Higher Jacobi identities

TL;DR

This paper explores higher-order Jacobi identities in Lie algebras, constructing specific permutation sets that generate new identities beyond the classical ones.

Contribution

It introduces a family of permutation subsets in symmetric groups that produce new identities valid in all Lie algebras.

Findings

Constructed sets $T_{k,l,n}$ of permutations

Derived new identities for Lie algebras

Extended classical Jacobi identities to higher orders

Abstract

By definition the identities $[x_1,x_2]+[x_2,x_1]=0$ and $[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0$ hold in any Lie algebra. It is easy to check that the identity $[x_1,x_2,x_3,x_4]+[x_2,x_1,x_4,x_3]+[x_3,x_4,x_1,x_2]+[x_4,x_3,x_2,x_1] = 0$ holds in any Lie algebra as well. We investigate sets of permutations that give identities of this kind. In particular, we construct a family of such subsets $T_{k,l,n}$ of the symmetric group $S_n,$ and hence, a family of identities that hold in any Lie algebra.