Batalin-Vilkovisky structures on the cohomologies of tensor Poisson algebras

It has been proved that the Poisson cohomology ring of a Poisson algebra has a Batalin-Vilkovisky algebra structure iff the Poisson algebra is pseudo-unimodular. In this paper, it is proved that the tensor algebra of two pseudo-unimodular Poisson algebras is also pseudo-unimodular, thus its Poisson cohomology ring is still a Batalin-Vilkovisky algebra. Furthermore, This Batalin-Vilkovisky algebra is isomorphic to the tensor Batalin-Vilkovisky algebra of the respective Poisson cohomology rings of the two pseudo-unimodular Poisson algebras.