The unipotent radical obstruction for the type-E6 minuscule carrier

Let H be the coordinate Hopf algebra of the specialized full-weight type-E₆ minuscule carrier. Over an algebraically closed field, if H is reduced, then every normal smooth unipotent closed subgroup of the carrier is trivial. Consequently its unipotent radical is trivial.

The mathematical input is the carrier's standard 27-dimensional comodule. It is simple, hence completely reducible, and faithful. The generic normal-unipotent elimination theorem TauCeti.HopfIdeal.eq_augmentation_of_isNormal_of_smoothUnipotent_of_isFaithful then applies: normality makes the fixed vectors of a smooth unipotent closed subgroup an ambient subcomodule, Kolchin's fixed-vector theorem forces the subgroup to act trivially, and faithfulness identifies its defining ideal with the augmentation ideal.

Reducedness is stated explicitly. No smoothness or connectedness of the carrier is asserted here, so the result is the normal-unipotent obstruction needed for reductivity rather than a proof that the carrier is reductive.

Main declarations

• TauCeti.E6Minuscule.eq_augmentation_of_isNormal_of_smoothUnipotent: every normal smooth unipotent closed subgroup of a reduced specialized carrier is trivial.
• TauCeti.E6Minuscule.unipotentRadicalDefiningIdeal_eq_augmentation: the unipotent radical of a reduced specialized carrier is trivial.

References

• J. E. Humphreys, Linear Algebraic Groups, §§19 and 26.
• J. C. Jantzen, Representations of Algebraic Groups, I.2 and II.2.

This is the type-E₆ counterpart of the argument for the full-weight type-E₇ minuscule carrier in TauCeti.Algebra.Lie.E7.Minuscule.UnipotentRadical.

theorem TauCeti.E6Minuscule.eq_augmentation_of_isNormal_of_smoothUnipotent (k : Type u) [Field k] [IsAlgClosed k] [IsReduced ↑(coordinateHopfAlgebra k)] (I : HopfIdeal k ↑(coordinateHopfAlgebra k)) (hI : I.IsNormal) (hU : smoothUnipotentCommHopfAlgProperty k ((finiteTypeCoordinateHopfAlgebra k).quotient I)) : I = HopfIdeal.augmentation k ↑(coordinateHopfAlgebra k)

Every normal smooth unipotent closed subgroup of a reduced specialized type-E₆ minuscule carrier is trivial.

The conclusion is stated contravariantly: the subgroup's defining Hopf ideal is the augmentation ideal of the carrier's coordinate algebra.

theorem TauCeti.E6Minuscule.unipotentRadicalDefiningIdeal_eq_augmentation (k : Type u) [Field k] [IsAlgClosed k] [IsReduced ↑(coordinateHopfAlgebra k)] : (finiteTypeCoordinateHopfAlgebra k).unipotentRadicalDefiningIdeal = HopfIdeal.augmentation k ↑(coordinateHopfAlgebra k)

The unipotent radical of a reduced specialized type-E₆ minuscule carrier over an algebraically closed field is trivial.