Quandle varieties, generalized symmetric spaces and $\varphi$-spaces

TL;DR

This paper introduces quandle varieties as algebraic structures generalizing symmetric spaces, establishing their connection to algebraic groups and automorphisms under certain conditions.

Contribution

It defines quandle varieties in algebraic geometry and proves their structure relates to algebraic groups with automorphisms, extending the theory of symmetric spaces.

Findings

Existence of an algebraic group G associated with quandle varieties

Each orbit is isomorphic to a quandle constructed from G, an automorphism, and a subgroup

Provides a framework linking algebraic varieties, quandles, and algebraic groups

Abstract

We define a quandle variety as an irreducible algebraic variety $Q$ endowed with an algebraically defined quandle operation $\rhd$. It can also be seen as an analogue of a generalized affine symmetric space or a regular $s$-manifold in algebraic geometry. Assume that $Q$ is normal as an algebraic variety and that the action of the inner automorphism group has a dense orbit. Then we show that there is an algebraic group $G$ such that each orbit is isomorphic to the quandle $(G/H, \rhd_\varphi)$ associated to the group $G$, an automorphism $\varphi$ of $G$ and a subgroup $H$ of $G^\varphi$.