openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Geometry /ActualRotationalModification.lean
| import OAI.GroupTheory.PolycyclicRecognition.Geometry.FactorRotationCharacter | |
| namespace OAI | |
| /-! # The actual compact rotational modification of the original group -/ | |
| noncomputable section | |
| namespace PolycyclicRecognition | |
| open scoped Manifold | |
| variable {E G : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [FiniteDimensional ℝ E] [MetricSpace G] [ChartedSpace E G] [Group G] | |
| [LieGroup (𝓘(ℝ,E)) ⊤ G] | |
| {a : ℕ} {T : ComplexTriangularAdjoint (GroupLieAlgebra (𝓘(ℝ,E)) G) a} | |
| {H : LieSubalgebra ℝ (GroupLieAlgebra (𝓘(ℝ,E)) G)} [H.IsCartanSubalgebra] | |
| {Exp : GroupLieAlgebra (𝓘(ℝ,E)) G → G} | |
| namespace ComplexWeightFactorModel | |
| variable (F : ComplexWeightFactorModel T H Exp) (hExp : IsGroupExponential Exp) | |
| /-- The compact action on the original group, now with its invariant character. -/ | |
| def compactRotationAction : realRotationTorus (GroupLieAlgebra (𝓘(ℝ,E)) G) H →* MulAut G := | |
| (F.exists_invariant_compact_rotations hExp).choose | |
| theorem continuous_compactRotationAction : | |
| Continuous (fun p : realRotationTorus (GroupLieAlgebra (𝓘(ℝ,E)) G) H × G => | |
| F.compactRotationAction hExp p.1 p.2) := | |
| (F.exists_invariant_compact_rotations hExp).choose_spec.1 | |
| theorem compactRotationAction_invariant : | |
| ∀ k g, F.rotationCharacter (F.compactRotationAction hExp k g) = F.rotationCharacter g := | |
| (F.exists_invariant_compact_rotations hExp).choose_spec.2 | |
| attribute [local instance] semidirectProductTopology | |
| /-- The concrete graph subgroup that removes the compact rotations. -/ | |
| abbrev RotationalModification := | |
| ↥(rotationalTwist (F.compactRotationAction hExp) F.rotationCharacter | |
| (F.compactRotationAction_invariant hExp)) | |
| /-- The modification preserves the original underlying topology. -/ | |
| def rotationalModificationHomeomorph : F.RotationalModification hExp ≃ₜ G := | |
| rotationalTwistHomeomorph (F.compactRotationAction hExp) F.rotationCharacter | |
| (F.compactRotationAction_invariant hExp) F.continuous_rotationCharacter | |
| instance rotationalExtensionTopologicalGroup : | |
| IsTopologicalGroup (G ⋊[F.compactRotationAction hExp] | |
| realRotationTorus (GroupLieAlgebra (𝓘(ℝ,E)) G) H) := by | |
| let : IsTopologicalGroup G := topologicalGroup_of_lieGroup (𝓘(ℝ,E)) ⊤ | |
| exact semidirectProductTopologicalGroup _ (F.continuous_compactRotationAction hExp) | |
| instance rotationalModificationTopologicalGroup : | |
| IsTopologicalGroup (F.RotationalModification hExp) := inferInstance | |
| /-- The original group is properly and cocompactly included in the compact extension. -/ | |
| def originalToRotationalExtension : | |
| ProperCocompactHom G (G ⋊[F.compactRotationAction hExp] | |
| realRotationTorus (GroupLieAlgebra (𝓘(ℝ,E)) G) H) := by | |
| let : IsTopologicalGroup G := topologicalGroup_of_lieGroup (𝓘(ℝ,E)) ⊤ | |
| exact semidirect_inl_properCocompact (F.compactRotationAction hExp) | |
| /-- The actual modification has the other proper cocompact inclusion. -/ | |
| def modificationToRotationalExtension : | |
| ProperCocompactHom (F.RotationalModification hExp) | |
| (G ⋊[F.compactRotationAction hExp] realRotationTorus (GroupLieAlgebra (𝓘(ℝ,E)) G) H) := by | |
| let : IsTopologicalGroup G := topologicalGroup_of_lieGroup (𝓘(ℝ,E)) ⊤ | |
| exact rotationalTwist_properCocompact (F.compactRotationAction hExp) F.rotationCharacter | |
| (F.compactRotationAction_invariant hExp) F.continuous_rotationCharacter | |
| end ComplexWeightFactorModel | |
| end PolycyclicRecognition | |
| end | |
| end OAI | |