$\xi$-Groups and Hu-Liu Leibniz Algebras

We initiate the study of $\xi$-groups and Hu-Liu Leibniz algebras, claim that alomost all simple Leibniz algebras and simple Hu-Liu Leibniz algebras are linear, and establish two passages. One is the passage from a special ${\mathcal{Z}}_2$-graded associative algebra to a Hu-Liu Leibniz algebra. The other one is the passage from a linear $\xi$-group to its tangent space which is a Hu-Liu Leibniz algebra.