Graded Lie Agebroids of Poisson Almost Commutative Algebras

Abstract

We introduce and study the notion of abelian groups graded Lie algebroid structures on almost commutative algebras $\mathcal A$ and show that any graded Poisson bracket on $\mathcal A$ induces a graded Lie algebroid structure on the $\mathcal A$-module of 1-forms on $\mathcal A$ as in the classical Poisson manifolds. We also derive from our formalism the graded Poisson cohomology.