m at h . R A ] 3 1 D ec 2 00 6 Even ( Odd ) Roots and Simple Leibniz Algebras with Lie Factor

We introduce 3-irreducible modules, even roots and odd roots for Leibniz algebras, produce a basis for a root space of a Leibniz algebra with a semisimple Lie factor, and classify finite dimensional simple Leibniz algebras with Lie factor sl2. In this paper, all vector spaces are vector spaces over an algebraically closed field k of characteristic 0. This paper consists of three sections. In Section 1, we introduce 3-irreducible modules which are natural building blocks among the modules over Leibniz algebras. In Section 2, we produce a basis of a weight space of a module over a right nilpotent Leibniz algebra by using the Extended Lie’s Theorem, and construct the root space decomposition of a Leibniz algebra by using its Cartan subalgebra, even roots and odd roots. In Section 3, we classify finite dimensional simple Leibniz algebras with Lie factor sl2. 1 3-Irreducible Modules We begin this section with the definition of a (right) Leibniz algebra ([7]). Definition 1.1 A vector space L is called a (right) Leibniz algebra if there exists a binary operation 〈 , 〉: L×L → L, called the angle bracket, such that the (right) Leibniz identity holds: 〈〈x, y〉, z〉 = 〈x, 〈y, z〉〉+ 〈〈x, z〉, y〉 for x, y, z ∈ L. (1) A Leibniz algebra L with an angle bracket 〈 , 〉 is also denoted by (L, 〈 , 〉). Definition 1.2 Let I be a subspace of a Leibniz algebra (L, 〈 , 〉).