Smoothness of the generated type-E₆ minuscule subgroup

The numbered simple-root subgroups and the weight torus of the full-weight type-E₆ minuscule carrier have reduced coordinate algebras over a field: a polynomial algebra on one generator and a group algebra of a free abelian group. Over an algebraically closed field the subgroup they generate therefore has reduced coordinate algebra, and a reduced affine group scheme of finite type over such a field is smooth.

Smoothness is one of the two geometric inputs, connectedness being the other, to recognizing this explicit carrier as a split reductive group of type E₆. The result concerns the generated subgroup; it says nothing about the base change of the integral carrier, which contains it.

References

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

theorem TauCeti.E6Minuscule.isReduced_generatedCoordinateHopfAlgebra (k : Type v) [Field k] [IsAlgClosed k] : IsReduced ↑(generatedCoordinateHopfAlgebra k)

The subgroup generated by the numbered type-E₆ minuscule root subgroups and weight torus has reduced coordinate algebra over an algebraically closed field.

theorem TauCeti.E6Minuscule.smoothCommHopfAlgProperty_generatedCoordinateHopfAlgebra (k : Type v) [Field k] [IsAlgClosed k] : smoothCommHopfAlgProperty k (generatedCoordinateHopfAlgebra k)

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