VARIETIES OF ANTICOMMUTATIVE N-ARY ALGEBRAS

A fundamental problem in the theory of n-ary algebras is to determine the correct generalization of the Jacobi identity. This paper describe s some computational results on this problem using representations of the symmetric group. It is well known that over a field of characteris tic 0 any variety of n-ary algebras can be defined by multilinear iden tities. In the anticommutative case, it is shown that for n less than or equal to 8 the (2n-1/n)-dimensional S2n-1-module of multilinear ide ntities in which each term involves two n-ary products (i.e., two pair s of n-ary anticommutative brackets) decomposes as the direct sum of t he n distinct simple modules labelled by the n partitions of 2n-1 in w hich only 1 and 2 occur as parts. In the cases n = 3 (resp. n = 4), th e kernel of the commutator expansion map and a generator for each of t he 7 (resp. 15) nonzero submodules are determined. The paper concludes with some conjectures for n greater than or equal to 5. (C) 1997 Acad emic Press.