On the solvable Poisson algebras

Abstract

In this manuscript, we study nilpotent and solvable Poisson algebras of dimension n. In the first part, we establish classical results such as Engel's theorem and Lie's theorem for Poisson algebras, and we examine the role of idempotents in these algebras. We also address the construction of nilpotent and solvable Poisson algebras, exploring the existence of Poisson algebras associated with a fixed Lie algebra, the study of filiform Poisson algebras, and constructions involving the tensor product and generalized Jacobians. Furthermore, we show that, under mild restrictions, the solvability and nilpotency of a Poisson algebra are essentially determined by those of the Lie bracket. This motivates a deeper investigation into Poisson algebra structures on solvable Lie algebras. In the second part, we provide criteria for the non existence of Poisson algebra structures on solvable extensions of nilpotent Lie algebras by a torus. In particular, we prove that complete solvable Lie algebras do not admit a Poisson algebra structure, and other related results. Additionally, we present results and examples illustrating the diversity of Poisson algebras arising in solvable Lie algebras that are non-maximal solvable extensions of nilpotent Lie algebras and highlighting the difficulty in formulating a unified criterion for all solvable Lie algebras.