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6aac220 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 | 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
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