology seminar

Speaker: Johan Alm

Title: Noncommutative Gerstenhaber algebra structures on Hochschild cohomology

Abstract: The Hochschild cochain complex of an associative algebra is itself both a dg associative algebra and a dg Lie algebra. Moreover, the Lie algebra has a representation via the so-called braces operations as a Lie algebra of derivations of the product. On the Hochschild cohomology these operations induce the usual cup product, the Gerstenhaber Lie bracket, and the adjoint action of the Gerstenhaber bracket. We have shown that the structure on the Hochschild cochain complex is not  formal: there is (generically) no strong homotopy quasi-isomorphism between it and the corresponding structure on the Hochschild cohomology. Thus, the triple of operations (product, bracket, adjoint action) on Hochschild cohomology should be augmented by higher operations. We will discuss how this relates to Kontsevich's and Tamarkin's formality theorems and, also, how to write explicit formulas for the higher homotopies. The results have connections to deformation quantization, string topology, mirror symmetry, multiple zeta values and the Grothendieck-Teichmüller group.