Diagonalizable modular derivations on bigraded Poisson spaces and Koszul duality

For two bigraded spaces $k^{m\mathrm{\mid }\mathrm{\mid }n}$ and $k^{n\wedge m}$ equipped with appropriate quadratic Poisson structures, Koszul duality identifies the mixed Poisson differential calculi and, in the unimodular case, their Poisson cohomologies as Batalin–Vilkovisky algebras. In the current paper we prove that Koszul duality induces an isomorphism of Batalin–Vilkovisky algebras between their Poisson cohomologies when the modular derivation of one algebra is diagonalizable, including non-unimodular cases.