Classifying Leibniz racks

Abstract

Lie quandles were defined by T. Fritz as a nonlinear generalization of Lie algebras, to serve as a nonlinear algebras of observables in generalized form of Hamiltonian mechanics. These objects were subsequently generalized in a prequel article to objects referred to as smooth Lie group families of smooth racks by the authors. In this article, we investigate a particular system of nonlinear functional differential equations whose solution in general provides a pathway to the classification of smooth Lie group families of smooth rack structures on any given smooth manifold. We then discuss a particularly practical ansatz for solving these equations, and use it to construct new families of examples, as well as prove a result on the sense in which the Bloch sphere is the unique Lie quandle structure on the sphere.