| Résume||For a commutative smooth $k$-algebra $A$, the Hochschild-Konstant-Rosenberg theorem identifies Hochschild homology of $A$ with forms $\Lambda^*(\Omega(A))$ on $A$ and Hochschild cohomology of $A$ with polyvector fields $\Lambda^*(Der(A))$ on $A$. These two new spaces have a very rich algebraic structure coming from the geometric Cartan calculus on smooth varieties: a Lie bracket on fields, an exterior product on forms and polyvector fields, an interior product of forms with fields and a deRham differential.
There is a non-commutative counterpart to this story for an associative algebra $A$, introduced by B. Tsygan and D. Tamarkin, now called the Tamarkin-Tsygan calculus of $A$. I will explain how to compute it through a dg model of $A$, by giving first a model for the spaces of forms and fields that give rise to homology and cohomology, and then provide explicit formulas for the Gerstenhaber bracket and cup product on fields, the contraction of forms by fields and the boundary of $A$. Connes on forms. This extends the work of J. Stasheff - who originally gave a definition of the Gerstenhaber bracket of $A$ intrinsic to the category of $dg$ algebras - to the whole Tamarkin-Tsygan calculus of an algebra.|