Flat connected finite quandles

TL;DR

This paper introduces the concept of flat quandles, inspired by symmetric spaces, and provides a classification of all flat connected finite quandles, advancing the understanding of their geometric structure.

Contribution

It defines flat quandles based on Riemannian symmetric space theory and classifies all flat connected finite quandles, a novel contribution to quandle theory.

Findings

Classification of all flat connected finite quandles

Introduction of the notion of flatness in quandles

Extension of symmetric space concepts to algebraic structures

Abstract

Quandles can be regarded as generalizations of symmetric spaces. In the study of symmetric spaces, the notion of flatness plays an important role. In this paper, we define the notion of flat quandles, by referring to the theory of Riemannian symmetric spaces, and classify flat connected finite quandles.