# Generalized Jacobi identities and Jacobi elements of the group ring of the symmetric group

## TL;DR

This paper introduces the concept of Jacobi elements in the group ring of the symmetric group and provides a combinatorial criterion for identifying Jacobi subsets, extending classical Lie algebra identities.

## Contribution

It defines Jacobi elements in the group ring of symmetric groups and characterizes Jacobi subsets through combinatorial conditions, generalizing Lie algebra identities.

## Key findings

- Introduced Jacobi elements in the group ring of $S_n$.
- Provided a combinatorial necessary and sufficient condition for Jacobi subsets.
- Extended classical Lie algebra identities to broader algebraic structures.

## 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. I. Alekseev in his recent work introduced the notion of Jacobi subset of the symmetric group $S_n$. It is a subset of $S_n$ that gives an identity of this kind. We introduce a notion of Jacobi element of the group ring $\mathbb{Z}[S_n]$ and describe them on the language of equations on coefficients. Using this description we obtain a purely combinatorial necessary and sufficient condition for a subset to be Jacobi.