Nilpotency, Solvability and Frattini Theory for Poisson algebras

TL;DR

The paper proves that a finite-dimensional Poisson algebra is nilpotent iff it is both associative-nilpotent and Lie-nilpotent, establishing the existence of the nilradical and relating it to the solvable radical via $R_A^2\subseteq N(\mathcal{P})$ (with $R^2$ nilpotent in characteristic zero). It develops a Frattini theory for dialgebras and specializes it to Poisson algebras, identifying how Frattini ideals govern nilpotency, solvability, and structural decompositions, aided by Engel subalgebras and the $S_{\mathcal{P}}(a,\mathbb{F})$ framework. The results provide splitting criteria and a classification of Poisson algebras where every maximal subalgebra is an ideal, linking radical theory with Frattini theory in the Poisson/dialgebra setting and offering tools for representation and deformation contexts.

Abstract

This paper shows that a Poisson algebra is nilpotent if and only if it is both associative and Lie nilpotent and examines various properties of the nilradical and the solvable radical. It introduces a basic Frattini theory for dialgebras and then investigates a more detailed theory for Poisoon algebras.