Abstract:
In this talk, we show under what additional ingredients a left module in
negative degrees 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 calculus structure (or Batalin-Vilkovisky module) over the
cohomology of the operad, which is in some sense a dual picture to the
relationship between cyclic operads and Batalin-Vilkovisky algebras. In
particular, we obtain explicit expressions for a generalised Lie
derivative as well as a generalised (cyclic) cap product that obey a
Cartan-Rinehart homotopy formula. Examples include the calculi known for
the Hochschild theory of associative algebras, for Poisson structures, but
above all the calculus for a general left Hopf algebroid with respect to
general coefficients (in which the classical calculus of vector fields and
differential forms is contained).