  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.
