Permutahedra, HKR isomorphism and polydifferential Gerstenhaber-Schack complex

This paper aims to give a short but self-contained introduction into the theory of (wheeled) props, properads, dioperads and operads, and illustrate some of its key ideas in terms of a prop(erad)ic interpretation of simplicial and permutahedra cell complexes with subsequent applications to the HochschildKostant-Rosenberg type isomorphisms. Let V be a graded vector space over a field K and OV := •V ∗ the free graded commutative algebra generated by the dual vector space V ∗ := HomK(V,K). One can interpret OV as the algebra of polynomial functions on the space V . The classical Hochschild-Kostant-Rosenberg theorem asserts that the Hochschild cohomology of OV (with coefficients in OV ) is isomorphic to the space, ∧TV , of polynomial polyvector fields on V which in turn is isomorphic as a vector space to ∧•V ⊗ •V ∗,