On split regular BiHom-Poisson color algebras

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 LL 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 LL, 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 LL is the direct sum of the family of its simple (graded) ideals.

Keywords

• bihom-lie color algebra
• bihom-poisson algebra
• root space
• root system
• 17b75
• 17a60
• 17b22
• 17b65