Connectedness 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 connected prime spectra over a field, the affine line and a split torus. Over an
algebraically closed field the subgroup they generate is therefore geometrically connected.
Connectedness is one of the two geometric inputs, smoothness 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. S. Milne, Algebraic Groups (2017), Propositions 2.37 and 2.48.
- R. Steinberg, Lectures on Chevalley Groups, §3.
theorem
TauCeti.E6Minuscule.geometricallyConnectedCommHopfAlgProperty_generatedCoordinateHopfAlgebra
(k : Type v)
[Field k]
[IsAlgClosed k]
:
The subgroup generated by the numbered type-E₆ minuscule root subgroups and weight torus is
geometrically connected over an algebraically closed field.