openAI-math-dataset / lean /OAI /GroupTheory /PolycyclicRecognition /Geometry /AlgebraExponential.lean
| import OAI.GroupTheory.PolycyclicRecognition.Geometry.ConjugationFlow | |
| namespace OAI | |
| /-! # Algebra exponentials as actual group elements | |
| The unit-valued exponential keeps the inverse explicit. This bridges the | |
| operator conjugation identity to a faithful group representation. | |
| -/ | |
| noncomputable section | |
| namespace PolycyclicRecognition | |
| variable {A : Type*} [NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A] | |
| def algebraExpUnit (x : A) : Aˣ := by | |
| let : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℝ A | |
| exact | |
| { val := NormedSpace.exp x | |
| inv := NormedSpace.exp (-x) | |
| val_inv := by | |
| rw [← NormedSpace.exp_add_of_commute (Commute.refl x).neg_right, | |
| add_neg_cancel, NormedSpace.exp_zero] | |
| inv_val := by | |
| rw [← NormedSpace.exp_add_of_commute (Commute.refl x).neg_left, | |
| neg_add_cancel, NormedSpace.exp_zero] } | |
| @[simp] theorem coe_algebraExpUnit (x : A) : | |
| (algebraExpUnit x : A) = NormedSpace.exp x := rfl | |
| @[simp] theorem algebraExpUnit_neg (x : A) : | |
| algebraExpUnit (-x) = (algebraExpUnit x)⁻¹ := by | |
| apply Units.ext | |
| rfl | |
| @[simp] theorem algebraExpUnit_zero : algebraExpUnit (0 : A) = 1 := by | |
| apply Units.ext | |
| exact NormedSpace.exp_zero | |
| theorem algebraExpUnit_add_of_commute {x y : A} (hxy : Commute x y) : | |
| algebraExpUnit (x + y) = algebraExpUnit x * algebraExpUnit y := by | |
| let : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℝ A | |
| apply Units.ext | |
| exact NormedSpace.exp_add_of_commute hxy | |
| theorem algebraExpUnit_conj (u : Aˣ) (x : A) : | |
| algebraExpUnit (u * x * (u⁻¹ : Aˣ) : A) = u * algebraExpUnit x * u⁻¹ := by | |
| let : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℝ A | |
| apply Units.ext | |
| exact NormedSpace.exp_units_conj u x | |
| theorem algebraExpUnit_continuous : Continuous (algebraExpUnit : A → Aˣ) := by | |
| let : NormedAlgebra ℚ A := NormedAlgebra.restrictScalars ℚ ℝ A | |
| apply Units.continuous_iff.mpr | |
| exact ⟨NormedSpace.exp_continuous, NormedSpace.exp_continuous.comp continuous_neg⟩ | |
| end PolycyclicRecognition | |
| end | |
| end OAI | |