A classification of polynomial functions satisfying the Jacobi identity over integral domains

TL;DR

This paper classifies all bivariate polynomial functions over infinite integral domains that satisfy the Jacobi identity, revealing they are of degree at most one in each variable, with results depending on the domain's characteristic.

Contribution

It provides a complete classification of polynomial solutions to the Jacobi identity over infinite integral domains, detailing their degree constraints and characteristic dependence.

Findings

All solutions are of degree at most one in each variable.

Classification depends on the characteristic of the domain.

Complete description of such polynomials over infinite integral domains.

Abstract

The Jacobi identity is one of the properties that are used to define the concept of Lie algebra and in this context is closely related to associativity. In this paper we provide a complete description of all bivariate polynomials that satisfy the Jacobi identity over infinite integral domains. Although this description depends on the characteristic of the domain, it turns out that all these polynomials are of degree at most one in each indeterminate.