Solvable reductive groups are commutative #
A reductive affine group with solvable geometric points is commutative, over any field and in every characteristic. Pass to an algebraic closure: the derived subgroup is connected, normal, smooth, and unipotent, so reductivity makes it trivial. Cocommutativity of the coordinate Hopf algebra then descends to the ground field.
This is the commutativity step in the classification of connected solvable reductive groups as tori. It does not identify the character lattice or assert that the group splits over the original field.
References #
- A. Borel, Linear Algebraic Groups, §11.
- J. S. Milne, Algebraic Groups (2017), §§16 and 19.
theorem
TauCeti.reductiveCommHopfAlgProperty.isCocomm_of_geometricallySolvable
{k : Type u}
[Field k]
{H : FiniteTypeCommHopfAlgCat k}
(hH : reductiveCommHopfAlgProperty k H)
(hsolv : geometricallySolvablePointsCommHopfAlgProperty k H.obj)
:
Coalgebra.IsCocomm k ↑H.obj
A solvable reductive affine group is commutative. Equivalently, its coordinate Hopf algebra is cocommutative. No perfectness or characteristic assumption on the base field is needed.