Quandle varieties, generalized symmetric spaces and φ-spaces

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We define a quandle variety as an irreducible algebraic variety Q endowed with an algebraically defined quandle operation. 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, φ) associated to the group G, an automorphism φ of G and a subgroup H of Gφ.