Batalin–Vilkovisky algebra structures on (Co)Tor(Co)Tor and Poisson bialgebroids

In this article, we extend our preceding studies on higher algebraic structures of (co)homology theories defined by a left bialgebroid (U,A)(U,A). For a braided commutative Yetter–Drinfel'd algebra N  , explicit expressions for the canonical Gerstenhaber algebra structure on ExtU(A,N)ExtU(A,N) are given. Similarly, if (U,A)(U,A) is a left Hopf algebroid where A is an anti-Yetter–Drinfel'd module over U  , it is shown that the cochain complex computing CotorU(A,N)CotorU(A,N) defines a cyclic operad with multiplication and hence the groups CotorU(A,N)CotorU(A,N) form a Batalin–Vilkovisky algebra. In the second part of this article, Poisson structures and the Poisson bicomplex for bialgebroids are introduced, which simultaneously generalise, for example, classical Poisson as well as cyclic homology. In case the bialgebroid U is commutative, a Poisson structure on U   leads to a Batalin–Vilkovisky algebra structure on TorU(A,A)TorU(A,A). As an illustration, we show how this generalises the classical Koszul bracket on differential forms, and conclude by indicating how classical Lie–Rinehart bialgebras (or, geometrically, Lie bialgebroids) arise from left bialgebroids.