Lie Quandles, Leibniz Racks and Noether's First Theorem

TL;DR

This paper explores the mathematical structures called Lie Quandles and Leibniz Racks, their relation to Lie algebras, and develops a nonlinear analogue of Noether's first theorem inspired by physics.

Contribution

It introduces a classification of generalized structures related to Lie Quandles and establishes results towards a nonlinear version of Noether's first theorem.

Findings

Classified a class of generalizations of Lie Quandles and Leibniz Racks.

Established results towards a nonlinear analogue of Noether's first theorem.

Explored the correspondence between linear and nonlinear structures in this context.

Abstract

In [Self-distributive structures in physics. Internat. J. Theoret. Phys. 64 (2025), no. 3, Paper No. 73], Fritz was motivated by the structure of Hamiltonian/Heisenberg mechanics to define the notion of "Lie Quandle", which he argued are nonlinear generalizations of finite dimensional real Lie algebras. In this article, we will investigate a linear/nonlinear correspondence to which Fritz' is a special case, classify a class of generalizations of these objects, as well as describe some results in the direction of a nonlinear analogue of Noether's first theorem first described by Fritz.