Abelian Subalgebras and Ideals of Maximal Dimension in Poisson algebras

This paper studies the abelian subalgebras and ideals of maximal dimension of Poisson algebras P of dimension n. We introduce the invariants α and β 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 α(P) = n-1. We characterize the Poisson algebras with α(P) = n-2 over arbitrary fields. In particular, we characterize Lie algebras L with α(L) = n-2. We also show that α(P) = n-2 for nilpotent Poisson algebras implies β(P)=n-2. Finally, we study these invariants for various distinguished Poisson algebras, providing us with several examples and counterexamples.