The Jacobi Identity beyond Lie Algebras

TL;DR

This paper explores the Jacobi identity's broader applicability beyond Lie algebras using categorical and synthetic differential geometry frameworks, clarifying its conceptual significance.

Contribution

It demonstrates that double dualization functors and synthetic differential geometry offer a clear conceptual framework for understanding the Jacobi identity beyond Lie algebras.

Findings

Double dualization functor framework clarifies Jacobi identity

Synthetic differential geometry provides conceptual insights

Extends Jacobi identity understanding beyond Lie algebras

Abstract

Frolicher and Nijenhuis recognized well in the middle of the previous century that the Lie bracket and its Jacobi identity could and should exist beyond Lie algebras. Nevertheless the conceptual meaning of their discovery has been obscured by the messy techniques they exploited. The principal objective in this paper is to show that the double dualization functor in a cartesian closed category as well as synthetic differential geometry provides an adequate framework, in which their discovery's conceptual meaning appears lucid.