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.
Related
- Lie Quandles, Leibniz Racks and Noether's First Theorem
- On the Leibniz cohomology of vector fields
- NS-Leibniz bialgebras, the Yang--Baxter equation and relative Rota--Baxter operators
- Tensor products of Leibniz bimodules and Grothendieck rings -- an addendum
- Kupershmidt--Nijenhuis structures on pre-Leibniz algebras
BibTeX
@misc{elhamdadi2026classifying,
title = {{Classifying Leibniz racks}},
author = {Mohamed Elhamdadi and Bryce Virgin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35534},
eprint = {2610.35534},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35534},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classifying Leibniz racks
AU - Mohamed Elhamdadi
AU - Bryce Virgin
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35534
UR - https://arxiv.org/abs/2610.35534
AB - 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.
ER -