Triviality of the unipotent radical of a weight Levi #
Over an algebraically closed field, every normal smooth unipotent closed subgroup of a general-linear weight Levi is trivial. In particular, the unipotent radical is trivial. The statements allow repeated weights, arbitrary characteristic, and rank zero. They provide the normal-subgroup input for reductivity of block-diagonal Levi subgroups and for identifying the unipotent radical of their parabolics.
References #
- J. S. Milne, Algebraic Groups (2017), Chapters 13 and 19.
- T. A. Springer, Linear Algebraic Groups, §§2.2 and 2.4.
- Formal source:
TauCeti.Algebra.AlgebraicGroup.GeneralLinear.Reductive.
theorem
TauCeti.GeneralLinear.eq_augmentation_weightLevi_of_isNormal_of_smoothUnipotent
(k : Type u)
[Field k]
[IsAlgClosed k]
{N : ℕ}
(w : Fin N → ℤ)
(I : HopfIdeal k ↑(weightLeviFiniteTypeCoordinateHopfAlgebra k w).obj)
(hI : I.IsNormal)
(hU : smoothUnipotentCommHopfAlgProperty k ((weightLeviFiniteTypeCoordinateHopfAlgebra k w).quotient I))
:
Every normal smooth unipotent closed subgroup of a weight Levi over an algebraically closed field is trivial, including when distinct coordinates have the same weight.
@[simp]
theorem
TauCeti.GeneralLinear.unipotentRadicalDefiningIdeal_weightLeviFiniteTypeCoordinateHopfAlgebra
(k : Type u)
[Field k]
[IsAlgClosed k]
{N : ℕ}
(w : Fin N → ℤ)
:
The unipotent radical of every weight Levi over an algebraically closed field is trivial. Contravariantly, its defining ideal is the augmentation ideal.