Abelian Subalgebras and Ideals of Maximal Dimension in Poisson algebras

TL;DR

This paper investigates the maximal dimensions of abelian subalgebras and ideals in Poisson algebras, introducing invariants to classify their structure and characterizing cases with specific invariant values.

Contribution

It introduces invariants for Poisson algebras, characterizes algebras with certain invariant values, and explores their properties in nilpotent cases and various examples.

Findings

Invariants nd  coincide if  = n-1.

Poisson algebras with  = n-2 are characterized over arbitrary fields.

Nilpotent Poisson algebras with  = n-2 have  = n-2.

Abstract

This paper studies the abelian subalgebras and ideals of maximal dimension of Poisson algebras $\mathcal{P}$ of dimension $n$. We introduce the invariants $\alpha$ and $\beta$ for Poisson algebras, which correspond to the dimension of an abelian subalgebra and ideal of maximal dimension, respectively. We prove that these invariants coincide if $\alpha(\mathcal{P}) = n-1$. We characterize the Poisson algebras with $\alpha(\mathcal{P}) = n-2$ over arbitrary fields. In particular, we characterize Lie algebras $L$ with $\alpha(L) = n-2$. We also show that $\alpha(\mathcal{P}) = n-2$ for nilpotent Poisson algebras implies $\beta(\mathcal{P})=n-2$. Finally, we study these invariants for various distinguished Poisson algebras, providing us with several examples and counterexamples.