Varieties of Anticommutativen-ary Algebras Original Research Article

A fundamental problem in the theory ofn-ary algebras is to determine the correct generalization of the Jacobi identity. This paper describes some computational results on this problem using representations of the symmetric group. It is well known that over a field of characteristic 0 any variety ofn-ary algebras can be defined by multilinear identities. In the anticommutative case, it is shown that forn ≤ 8 the[formula]-dimensionalS2n − 1-module of multilinear identities in which each term involves twon-ary products (i.e., two pairs ofn-ary anticommutative brackets) decomposes as the direct sum of thendistinct simple modules labelled by thenpartitions of 2n − 1 in which only 1 and 2 occur as parts. In the casesn = 3 (resp.n = 4), the kernel of the commutator expansion map and a generator for each of the 7 (resp. 15) nonzero submodules are determined. The paper concludes with some conjectures forn ≥ 5.