Connected subgroups of a diagonalizable subgroup's normalizer

A morphism from a reduced connected affine group into the normalizer of a closed diagonalizable subgroup factors through its centralizer. Both groups are of finite type over an algebraically closed field, but the ambient group and the diagonalizable subgroup may be nonreduced. The conclusion is scheme-theoretic and controls every value algebra.

This is the connectedness step in comparing normalizer quotients with Weyl groups. The finite character action from BialgHom.finite_range_normalizerCharacterHom makes conjugation coefficients constant on the connected source. Their values at the identity then force the restricted conjugation to be trivial.

The coefficient argument also gives centrality of normal diagonalizable subgroups, as recorded in TauCeti.Algebra.AlgebraicGroup.DiagonalizableGroup.Normal. The final results identify the normalizer and centralizer tangent Lie algebras when the normalizer is reduced; no reducedness of the diagonalizable subgroup is required.

References

• J. S. Milne, Algebraic Groups (2017), §12 and §21.1.
• T. A. Springer, Linear Algebraic Groups, §7.1.

theorem BialgHom.centralizerHopfIdeal_le_ker_of_forall_mapDomain_mem_normalizerPoints {k : Type u_1} {H : Type u_2} {K : Type u_3} {X : Type u_4} [Field k] [IsAlgClosed k] [CommRing H] [HopfAlgebra k H] [Algebra.FiniteType k H] [CommGroup X] [CommRing K] [HopfAlgebra k K] [Algebra.FiniteType k K] [IsReduced K] [ConnectedSpace (PrimeSpectrum K)] (π : H →ₐc[k] MonoidAlgebra k X) (hπ : Function.Surjective ⇑π) (φ : H →ₐc[k] K) (hnorm : ∀ (g : WithConv (K →ₐ[k] k)), φ.mapDomain g ∈ π.normalizerPoints hπ) : π.centralizerHopfIdeal.toIdeal ≤ RingHom.ker φ

A morphism from a reduced connected affine group of finite type whose rational points normalize a closed diagonalizable subgroup factors through its scheme-theoretic centralizer. Neither the ambient group nor the diagonalizable subgroup needs to be reduced.

theorem BialgHom.centralizerHopfIdeal_le_of_le_normalizerHopfIdeal {k : Type u} [Field k] [IsAlgClosed k] {H : TauCeti.FiniteTypeCommHopfAlgCat k} {X : Type u} [CommGroup X] (π : ↑H.obj →ₐc[k] MonoidAlgebra k X) (hπ : Function.Surjective ⇑π) (I : TauCeti.HopfIdeal k ↑H.obj) [IsReduced ↑(H.quotient I).obj] [ConnectedSpace (PrimeSpectrum ↑(H.quotient I).obj)] (hI : π.normalizerHopfIdeal hπ ≤ I) : π.centralizerHopfIdeal ≤ I

A reduced connected closed subgroup of the scheme-theoretic normalizer of a closed diagonalizable subgroup lies in its scheme-theoretic centralizer. Ideal inclusions reverse the inclusions of subgroup schemes.

theorem BialgHom.centralizerHopfIdeal_le_identityComponent_normalizerHopfIdeal {k : Type u} [Field k] [IsAlgClosed k] {H : TauCeti.FiniteTypeCommHopfAlgCat k} {X : Type u} [CommGroup X] (π : ↑H.obj →ₐc[k] MonoidAlgebra k X) (hπ : Function.Surjective ⇑π) [IsReduced ↑(H.quotient (π.normalizerHopfIdeal hπ)).obj] : π.centralizerHopfIdeal ≤ (π.normalizerHopfIdeal hπ).identityComponent

If the scheme normalizer is reduced, its identity component centralizes the entire diagonalizable subgroup, including any infinitesimal structure in that subgroup.

@[simp]

theorem BialgHom.lieSubalgebra_normalizerHopfIdeal_eq_centralizerHopfIdeal {k : Type u} [Field k] [IsAlgClosed k] {H : TauCeti.FiniteTypeCommHopfAlgCat k} {X : Type u} [CommGroup X] (π : ↑H.obj →ₐc[k] MonoidAlgebra k X) (hπ : Function.Surjective ⇑π) [IsReduced ↑(H.quotient (π.normalizerHopfIdeal hπ)).obj] : (π.normalizerHopfIdeal hπ).lieSubalgebra = π.centralizerHopfIdeal.lieSubalgebra

A reduced scheme normalizer and the scheme centralizer of a closed diagonalizable subgroup have the same tangent Lie algebra. This holds in every characteristic; in characteristic zero the reducedness hypothesis follows from Cartier's theorem.