The decomposition of Hochschild cohomology and Gerstenhaber operations

Gerstenhaber Brackets on Hochschild Cohomology of Twisted Tensor Products

We construct the Gerstenhaber bracket on Hochschild cohomology of a twisted tensor product of algebras, and, as examples, compute Gerstenhaber brackets for some quantum complete intersections arising in work of Buchweitz, Green, Madsen, and Solberg. We prove that a subalgebra of the Hochschild cohomology ring of a twisted tensor product, on which the twisting is trivial, is isomorphic, as a Ger...

Dyer-Lashof-Cohen operations in Hochschild cohomology

In the paper we give explicit formulae for operations in Hochschild cohomology which are analogous to the operations in the homology of double loop spaces. As a corollary we obtain that any brace algebra in finite characteristics is always a restricted Lie algebra.

Crossed Interval Groups and Operations on the Hochschild Cohomology

We introduce crossed interval groups and construct a crossed interval analog IS of the Fiedorowicz-Loday symmetric category ∆S. We prove that the functor FS(−) of the free IS-extension of an I-object does not change homotopy type. We then observe that the operad B of natural operations on the Hochschild cohomology equals FS(T ), where T is an operad whose homotopy type is known. We conclude fro...

Gerstenhaber Brackets on Hochschild Cohomology of Quantum Symmetric Algebras and Their Group Extensions

We construct chain maps between the bar and Koszul resolutions for a quantum symmetric algebra (skew polynomial ring). This construction uses a recursive technique involving explicit formulae for contracting homotopies. We use these chain maps to compute the Gerstenhaber bracket, obtaining a quantum version of the Schouten-Nijenhuis bracket on a symmetric algebra (polynomial ring). We compute b...

Second Hochschild Cohomology of Convolution Algebras