openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Geometry /AnalyticDerivationFlow.lean
| import OAI.GroupTheory.PolycyclicRecognition.Geometry.AlgebraExponential | |
| import OAI.GroupTheory.PolycyclicRecognition.Geometry.LieWordDerivative | |
| import Mathlib.Algebra.Lie.Derivation.Basic | |
| namespace OAI | |
| /-! # The analytic exponential of any finite-dimensional real derivation | |
| Intertwining the bracket with the operator commutator proves bracket | |
| preservation. The derivation need not be nilpotent. | |
| -/ | |
| noncomputable section | |
| namespace PolycyclicRecognition | |
| variable {L : Type*} [RealNormedLieAlgebra L] [FiniteDimensional ℝ L] | |
| theorem derivation_bracket_intertwine (D : LieDerivation ℝ L L) (x : L) : | |
| normedLieBracket (D x) = | |
| adjointOperator (LinearMap.toContinuousLinearMap D.toLinearMap) (normedLieBracket x) := by | |
| apply ContinuousLinearMap.ext | |
| intro y | |
| change ⁅D x, y⁆ = D ⁅x, y⁆ - ⁅x, D y⁆ | |
| rw [D.apply_lie_eq_sub] | |
| have h := lie_skew (D x) y | |
| rw [← h] | |
| abel | |
| theorem analytic_derivation_exp_preserves_lie (D : LieDerivation ℝ L L) (x y : L) : | |
| NormedSpace.exp (LinearMap.toContinuousLinearMap D.toLinearMap) ⁅x, y⁆ = | |
| ⁅NormedSpace.exp (LinearMap.toContinuousLinearMap D.toLinearMap) x, | |
| NormedSpace.exp (LinearMap.toContinuousLinearMap D.toLinearMap) y⁆ := by | |
| let d := LinearMap.toContinuousLinearMap D.toLinearMap | |
| have h := exp_intertwine_apply (normedLieBracket (L := L)) d (adjointOperator d) | |
| (derivation_bracket_intertwine D) x | |
| rw [exp_adjointOperator_apply] at h | |
| have hi := congrArg (fun T : L →L[ℝ] L => T y) (algebraExpUnit d).inv_val | |
| change NormedSpace.exp (-d) (NormedSpace.exp d y) = y at hi | |
| have he := DFunLike.congr_fun h (NormedSpace.exp d y) | |
| change ⁅NormedSpace.exp d x, NormedSpace.exp d y⁆ = | |
| NormedSpace.exp d ⁅x, NormedSpace.exp (-d) (NormedSpace.exp d y)⁆ at he | |
| rw [hi] at he | |
| exact he.symm | |
| def analyticDerivationExp (D : LieDerivation ℝ L L) : L ≃ₗ⁅ℝ⁆ L := | |
| { (ContinuousLinearEquiv.ofUnit | |
| (algebraExpUnit (LinearMap.toContinuousLinearMap D.toLinearMap))).toLinearEquiv with | |
| map_lie' := fun {x y} => analytic_derivation_exp_preserves_lie D x y } | |
| @[simp] theorem analyticDerivationExp_apply (D : LieDerivation ℝ L L) (x : L) : | |
| analyticDerivationExp D x = | |
| NormedSpace.exp (LinearMap.toContinuousLinearMap D.toLinearMap) x := rfl | |
| def analyticDerivationFlow (D : LieDerivation ℝ L L) (t : ℝ) : L ≃ₗ⁅ℝ⁆ L := | |
| analyticDerivationExp (t • D) | |
| theorem analyticDerivationFlow_zero (D : LieDerivation ℝ L L) (x : L) : | |
| analyticDerivationFlow D 0 x = x := by | |
| rw [analyticDerivationFlow, zero_smul, analyticDerivationExp_apply] | |
| change NormedSpace.exp (0 : L →L[ℝ] L) x = x | |
| rw [NormedSpace.exp_zero] | |
| rfl | |
| theorem analyticDerivationFlow_add (D : LieDerivation ℝ L L) (s t : ℝ) (x : L) : | |
| analyticDerivationFlow D (s + t) x = | |
| analyticDerivationFlow D s (analyticDerivationFlow D t x) := by | |
| let := NormedAlgebra.restrictScalars ℚ ℝ (L →L[ℝ] L) | |
| let d := LinearMap.toContinuousLinearMap D.toLinearMap | |
| change NormedSpace.exp ((s + t) • d) x = | |
| NormedSpace.exp (s • d) (NormedSpace.exp (t • d) x) | |
| rw [add_smul, NormedSpace.exp_add_of_commute | |
| (((Commute.refl d).smul_left s).smul_right t)] | |
| rfl | |
| end PolycyclicRecognition | |
| end | |
| end OAI | |