Leibniz n-algebras

A Leibniz n-algebra is a vector space equipped with an n-ary operation which has the property of being a derivation for itself. This property is crucial in Nambu mechanics. For n ˆ 2 this is the notion of Leibniz algebra. In this paper we prove that the free Leibniz …n‡ 1†algebra can be described in terms of the n-magma, that is the set of n-ary planar trees. Then it is shown that the n-tensor power functor, which makes a Leibniz …n‡ 1†-algebra into a Leibniz algebra, sends a free object to a free object. This result is used in the last section, together with former results of Loday and Pirashvili, to construct a small complex which computes Quillen cohomology with coe1⁄2cients for any Leibniz n-algebra. 2000 Mathematics Subject Classi®cation: 17Axx, 70H05. 1 Introduction Leibniz algebras were introduced by the second author in [4]. They play an important role in Hochschild homology theory [4], [5] as well as in Nambu mechanics ([6][10], see also [1]). Let us recall that a Leibniz algebra is a vector space g equipped with a bilinear map ‰ÿ;ÿŠ : gn g! g satisfying the identity: …1:1† ‰x; ‰y; zŠŠ ˆ ‰‰x; yŠ; zŠ ÿ ‰‰x; zŠ; yŠ: One easily sees that Lie algebras are exactly Leibniz algebras satisfying the relation ‰x; xŠ ˆ 0. Hence Leibniz algebras are a non-commutative version of Lie algebras. Recently there have been several works dealing with various generalization of Lie structures by extending the binary bracket to an n-bracket (see [1], [2], [9], [11]). In this paper we introduce the notion of a Leibniz n-algebraÐa natural generalization of both concepts. For n ˆ 2 one recovers Leibniz algebras. Any Leibniz algebra g is also a Leibniz n-algebra under the following n-bracket: ‰x1; x2; . . . ; xnŠ :ˆ ‰x1; ‰x2; . . . ; ‰xnÿ1; xnŠ ŠŠ: Conversely, if L is a Leibniz …n‡ 1†-algebra, then on Dn…L† ˆLnn the following