openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Geometry /AnalyticTriangularRealization.lean
| 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 | |