Connectedness of the generated doubled type-E₆ subgroup

The numbered root subgroups and the weight torus of the doubled minuscule carrier generate a closed subgroup of GL₅₄. Over an algebraically closed field this subgroup is connected: the coordinate algebras of its generators have connected spectra, and the closed subgroup they generate inherits this property. This supplies the connectedness input for recognizing the explicit carrier as a pinned split group of type E₆.

The integral carrier is presently known to contain the generated subgroup. The result here concerns the generated subgroup; identifying it with the integral carrier after base change requires a separate comparison.

References

• J. S. Milne, Algebraic Groups (2017), Propositions 2.37 and 2.48.
• R. Steinberg, Lectures on Chevalley Groups, Section 3.
• TauCeti.Algebra.AlgebraicGroup.Connected.Generated, connectedness of common-kernel quotients.
• TauCeti.Algebra.Lie.D4.Tripled.GeneratedConnected, the analogous result for the tripled type-D₄ generated subgroup.

theorem TauCeti.E6DoubledMinuscule.geometricallyConnectedCommHopfAlgProperty_generatedCoordinateHopfAlgebra (k : Type u) [Field k] [IsAlgClosed k] : geometricallyConnectedCommHopfAlgProperty k (generatedCoordinateHopfAlgebra k)

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