Gerstenhaber and Batalin-Vilkovisky Structures on Modules over Operads

In this article, we show under what additional ingredients a comp (or opposite) module over an operad with multiplication can be given the structure of a cyclic k-module and how the underlying simplicial homology gives rise to a Batalin–Vilkovisky module over the cohomology of the operad. In particular, one obtains a generalized Lie derivative and a generalized (cyclic) cap product that obey a Cartan–Rinehart homotopy formula, and hence yield the structure of a noncommutative differential calculus in the sense of Nest, Tamarkin, Tsygan, and others. Examples include the calculi known for the Hochschild theory of associative algebras, for Poisson structures, but above all the calculus for general left Hopf algebroids with respect to general coefficients (in which the classical calculus of vector fields and differential forms is contained).