Geometric connectedness of the upper-triangular subgroup of SLₙ

The upper-triangular subgroup B of SLₙ is geometrically connected over every field. It is a retract, as a scheme, of the upper-triangular subgroup B' of GLₙ, which is geometrically connected: the inclusion B → B' has the left inverse

g ↦ g · diag((det g)⁻¹, 1, …, 1),

which rescales the first column so that the determinant becomes one and keeps the matrix upper triangular. This retraction is not a group homomorphism, but on coordinate rings it is an algebra homomorphism O(B) → O(B') with a left inverse. Hence O(B) embeds into O(B'), and geometric connectedness descends along such embeddings.

In rank zero both groups are trivial and the retraction is the identity.

Main declaration

• TauCeti.SpecialLinear.UpperTriangular. geometricallyConnectedCommHopfAlgProperty_coordinateHopfAlgebra: the upper-triangular subgroup of SLₙ is geometrically connected.

References

• J. S. Milne, Algebraic Groups (2017), Chapters 12 and 17.
• T. A. Springer, Linear Algebraic Groups, Sections 6.2--6.3.

theorem TauCeti.SpecialLinear.UpperTriangular.geometricallyConnectedCommHopfAlgProperty_coordinateHopfAlgebra (n : ℕ) (k : Type u) [Field k] : geometricallyConnectedCommHopfAlgProperty k (coordinateHopfAlgebra k n)

The upper-triangular subgroup of SLₙ is geometrically connected over every field.