I will discuss the algebraic structure of homology theories defined by a left Hopf algebroid U
over a possibly noncommutative base algebra A, such as for example Hochschild, Lie algebroid
(in particular Lie algebra and Poisson), or group and étale groupoid (co)homology:
Ext_U(A,A) is a Gerstenhaber algebra that acts via a generalised Lie derivative and cap product
on Tor^U(M,A), and there is a generalised Cartan homotopy formula that relates the Lie derivative
to cyclic homology. As an application, Ginzburg's result that the cohomology ring of a Calabi–Yau
algebra is a Batalin–Vilkovisky algebra is extended to twisted Calabi–Yau algebras.
(joint work with Niels Kowalzig)