Algebraic Groups I. Dynamic approach to algebraic groups

Let G be a smooth affine group over a field k, and λ : Gm → G a k-homomorphism (possibly trivial, though that case is not interesting). One often calls λ a 1-parameter k-subgroup of G, even when kerλ 6= 1. Such a homomorphism defines a left action of Gm on G via the functorial procedure t.g = λ(t)gλ(t)−1 for g ∈ G(R) and t ∈ R× for any k-algebra R. In lecture we introduced the following associated subgroup functors of G: for any k-algebra R,