Graded Regular BiHom-Lie Algebras

In the current work, we introduce the class of graded regular BiHom-Lie algebras as a natural extension of the class of graded Lie algebras, and hence of split Lie algebras. In particular, we show that an arbitrary graded regular BiHom-Lie algebra [Formula: see text] can be expressed as [Formula: see text], where [Formula: see text] is a linear subspace in [Formula: see text], 1 being the neutral element of the grading group, and any [Formula: see text] a well-described (graded) ideal of [Formula: see text], satisfying [Formula: see text] if [Formula: see text]. Moreover, under some conditions, we characterize the simplicity of [Formula: see text] and we show that [Formula: see text] is the direct sum of the family of its simple (graded) ideals.