On split regular Hom-Lie superalgebras

We introduce the class of split regular Hom-Lie superalgebras as the natural extension of the one of split Hom-Lie algebras and Lie superalgebras, and study its structure by showing that an arbitrary split regular Hom-Lie superalgebra ${\frak L}$ is of the form ${\frak L} = U + \sum_j I_j$ with $U$ a linear subspace of a maximal abelian graded subalgebra $H$ and any $I_j$ a well described (split) ideal of ${\frak L}$ satisfying $[I_j,I_k] = 0$ if $j \neq k$. Under certain conditions, the simplicity of ${\frak L}$ is characterized and it is shown that ${\frak L}$ is the direct sum of the family of its simple ideals.