Smoothness of the subgroup generated by the doubled type-E6 pinning candidates

The twelve numbered root subgroups and the weight torus of the doubled minuscule type-E₆ carrier generate a closed subgroup of GL₅₄. Its defining Hopf ideal is the common kernel of their coordinate maps. Over an algebraically closed field, the generated subgroup is smooth: its coordinate algebra is reduced because each generator has reduced coordinate algebra.

The base change of the integral toral-closure carrier contains this generated subgroup. The two schemes are not identified here; neither reductivity nor an identification with the pinned simply connected type-E₆ group scheme follows from smoothness alone.

The construction follows the generated subgroup API of TauCeti.Algebra.Lie.E7.Minuscule.Generated.Smooth.

References

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

theorem TauCeti.E6DoubledMinuscule.isReduced_generatedCoordinateHopfAlgebra (k : Type u) [Field k] [IsAlgClosed k] : IsReduced ↑(generatedCoordinateHopfAlgebra k)

The generated subgroup's coordinate algebra is reduced over an algebraically closed field.

theorem TauCeti.E6DoubledMinuscule.smoothCommHopfAlgProperty_generatedCoordinateHopfAlgebra (k : Type u) [Field k] [IsAlgClosed k] : smoothCommHopfAlgProperty k (generatedCoordinateHopfAlgebra k)

The subgroup generated by the doubled type-E₆ root subgroups and torus is smooth over an algebraically closed field.