Constructions and representation theory of BiHom-post-Lie algebras

Abstract

The main goal of this paper is to give some construction results of BiHom-post-Lie algebras which are a generalization of both post-Lie-algebras and Hom-post-Lie algebras. They are the algebraic structures behind the weighted $${\mathcal {O}}$$ -operator of BiHom-Lie algebras. They can be also regarded as the splitting into three parts of the structure of a BiHom-Lie-algebra. Moreover we develop the representation theory of BiHom-post-Lie algebras on a vector space V. We show that there is naturally an induced representation of its sub-adjacent Lie algebra. We give also all 2-dimensional BiHom-post-Lie algebras. We exhibit in this work some important examples of post-Lie algebras and Hom-post-Lie algebras.