Higher-Dimensional general Jacobi identities I

TL;DR

This paper explores a four-dimensional generalization of the Jacobi identity within synthetic differential geometry, aiming to extend the foundational algebraic structures underlying vector fields and Lie algebras.

Contribution

It provides a detailed investigation of a four-dimensional generalization of the Jacobi identity, building on prior work and setting the stage for a unified approach to higher-dimensional identities.

Findings

Detailed analysis of four-dimensional Jacobi identity

Connections to synthetic differential geometry and vector fields

Framework for future higher-dimensional generalizations

Abstract

It was shown by the author [International Journal of Theoretical Physics 36 (1997), 1099-1131] in synthetic differential geometry that what is called the general Jacobi identity obtaining in microcubes underlies the Jacobi identity of vector fields. It is well known in the theory of Lie algebras that a plethora of higher-dimensional generalizations of the Jacobi identity hold, though it is usually established not as a direct derivation from the axioms of Lie algebras but by making an appeal to the so-called Poincar\'e-Birkhoff-Witt theorem. The general Jacobi identity was rediscovered by Kirill Mackenzie in the second decade of this century [Geometric Methods in Physics, 357-366, Birkh\"auser/Springer 2013]. The principal objective in this paper is to investigate a four-dimensional generalization of the general Jacobi identity in detail. In a subsequent paper we will propose a uniform method for establishing a bevy of higher-dimensional generalizations of the Jacobi identity under a single umbrella.