openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Geometry /AdjointMatrixHeight.lean
| import OAI.GroupTheory.PolycyclicRecognition.Geometry.AdjointExponentialRepresentation | |
| import OAI.GroupTheory.PolycyclicRecognition.Geometry.TriangularMatrixExponential | |
| import OAI.GroupTheory.PolycyclicRecognition.Geometry.TriangularHeight | |
| namespace OAI | |
| /-! # Logarithmic diagonal characters of the actual adjoint action -/ | |
| noncomputable section | |
| namespace PolycyclicRecognition | |
| open scoped Manifold Matrix.Norms.Operator | |
| def matrixLogDiagonalHom {G : Type*} [Group G] {n : ℕ} | |
| (ρ : G →* Matrix (Fin n) (Fin n) ℝ) | |
| (hupper : ∀ g, (ρ g).IsUpperTriangular) (hpos : ∀ g i, 0 < ρ g i i) : | |
| G →* Multiplicative (Fin n → ℝ) where | |
| toFun g := Multiplicative.ofAdd (fun i => Real.log (ρ g i i)) | |
| map_one' := by | |
| apply Multiplicative.ext | |
| funext i | |
| simp | |
| map_mul' g h := by | |
| apply Multiplicative.ext | |
| funext i | |
| change Real.log (ρ (g * h) i i) = Real.log (ρ g i i) + Real.log (ρ h i i) | |
| rw [map_mul, triangular_mul_diag _ _ (hupper g) (hupper h), | |
| Real.log_mul (hpos g i).ne' (hpos h i).ne'] | |
| @[simp] theorem matrixLogDiagonalHom_apply {G : Type*} [Group G] {n : ℕ} | |
| (ρ : G →* Matrix (Fin n) (Fin n) ℝ) (hupper) (hpos) (g : G) (i : Fin n) : | |
| (matrixLogDiagonalHom ρ hupper hpos g).toAdd i = Real.log (ρ g i i) := rfl | |
| variable {E G : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [FiniteDimensional ℝ E] [TopologicalSpace G] [ChartedSpace E G] | |
| [Group G] [LieGroup (𝓘(ℝ, E)) ⊤ G] [T2Space G] | |
| def adjointMatrixHom {n : ℕ} (b : Module.Basis (Fin n) ℝ (GroupLieAlgebra (𝓘(ℝ, E)) G)) : | |
| G →* Matrix (Fin n) (Fin n) ℝ := | |
| (LinearMap.toMatrixAlgEquiv b).toMonoidHom.comp | |
| (ContinuousLinearMap.toLinearMapRingHom.toMonoidHom.comp conjugationDifferentialHom) | |
| variable {A : Type*} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] | |
| attribute [local instance 100] LieRing.ofAssociativeRing | |
| theorem adjointMatrixHom_exp {n : ℕ} | |
| (b : Module.Basis (Fin n) ℝ (GroupLieAlgebra (𝓘(ℝ, E)) G)) | |
| {Exp : GroupLieAlgebra (𝓘(ℝ, E)) G → G} (hExp : IsGroupExponential Exp) | |
| (j : GroupLieAlgebra (𝓘(ℝ, E)) G →ₗ⁅ℝ⁆ A) (ρ : G →* Aˣ) | |
| (hρ : Function.Injective ρ) (hρExp : ∀ x, ρ (Exp x) = algebraExpUnit (j x)) | |
| (x : GroupLieAlgebra (𝓘(ℝ, E)) G) : | |
| adjointMatrixHom b (Exp x) = | |
| NormedSpace.exp (LinearMap.toMatrix b b (LieAlgebra.ad ℝ _ x)) := by | |
| change LinearMap.toMatrix b b ((conjugationDifferentialHom (Exp x)).toLinearMap) = _ | |
| rw [conjugationDifferential_exp hExp j ρ hρ hρExp, toMatrix_adjointExponentialLinear] | |
| theorem adjointMatrixHom_upper {n : ℕ} | |
| (T : TriangularAdjoint (GroupLieAlgebra (𝓘(ℝ, E)) G) n) | |
| {Exp : GroupLieAlgebra (𝓘(ℝ, E)) G → G} (hExp : IsGroupExponential Exp) | |
| (hsurj : Function.Surjective Exp) | |
| (j : GroupLieAlgebra (𝓘(ℝ, E)) G →ₗ⁅ℝ⁆ A) (ρ : G →* Aˣ) | |
| (hρ : Function.Injective ρ) (hρExp : ∀ x, ρ (Exp x) = algebraExpUnit (j x)) | |
| (g : G) : (adjointMatrixHom T.basis g).IsUpperTriangular := by | |
| obtain ⟨x, rfl⟩ := hsurj g | |
| rw [adjointMatrixHom_exp T.basis hExp j ρ hρ hρExp] | |
| exact matrix_exp_upperTriangular _ (T.upper x) | |
| theorem adjointMatrixHom_exp_diagonal {n : ℕ} | |
| (T : TriangularAdjoint (GroupLieAlgebra (𝓘(ℝ, E)) G) n) | |
| {Exp : GroupLieAlgebra (𝓘(ℝ, E)) G → G} (hExp : IsGroupExponential Exp) | |
| (j : GroupLieAlgebra (𝓘(ℝ, E)) G →ₗ⁅ℝ⁆ A) (ρ : G →* Aˣ) | |
| (hρ : Function.Injective ρ) (hρExp : ∀ x, ρ (Exp x) = algebraExpUnit (j x)) | |
| (x : GroupLieAlgebra (𝓘(ℝ, E)) G) (i : Fin n) : | |
| adjointMatrixHom T.basis (Exp x) i i = Real.exp (T.character i x) := by | |
| rw [adjointMatrixHom_exp T.basis hExp j ρ hρ hρExp, | |
| matrix_exp_diagonal _ (T.upper x), T.character_eq_diag] | |
| /-- The actual adjoint diagonal integrates exactly the canonical height | |
| characters, using the already required representation and exponential chart. -/ | |
| def adjointCharacterHom {n : ℕ} | |
| (T : TriangularAdjoint (GroupLieAlgebra (𝓘(ℝ, E)) G) n) | |
| {Exp : GroupLieAlgebra (𝓘(ℝ, E)) G → G} (hExp : IsGroupExponential Exp) | |
| (hsurj : Function.Surjective Exp) | |
| (j : GroupLieAlgebra (𝓘(ℝ, E)) G →ₗ⁅ℝ⁆ A) (ρ : G →* Aˣ) | |
| (hρ : Function.Injective ρ) (hρExp : ∀ x, ρ (Exp x) = algebraExpUnit (j x)) : | |
| G →* Multiplicative (Fin n → ℝ) := | |
| matrixLogDiagonalHom (adjointMatrixHom T.basis) | |
| (adjointMatrixHom_upper T hExp hsurj j ρ hρ hρExp) | |
| (by | |
| intro g i | |
| obtain ⟨x, rfl⟩ := hsurj g | |
| rw [adjointMatrixHom_exp_diagonal T hExp j ρ hρ hρExp] | |
| exact Real.exp_pos _) | |
| theorem adjointCharacterHom_exp {n : ℕ} | |
| (T : TriangularAdjoint (GroupLieAlgebra (𝓘(ℝ, E)) G) n) | |
| {Exp : GroupLieAlgebra (𝓘(ℝ, E)) G → G} (hExp : IsGroupExponential Exp) | |
| (hsurj : Function.Surjective Exp) | |
| (j : GroupLieAlgebra (𝓘(ℝ, E)) G →ₗ⁅ℝ⁆ A) (ρ : G →* Aˣ) | |
| (hρ : Function.Injective ρ) (hρExp : ∀ x, ρ (Exp x) = algebraExpUnit (j x)) | |
| (x : GroupLieAlgebra (𝓘(ℝ, E)) G) (i : Fin n) : | |
| (adjointCharacterHom T hExp hsurj j ρ hρ hρExp (Exp x)).toAdd i = T.character i x := by | |
| change Real.log (adjointMatrixHom T.basis (Exp x) i i) = _ | |
| rw [adjointMatrixHom_exp_diagonal T hExp j ρ hρ hρExp, Real.log_exp] | |
| end PolycyclicRecognition | |
| end | |
| end OAI | |