Hello to all.

There is a well-known formalism in deformation-quantization which puts the algebraic structure of polyvector fields in a noncommutative setting. Tamarkin-Tsygan define a (pre)calculus to be the data $(G, M, L, i)$ where $(G,.,[,])$ is a Gerstenhaber algebra, $M$ is a complex, $i$ is an action of the graded algebra $(G,.)$ onto the graded module $M$ and $L$ an action of the graded Lie algebra $G[1], [,]$ onto the graded module $M$.

These two actions have to satisfy the following two compatibilities: $i_{[a,b]}= [L_a, i_b]$ and $L_{ab}= L_ai_b+(-1)^{\vert a\vert}i_aL_b$. Is there a way to interpret this second rule in a more natural way or does one just impose this rule because it is satisfied by the protopical examples (ie. polyvector fields and hochschild (co)homology)?

as a commutative algebra? $\endgroup$