ξ-Groups and Hu-Liu Leibniz Algebras

We initiate the study of ξ-groups and Hu-Liu Leibniz algebras, claim that almost all simple Leibniz algebras and simple Hu-Liu Leibniz algebras are linear, and establish two passages. One is the passage from a special Z2-graded associative algebra to a Hu-Liu Leibniz algebra. The other one is the passage from a linear ξ-group to its tangent space which is a Hu-Liu Leibniz algebra. The Lie correspondence between connected linear groups and linear Lie algebras is a central result of the Lie theory. Anyone who attempts at generalizing the Lie correspondence is bothered by the following problem: which kinds of algebraic objects should be used to replace groups and Lie algebras? After searching for the generalization of the Lie correspondence for many years, I am still not sure whether it is practical to live in hope of getting a complete generalization of the Lie correspondence by replacing groups and Lie algebras with some kinds of algebraic objects which are not groups and Lie algebras respectively. However, it is certain that the Lie correspondence and the basic Lie theory can be extended completely if some kinds of algebraic structures are added to groups and Lie algebras. In order to extend the Lie correspondence, the simplest replacements of groups and Lie algebras are ξ-groups and Hu-Liu Leibniz algebras, which are the algebraic objects obtained by adding more algebraic structures to groups and Lie algebras, respectively. The purpose of this paper is to give the basic properties of ξ-groups and Hu-Liu Leibniz algebras. This paper is organized as follows. First, in Section 1 I review some notions about Leibniz algebras, introduce the notion of a linear Leibniz algebra by using a special Z2-graded associative algebra, and claim that almost all simple Leibniz algebras are linear. Next, in Section 2 I introduce the notion of a HuLiu Leibniz algebra by adding a Lie algebra structure to a Leibniz algebra, establish the passage from a special Z2-graded associative algebra to a Hu-Liu Leibniz algebra, and claim that almost all simple Hu-Liu Leibniz algebras are linear. Finally, in Section 3 I introduce the notion of a ξ-group, and establish