Universal enveloping algebras of Leibniz algebras and (co)homology

The homology of Lie algebras is closely related to the cyclic homology of associative algebras [LQ]. In [L] the first author constructed a "noncommutative" analog of Lie algebra homology which is, similarly, related to Hochschild homology [C, L]. For a Lie algebra g this new theory is the homology of the complex C,(g) ... ~ ~| g|-+ ... ~1 ~ k, whose boundary map d is given by the formula d(gl|174 = ~ (-1)J(gl@'"|174174174 " l<i<j<n Note that d is a lifting of the classical Chevalley-Eilenberg boundary map d: Ang A~-tg. One striking point in the proof of d 2 = 0 is the following fact: the only property of the bracket, which is needed, is the so-called Leibniz identity [x, [y, z]] = [[x, y], z]-[[x, z], y] , for allx, y, zEg. So, it is natural to introduce new objects: the Leibniz algebras, which are modules over a commutative ring k, equipped with a bilinear map [- ,-]:9 • 9 "-* g satisfying the Leibniz identity. Since the Leibniz identity is equivalent to the classical Jacobi identity when the bracket is skew-symmetric, this notion is a sort of "non-commutative" analog of Lie algebras. Hence for any Leibniz algebra there is defined a homology theory (and dually a cohomology theory) HL, (9): = H, (C, (9), d). The principal aim of this paper is to answer affirmatively the following question. Is HL, (resp. HL*) a Tor-functor (resp. Ext-functor)? This leads naturally to the search for a universal enveloping algebra of a Leibniz algebra. In Sect. 1 we give examples of Leibniz algebras and we show that the underlying module of a free Leibniz algebra is a tensor module. Then we define the notion of