Smoothness of the subgroup generated by the tripled type-D₄ roots and torus

The numbered root subgroups and weight torus of the tripled type-D₄ carrier have reduced coordinate algebras over a field. Their common-kernel Hopf quotient is therefore reduced over an algebraically closed field, and the resulting finite-type group scheme is smooth. Together with the connectedness result, this gives two geometric properties needed to compare the generated subgroup with the integral carrier and ultimately with the pinned group of type D₄.

The construction applies to the subgroup generated after base change; it does not assert that this subgroup equals the base change of the integral carrier.

References

• J. E. Humphreys, Linear Algebraic Groups, §§26–27.
• J. S. Milne, Algebraic Groups (2017), §2.h.

theorem TauCeti.D4Tripled.isReduced_generatedCoordinateHopfAlgebra (k : Type u) [Field k] [IsAlgClosed k] : IsReduced ↑(generatedCoordinateHopfAlgebra k)

The subgroup generated by the tripled type-D₄ roots and torus has reduced coordinate algebra over an algebraically closed field.

theorem TauCeti.D4Tripled.smoothCommHopfAlgProperty_generatedCoordinateHopfAlgebra (k : Type u) [Field k] [IsAlgClosed k] : smoothCommHopfAlgProperty k (generatedCoordinateHopfAlgebra k)

The subgroup generated by the tripled type-D₄ roots and torus is smooth over an algebraically closed field.