Connectedness of the generated short-root type-G₂ carrier

After scalar extension to an algebraically closed field of characteristic three, the four numbered root subgroups and the rank-two weight torus generate a geometrically connected closed subgroup of GL₇. Their coordinate algebras are scalar extensions of a polynomial algebra and a Laurent polynomial algebra, respectively, so each has connected spectrum. The common-kernel construction then preserves connectedness of the generated subgroup. The scalar-extension identification transfers this property to the prime-field carrier itself.

This supplies a geometric property of the generated subgroup needed when comparing the short-root carrier with a pinned simply connected type-G₂ group scheme.

References

• J. S. Milne, Algebraic Groups (2017), Propositions 2.37 and 2.48.
• TauCeti.Algebra.Lie.F4.ShortRoot.PrimeField.Generated.Connected: the carrier/base-change connectedness comparison.

The formal argument follows TauCeti.Algebra.Lie.D4.Tripled.GeneratedConnected.

theorem TauCeti.G2ShortRoot.PrimeField.geometricallyConnectedCommHopfAlgProperty_generatedCoordinateHopfAlgebra (k : Type u) [Field k] [Algebra (ZMod 3) k] [IsAlgClosed k] : geometricallyConnectedCommHopfAlgProperty k (generatedCoordinateHopfAlgebra k)

The closed subgroup generated by the scalar-extended short-root type-G₂ root subgroups and weight torus is geometrically connected over an algebraically closed field.

theorem TauCeti.G2ShortRoot.PrimeField.geometricallyConnectedCommHopfAlgProperty_carrierAlgebra : geometricallyConnectedCommHopfAlgProperty (ZMod 3) carrierAlgebra

The short-root type-G₂ carrier over 𝔽₃ is geometrically connected.