We show that there is an action of the symmetric group on Hochschild cochain complex a twisted algebra with coefficients in bimodule. This allows us to define cohomology algebras, similarly th construction due Staic. give explicit embeddings and connecting homomorphisms between spaces algebras.

Abstract Coisotropic algebras consist of triples for which a reduction can be defined and unify in very algebraic fashion coisotropic several settings. In this paper, we study the theory (formal) deformation showing that deformations are governed by suitable DGLAs. We define functor prove it commutes with reduction. Finally, obstructions to existence uniqueness present some geometric examples.

We show that if (M,⊗, I) is a monoidal model category then REnd M (I) is a (weak) 2-monoid in sSet. This applies in particular when M is the category of A-bimodules over a simplicial monoid A: the derived endomorphisms of A then form its Hochschild cohomology, which therefore becomes a simplicial 2-monoid.

In this paper, we compute the Gerstenhaber bracket on the Hochschild cohomology of C∞(M)⋊Γ. Using this computation, we classify all the noncommutative Poisson structures on C∞(M) ⋊ Γ when M is a symplectic manifold. We provide examples of deformation quantizations of these noncommutative Poisson structures.