The radical of a reductive group

The solvable radical of a reductive affine group over any field is a central torus, in every characteristic. The torus need not split over the ground field, and centrality is scheme-theoretic: it controls points over arbitrary commutative value algebras.

The radical is a smooth geometrically connected normal subgroup, so TauCeti.reductiveCommHopfAlgProperty.quotient_of_isNormal makes it reductive. The classification of solvable reductive groups in TauCeti.reductiveCommHopfAlgProperty.torus_of_geometricallySolvable identifies it as a torus. Centrality then follows from TauCeti.HopfIdeal.IsNormal.isCentral_of_multiplicativeType.

These results supply the central torus in the reduction of a reductive group to its semisimple quotient.

References

• J. S. Milne, Algebraic Groups (2017), Chapters 12 and 19.
• A. Borel, Linear Algebraic Groups, §§11 and 14.

theorem TauCeti.reductiveCommHopfAlgProperty.torus_solvableRadical {k : Type u} [Field k] {H : FiniteTypeCommHopfAlgCat k} (hH : reductiveCommHopfAlgProperty k H) : torusCommHopfAlgProperty k H.solvableRadical

The solvable radical of a reductive affine group is a torus, in every characteristic and including over fields over which the group does not split.

theorem TauCeti.reductiveCommHopfAlgProperty.isCentral_solvableRadicalDefiningIdeal {k : Type u} [Field k] {H : FiniteTypeCommHopfAlgCat k} (hH : reductiveCommHopfAlgProperty k H) : H.solvableRadicalDefiningIdeal.IsCentral

The solvable radical of a reductive affine group is contained in the scheme-theoretic center, in every characteristic. The defining Hopf ideal represents the radical as a closed subgroup.