k-Leibniz algebras from lower order ones: from Lie triple to Lie l-ple systems

Two types of higher order Lie $\ell$-ple systems are introduced in this paper. They are defined by brackets with $\ell > 3$ arguments satisfying certain conditions, and generalize the well known Lie triple systems. One of the generalizations uses a construction that allows us to associate a $(2n-3)$-Leibniz algebra \fL with a metric $n$-Leibniz algebra \tilde{\fL} by using a $2(n-1)$-linear Kasymov trace form for \tilde{\fL}. Some specific types of $k$-Leibniz algebras, relevant in the construction, are introduced as well. Both higher order Lie $\ell$-ple generalizations reduce to the standard Lie triple systems for $\ell=3$.Comment: 22 pages, no figure