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

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 $G^\varphi$.