Normal diagonalizable subgroups are central

A normal diagonalizable closed subgroup of a reduced connected affine group of finite type over an algebraically closed field is central. The diagonalizable subgroup need not be reduced: the conclusion concerns its full scheme structure, including infinitesimal subgroups.

The normalizer acts through a finite group of character automorphisms. Conjugation coefficients are regular functions on the ambient group, so connectedness makes them constant. Specializing at the identity makes the restricted conjugation trivial. Point separation is used only on the reduced ambient coordinate ring, never on the diagonalizable subgroup.

References

• J. S. Milne, Algebraic Groups (2017), Corollary 12.38.

The central input is BialgHom.centralizerHopfIdeal_le_ker_of_forall_mapDomain_mem_normalizerPoints: the identity morphism factors through the normalizer when the diagonalizable subgroup is normal.

theorem BialgHom.isCentral_kerOfSurjective_of_isNormal {k : Type u_1} {H : Type u_2} {X : Type u_3} [Field k] [CommRing H] [HopfAlgebra k H] [CommGroup X] [IsAlgClosed k] [Algebra.FiniteType k H] [IsReduced H] [ConnectedSpace (PrimeSpectrum H)] (π : H →ₐc[k] MonoidAlgebra k X) (hπ : Function.Surjective ⇑π) (hN : (TauCeti.HopfIdeal.kerOfSurjective π hπ).IsNormal) : (TauCeti.HopfIdeal.kerOfSurjective π hπ).IsCentral

A normal diagonalizable closed subgroup of a reduced connected finite-type affine group over an algebraically closed field is central as a subgroup scheme. No smoothness or reducedness of the diagonalizable subgroup is assumed.