The structure of split regular Hom-Poisson algebras
Volume 145 / 2016
Abstract
We introduce the class of split regular Hom-Poisson algebras formed by those Hom-Poisson algebras whose underlying Hom-Lie algebras are split and regular. This class is the natural extension of the ones of split Hom-Lie algebras and of split Poisson algebras. We show that the structure theorems for split Poisson algebras can be extended to the more general setting of split regular Hom-Poisson algebras. That is, we prove that an arbitrary split regular Hom-Poisson algebra ${\mathfrak P}$ is of the form ${\mathfrak P}=U + \sum _{j}{I}_{j}$ with $U$ a linear subspace of a maximal abelian subalgebra $H$ and any ${I}_{j}$ a well described (split) ideal of ${\mathfrak P}$, satisfying $\{{ I}_j , { I}_k\}+{ I}_j { I}_k=0$ if $j\not =k$. Under certain conditions, the simplicity of ${\mathfrak P}$ is characterized, and it is shown that ${\mathfrak P}$ is the direct sum of the family of its simple ideals.
