The structure of split regular Hom-Poisson algebras

María J. Aragón Periñán ¹   ; Antonio J. Calderón Martín ¹

¹ Department of Mathematics Faculty od Sciences University of Cádiz 11510 Puerto Real, Cádiz, Spain

Colloquium Mathematicum, Tome 145 (2016) no. 1, pp. 1-13

Résumé

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.

Détail

DOI : 10.4064/cm6568-9-2015

Keywords: introduce class split regular hom poisson algebras formed those hom poisson algebras whose underlying hom lie algebras split regular class natural extension split hom lie algebras split poisson algebras structure theorems split poisson algebras extended general setting split regular hom poisson algebras prove arbitrary split regular hom poisson algebra mathfrak form mathfrak sum linear subspace maximal abelian subalgebra described split ideal mathfrak satisfying under certain conditions simplicity mathfrak characterized shown mathfrak direct sum family its simple ideals