Leibniz Algebras Defined By Tensor Product Of Lie Algebras

In this paper, motivated by the article written by R. Kurdiani and T. Pirashvili, we describe the property of the Leibniz algebra defined by tensor product of Lie algebra. At first, we compute the difference of dimensions between HL~2((?),F) and H~2((?),F), where (?) is a Borel subalgebra of a semi-simple Lie algebra over a field of characteristic zero. Then we give the definition of invariant symmetric of bilinear forms on a Leibniz algebra (?) and prove that the dimensions of B((?),F) and of B((?),F) are the same when (?)=[(?),(?)], and (?)=[(?),(?)].We also prove that the dimension of Der((?)) is less then the dimension of Der((?)). At the third part, we give some properties of (?).(?) is isomorphic to (?) when (?) is a semi-simple Lie algebra over a field of characteristic zero. At last, we give an exact sequence on (?).