On split regular BiHom-Poisson color algebras

Yaling Tao,Yan Cao

doi:10.1515/math-2021-0039

Abstract

Abstract The purpose of this paper is to introduce the class of split regular BiHom-Poisson color algebras, which can be considered as the natural extension of split regular BiHom-Poisson algebras and of split regular Poisson color algebras. Using the property of connections of roots for this kind of algebras, we prove that such a split regular BiHom-Poisson color algebra L L is of the form L = ⊕ [ α ] ∈ Λ / ∼ I [ α ] L={\oplus }_{\left[\alpha ]\in \Lambda \text{/} \sim }{I}_{\left[\alpha ]} with I [ α ] {I}_{\left[\alpha ]} a well described (graded) ideal of L L , satisfying [ I [ α ] , I [ β ] ] + I [ α ] I [ β ] = 0 \left[{I}_{\left[\alpha ]},{I}_{\left[\beta ]}]+{I}_{\left[\alpha ]}{I}_{\left[\beta ]}=0 if [ α ] ≠ [ β ] \left[\alpha ]\ne \left[\beta ] . In particular, a necessary and sufficient condition for the simplicity of this algebra is determined, and it is shown that L L is the direct sum of the family of its simple (graded) ideals.

Highlights

The interest in Poisson algebras has grown in the last few years, motivated especially by their applications in geometry and mathematical physics
Poisson algebras play a fundamental role in deformation of commutative associative algebras [1]
The cohomology group, deformation, tensor product and Γ-graded of Poisson algebras have been studied by many authors in [2,3,4,5]

Summary

Introduction

The interest in Poisson algebras has grown in the last few years, motivated especially by their applications in geometry and mathematical physics. The structure of different classes of split algebras such as split regular Hom-Poisson algebras, split regular Hom-Poisson color algebras, split regular BiHom-Lie superalgebras, split BiHom-Leibniz superalgebras and split Leibniz triple systems have been studied by using techniques of connections of roots (see for instance [14,15,16,17,18,19,20,21,22,23,24,25,26]). The purpose of this paper is to consider the decomposition and simplicity of split regular BiHom-Poisson color algebras by the techniques of connections of roots

Preliminaries

Decompositions

The simple components