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 and 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 (P) = n-1. We characterize the Poisson algebras with (P) = n-2 over arbitrary fields. In particular, we characterize Lie algebras L with (L) = n-2. We also show that (P) = n-2 for nilpotent Poisson algebras implies (P) =n-2. Finally, we study these invariants for various distinguished Poisson algebras, providing us with several examples and counterexamples.