Higher-dimensional general Jacobi identities I

Abstract

It was shown by the author [International Journal of Theoretical Physics 36 (1997), 1099{1131] that what is called the general Jacobi identity, obtaining in microcubes, underlies the Jacobi identity of vector elds. It is well known in the theory of Lie algebras that a plethora of higher-dimensional generalizations of the Jacobi identity hold, though they are usually established not as a derivation on the nose from the axioms of Lie algebras but by making an appeal to the socalled Poincar e{Birkho {Witt theorem and the like. The general Jacobi identity was rediscovered by Kirill Mackenzie in the second decade of this century [Geometric Methods in Physics, Birkh auser/Springer, 2013, 357{366]. The principal objective of 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 general Jacobi identity under a single umbrella.