Quandle varieties, generalized symmetric spaces, and \(\phi\)-spaces (Q292182)

The author defines a quandle variety as an irreducible algebraic variety \(Q\) endowed with an algebraically defined quandle operation \(\vartriangleright\). 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 its inner automorphism group Inn(\(Q\)) has a dense orbit. Then can be shown that there is an algebraic group \(G\) acting on \(Q\) with the same orbits as Inn(\(Q\)) such that each \(G\)-orbit is isomorphic to the quandle (\(G/H,\vartriangleright \varphi\)) associated to the group \(G\), an automorphism \(\varphi\) of \(G\) and a subgroup \(H\) of \(G\varphi\).