Varieties of Anticommutativen-ary Algebras

✍ Scribed by Murray Bremner

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 188 KB
• Volume: 191
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1996.6895

✦ Synopsis

A fundamental problem in the theory of n-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 of n-ary algebras can be defined by multilinear identities. In the anticommutative case, it is shown that for 2 n y 1 Ž . nF8 the -dimensional S -module of multilinear identities in which 2 ny1 n Ž each term involves two n-ary products i.e., two pairs of n-ary anticommutative . brackets decomposes as the direct sum of the n distinct simple modules labelled by the n partitions of 2 n y 1 in which only 1 and 2 occur as parts. In the cases Ž . n s 3 resp. n s 4 , the kernel of the commutator expansion map and a generator Ž . for each of the 7 resp. 15 nonzero submodules are determined. The paper concludes with some conjectures for n G 5.