**
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

**Keywords**:
*n*-algebra, *n*-Lie algebra, Nambu tensor

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Acta Mathematica Universitatis Comenianae

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