c © 2002 Heldermann Verlag A Leibniz Algebra Structure on the Second Tensor Power

Abstract. For any Lie algebra g, the bracket [x ⊗ y, a ⊗ b]: = [x, [a, b]] ⊗ y + x ⊗ [y, [a, b]] defines a Leibniz algebra structure on the vector space g ⊗ g. We let g⊗g be the maximal Lie algebra quotient of g ⊗ g. We prove that this particular Lie algebra is an abelian extension of the Lie algebra version of the nonabelian tensor product g g of Brown and Loday [1] constructed by Ellis [2], [3]. We compute this abelian extension and Leibniz homology of g ⊗ g in the case, when g is a finite dimensional semi-simple Lie algebra over a field of characteristic zero. Let g be a Lie algebra. We define the following bracket [x ⊗ y, a ⊗ b]: = [x, [a, b]] ⊗ y + x ⊗ [y, [a, b]] (1) on g ⊗ g. It turns out that g ⊗ g equipped with this bracket is not in general a Lie algebra but only a Leibniz algebra. Let us recall that Leibniz algebras ar