loday@math.u-strasbg.fr, pira@mathematik.Uni-Bielefeld.DE

Submission: 2000, Mar. 06

A {\it 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 $n=2$ 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 coefficients for any Leibniz $n$-algebra.

2000 Mathematics Subject Classification: 17Axx, 70H05.

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