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The structure of split regular Hom-Poisson algebras

Volume 145 / 2016

María J. Aragón Periñán, Antonio J. Calderón Martín Colloquium Mathematicum 145 (2016), 1-13 MSC: Primary 17A30, 17B63; Secondary 17A60, 17B22. DOI: 10.4064/cm6568-9-2015 Published online: 10 March 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.

Authors

  • María J. Aragón PeriñánDepartment of Mathematics
    Faculty od Sciences
    University of Cádiz
    11510 Puerto Real, Cádiz, Spain
    e-mail
  • Antonio J. Calderón MartínDepartment of Mathematics
    Faculty od Sciences
    University of Cádiz
    11510 Puerto Real, Cádiz, Spain
    e-mail

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