Poisson $n$-Lie algebras: constructions and the structure of solvable algebras

TL;DR

This paper develops constructions and structural analysis of Poisson n-Lie algebras, establishing their relationships with n-Lie algebras of Jacobians, and characterizes solvable and nilpotent cases.

Contribution

It introduces new methods to construct Poisson n-Lie algebras from n-Lie Jacobian algebras and explores their solvability, nilpotency, and ideal structure.

Findings

Constructed Poisson n-Lie algebras from n-Lie Jacobian algebras.

Established conditions for when these constructions yield valid Poisson n-Lie algebras.

Characterized solvable and nilpotent Poisson n-Lie algebras and introduced hypo-nilpotent ideals.

Abstract

In this paper, we develop a construction of Poisson $n$-Lie algebras arising from $n$-Lie algebras of Jacobians and establish conditions under which this construction yields a Poisson $n$-Lie algebra. We also formulate a general conjecture in the unital case. In addition, we show that tensor products of Poisson algebras admit natural Poisson $n$-Lie structures via suitable quotient constructions. Conversely, we construct a Poisson algebra from a given Poisson $n$-Lie algebra, thereby establishing a correspondence between these classes of algebras. Furthermore, we obtain analogues of Engel's and Lie's theorems and provide a characterization of solvable and nilpotent Poisson $n$-Lie algebras in terms of the underlying algebraic structures. We also introduce the notion of hypo-nilpotent ideals and prove results concerning maximal hypo-nilpotent ideals in finite-dimensional solvable Poisson $n$-Lie algebras. Finally, we show that generalized eigenspaces of multiplication operators form ideals.