On $n$-ary Lie algebras of type $(r,l)$

TL;DR

This paper explores the generalization of Lie algebras to n-ary structures of type (r,l), compares different definitions, and proves the non-existence of certain simple n-ary Lie algebras.

Contribution

It introduces a comparison of definitions for n-ary Lie algebras and establishes a non-existence result for simple algebras of specific types.

Findings

No simple n-ary Lie algebras of type (n-1,l) for l>0.

Comparison of usual and invariant definitions of n-ary Lie algebras.

Generalization of classical Lie algebra concepts to n-ary structures.

Abstract

These notes are devoted to the multiple generalization of a Lie algebra introduced by A.M.Vinogradov and M.M.Vinogradov. We compare definitions of such algebras in the usual and invariant case. Furthermore, we show that there are no simple $n$-ary Lie algebras of type $(n-1,l)$ for $l>0$.