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6aac220 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 | import OAI.GroupTheory.PolycyclicRecognition.Geometry.IntrinsicTriangularRealization
import OAI.GroupTheory.PolycyclicRecognition.Geometry.MatrixDifferentialFromSmoothness
import OAI.GroupTheory.PolycyclicRecognition.Analysis.ConstructChainMetric
import OAI.GroupTheory.PolycyclicRecognition.Analysis.LocallyCompactGroupEmbedding
namespace OAI
/-! # Closed triangular realizations of analytic solvable Lie models -/
noncomputable section
namespace PolycyclicRecognition
open scoped Manifold Matrix.Norms.Operator
theorem realTriangulable_of_triangularAdjoint (S : SimplyConnectedSolvableLieModel)
(hT : ∃ a, Nonempty (TriangularAdjoint
(GroupLieAlgebra (𝓘(ℝ,S.modelSpace)) S.carrier) a)) : IsRealTriangulable S := by
obtain ⟨d⟩ := S.nonempty_properChainMetric geodesicMetricExistenceInput
let metric : MetricSpace S.carrier := d.compatibleMetric
let tg : IsTopologicalGroup S.carrier := topologicalGroup_of_lieGroup (𝓘(ℝ,S.modelSpace)) ⊤
let lc : LocallyCompactSpace S.carrier :=
Manifold.locallyCompact_of_finiteDimensional (𝓘(ℝ,S.modelSpace))
obtain ⟨a,⟨T⟩⟩ := hT
obtain ⟨m,hm,σ,hσ,_⟩ := onshuusPostRealizationInput_intrinsic T
(groupExponential (E := S.modelSpace) (G := S.carrier)) groupExponential_isGroupExponential
let ρ : S.carrier →* (Matrix (Fin m) (Fin m) ℝ)ˣ := (positiveTriangular m).subtype.comp σ
have hind : Topology.IsInducing ρ :=
Units.isOpenEmbedding_val.isInducing.of_comp_iff.mp hσ
have hinj : Function.Injective ρ := by
intro x y h
apply hσ.injective
change ((ρ x : (Matrix (Fin m) (Fin m) ℝ)ˣ) : Matrix (Fin m) (Fin m) ℝ) =
((ρ y : (Matrix (Fin m) (Fin m) ℝ)ˣ) : Matrix (Fin m) (Fin m) ℝ)
exact congrArg (fun u : (Matrix (Fin m) (Fin m) ℝ)ˣ => (u : Matrix (Fin m) (Fin m) ℝ)) h
exact ⟨m,ρ,closedEmbedding_of_locallyCompact_group ρ ⟨hind,hinj⟩,
fun g => (σ g).property.1⟩
theorem realTriangulable_of_faithful_triangular_hom (S : SimplyConnectedSolvableLieModel)
{m : ℕ} (ρ : S.carrier →* (Matrix (Fin m) (Fin m) ℝ)ˣ)
(hc : Continuous ρ) (hi : Function.Injective ρ)
(ht : ∀ g, (ρ g : Matrix (Fin m) (Fin m) ℝ).IsUpperTriangular) : IsRealTriangulable S := by
obtain ⟨j,hj⟩ := matrixLieDifferentialInput S m ρ hc hi
apply realTriangulable_of_triangularAdjoint S
apply nonempty_triangularAdjoint_of_triangular_representation j
(matrixLieDifferential_injective S m ρ hi j hj)
exact fun v => hasMFDerivAt_upperTriangular hj ht v
end PolycyclicRecognition
end
end OAI
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