ACTA MATHEMATICA UNIVERSITATIS COMENIANAE

Vol. LXXII, 1(2003)
p. 23 – 44

Irreducible Identities of n-algebras
M. Rotkiewicz

Abstract.  One can generalize the notion of $n$-Lie algebra (in the sense of Fillipov) and define ''weak $n$-Lie algebra'' to be an anticommutative $n$-ary algebra $(A,[\cdot ,\ldots ,\cdot ])$ such that the $% (n-1)$-ary bracket $[\cdot ,\ldots ,\cdot ]_a =[\cdot ,\ldots ,\cdot ,a]$ is an $(n-1)$-Lie bracket on $A$ for all $a$ in $A$. It is well known that every $n$-Lie algebra is weak $n$-Lie algebra. Under some additional assumptions these notions coincide. We show that it is not the case in general. By means of representation theory of symmetric groups a full description of $n$-bracket multilinear identities of degree $2$ that can be satisfied by an anticommutative $n$-ary algebra is obtained. This is a solution to the conjectures proposed by M. Bremner. These methods allow us to prove that the dual representation of an $n$-Lie algebra is in fact a representation in the sense of Kasymov. We also consider the generalizations of $n$-Lie algebra proposed by A. Vinogradov, M. Vinogradov and Gautheron. Some correlation between these generalizations can be easily seen. We also describe the kernel of the expansion map.

AMS subject classification:  16W55, 17B01, 17B99
Keywordsn-algebra, n-Lie algebra, Nambu tensor