Borel subgroups and the identity component

Over an algebraically closed field, an affine group of finite type and its identity component have the same Borel subgroups. More generally, a Borel subgroup contained in a closed subgroup is Borel in that subgroup's identity component. These comparisons allow conjugacy arguments to pass to connected ambient groups.

Containment in the identity component follows from CommHopfAlgCat.identityComponentHopfIdeal_toIdeal_le_ker_of_connected and HopfIdeal.identityComponent_le_of_le. Borel maximality is transported by the closed-subgroup comparison in Borel.Subgroup.

References

• A. Borel, Linear Algebraic Groups, 2nd ed. (1991), §11.1.
• J. S. Milne, Algebraic Groups (2017), §§2.a and 17.a.

theorem TauCeti.HopfIdeal.isBorelOverAlgClosed_comap_identityComponent_iff {k : Type u} [Field k] [IsAlgClosed k] {H : FiniteTypeCommHopfAlgCat k} (J : HopfIdeal k ↑H.identityComponent.obj) : IsBorelOverAlgClosed k H (J.comapOfSurjective (Bialgebra.Quotient.mkBialgHom HopfAlgebra.identityComponentHopfIdeal.toIdeal) ⋯) ↔ IsBorelOverAlgClosed k H.identityComponent J

A closed subgroup of the identity component is Borel there exactly when it is Borel in the whole group. No smoothness hypothesis on the ambient group is needed.

theorem TauCeti.HopfIdeal.IsBorelOverAlgClosed.map_identityComponent_of_le {k : Type u} [Field k] [IsAlgClosed k] {H : FiniteTypeCommHopfAlgCat k} {P I : HopfIdeal k ↑H.obj} (hI : IsBorelOverAlgClosed k H I) (hPI : P ≤ I) : IsBorelOverAlgClosed k (H.quotient P.identityComponent) (I.map (Bialgebra.Quotient.mkBialgHom P.identityComponent.toIdeal))

A Borel subgroup contained in P remains Borel when the ambient group is restricted to the identity component of P.