We introduce the class of split regular Hom-Leibniz algebras as the natural generalization of split Leibniz algebras and split regular Hom-Lie algebras. By developing techniques of connections of roots for this kind of algebras, we show that such a split regular Hom-Leibniz algebra $L$ is of the form $L = U + \sum\limits_{ [j] \in \Lambda/\sim}I_{ [j]}$ with $U$ a subspace of the abelian subalgebra $H$ and any $I_{ [j]}$, a well described ideal of $L$, satisfying $[I_{ [j]}, I_{ [k]}] = 0$ if $[j]\neq [k]$. Under certain conditions, in the case of $L$ being of maximal length, the simplicity of the algebra is characterized.
Categories: math.RA