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arXiv · 2405.05859

Abelian Subalgebras and Ideals of Maximal Dimension in Poisson algebras

Amir Fernández Ouaridi·Rosa María Navarro·David A. Towers
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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.

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Amir Fernández Ouaridi, Rosa María Navarro, David A. Towers. 2024-05-09. Abelian Subalgebras and Ideals of Maximal Dimension in Poisson algebras. https://arxiv.org/abs/2405.05859

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