Geometric connectedness of the special orthogonal groups #

The coordinate Hopf algebra of the standard group SOₙ is geometrically connected in every dimension away from characteristic two, and in dimension two over every field. Connectedness is tested by idempotents: over an algebraically closed extension an idempotent regular function is constant once right translation by every rational point fixes it, and a rational point which is joined to the identity by a path (a point over a domain specializing to it at one parameter and to the identity at another) translates every idempotent to itself.

In every dimension, away from characteristic two, the paths come from reflections. Cartan-Dieudonné writes a special orthogonal matrix as a product of pairs of reflections, and the points fixing a given idempotent form a subgroup, so it is enough to join each product of two reflections to the identity. Reflecting in a fixed anisotropic vector v and then in a vector moving along a Laurent-polynomial family through the plane spanned by v and w does exactly that, by TauCeti.exists_laurentPath_reflectionMatrix_mul.

Rank two is treated separately because that argument needs an invertible two, while the determinant-one model of SO₂ is geometrically connected over every field. Away from characteristic two, after extending to an algebraically closed field, the explicit equivalence SO₂(K) ≃* Kˣ writes every rational point on a Laurent-polynomial path from the identity; in characteristic two, every rational point lies on the affine-line path of matrices !![1 + tb, tb; -tb, 1 + tb]. That model is nonreduced in characteristic two, but its underlying space is still connected.