The identity component is characteristic

Every automorphism of an affine group preserves its identity component. In coordinate algebras, the defining Hopf ideal is fixed by inverse image along every automorphism. This allows the identity component of a normal closed subgroup to be treated as a normal subgroup of the ambient group, using the characteristic-subgroup criterion.

References

• J. S. Milne, Algebraic Groups (2017), Proposition 2.37.

@[simp]

theorem TauCeti.HopfAlgebra.comapOfSurjective_identityComponentHopfIdeal {k : Type u} [Field k] [IsAlgClosed k] {H : CommHopfAlgCat k} [Algebra.FiniteType k ↑H] (e : H ≅ H) : identityComponentHopfIdeal.comapOfSurjective (CommHopfAlgCat.Hom.hom e.hom) ⋯ = identityComponentHopfIdeal

Every automorphism of an affine group preserves its identity component.