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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 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 | import OAI.GroupTheory.PolycyclicRecognition.Geometry.ExponentialProduct
import OAI.GroupTheory.PolycyclicRecognition.Algebra.ContinuousOneParameter
import OAI.GroupTheory.PolycyclicRecognition.Geometry.LocalParameterSmoothness
namespace OAI
/-! # Automatic smoothness of continuous homomorphisms into algebra units
Products of one-parameter subgroups supply coordinates of the second kind.
The integral argument for continuous algebra-valued one-parameter groups
then proves smoothness in those coordinates.
-/
noncomputable section
namespace PolycyclicRecognition
open Function
open scoped Manifold ContDiff
variable {E G A : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
[TopologicalSpace G] [ChartedSpace E G] [Group G] [LieGroup (𝓘(ℝ, E)) ⊤ G]
[NormedRing A] [NormedAlgebra ℝ A] [CompleteSpace A]
theorem contDiff_comp_exponentialProduct
{Exp : GroupLieAlgebra (𝓘(ℝ, E)) G → G} (hExp : IsGroupExponential Exp)
(ρ : G →* Aˣ) (hρ : Continuous (fun g => (ρ g : A)))
{n : ℕ} (v : Fin n → E) :
ContDiff ℝ ∞ (fun x => (ρ (exponentialProduct Exp v x) : A)) := by
induction n with
| zero =>
simpa only [exponentialProduct, map_one, Units.val_one] using
(contDiff_const : ContDiff ℝ ∞ (fun _ : Fin 0 → ℝ => (1 : A)))
| succ n ih =>
have hc : ContDiff ℝ ∞ (fun t : ℝ => (ρ (Exp (t • v 0)) : A)) := by
apply contDiff_continuous_oneParameter
· exact hρ.comp (hExp.contMDiff_curve (v 0)).continuous
· rw [zero_smul]
have hz : Exp (0 : E) = 1 := hExp.map_zero
rw [hz, map_one, Units.val_one]
· intro s t
have heq : Exp ((s + t) • v 0) = Exp (s • v 0) * Exp (t • v 0) :=
hExp.map_add_smul (v 0) s t
rw [heq, map_mul, Units.val_mul]
have hh := hc.comp (ContinuousLinearMap.proj (0 : Fin (n + 1)) :
(Fin (n + 1) → ℝ) →L[ℝ] ℝ).contDiff
have ht := (ih (v ∘ Fin.succ)).comp (coordinateTail n).contDiff
have hp := hh.mul ht
change ContDiff ℝ ∞ (fun x : Fin (n + 1) → ℝ =>
(ρ (Exp (x 0 • v 0)) : A) *
(ρ (exponentialProduct Exp (v ∘ Fin.succ) (x ∘ Fin.succ)) : A)) at hp
simpa only [exponentialProduct, map_mul, Units.val_mul] using hp
theorem contMDiffAt_one_of_continuous_unitsHom [FiniteDimensional ℝ E]
{Exp : GroupLieAlgebra (𝓘(ℝ, E)) G → G} (hExp : IsGroupExponential Exp)
(ρ : G →* Aˣ) (hρ : Continuous (fun g => (ρ g : A))) :
ContMDiffAt (𝓘(ℝ, E)) (𝓘(ℝ, A)) ∞ (fun g => (ρ g : A)) 1 := by
let b := Module.finBasis ℝ E
let d := b.equivFun.toContinuousLinearEquiv.symm
have hcomb : tangentCombination b = d.toContinuousLinearMap := by
ext x
simp only [tangentCombination_apply]
exact (b.equivFun_symm_apply x).symm
have hd : HasMFDerivAt (𝓘(ℝ, Fin (Module.finrank ℝ E) → ℝ)) (𝓘(ℝ, E))
(exponentialProduct Exp b) 0 d.toContinuousLinearMap := by
apply (hExp.hasMFDerivAt_exponentialProduct_zero b).congr_mfderiv
exact hcomb
have hh := contMDiffAt_of_local_parameterization (f := fun g => (ρ g : A))
(hExp.contMDiff_exponentialProduct b).contMDiffAt hd
(contDiff_comp_exponentialProduct hExp ρ hρ b).contDiffAt
simpa only [hExp.exponentialProduct_zero] using hh
theorem contMDiff_of_continuous_unitsHom [FiniteDimensional ℝ E]
{Exp : GroupLieAlgebra (𝓘(ℝ, E)) G → G} (hExp : IsGroupExponential Exp)
(ρ : G →* Aˣ) (hρ : Continuous (fun g => (ρ g : A))) :
ContMDiff (𝓘(ℝ, E)) (𝓘(ℝ, A)) ∞ (fun g => (ρ g : A)) := by
have hbase := contMDiffAt_one_of_continuous_unitsHom hExp ρ hρ
intro g
have hl : ContMDiffAt (𝓘(ℝ, E)) (𝓘(ℝ, E)) ∞ (fun x : G => g⁻¹ * x) g :=
(contMDiff_mul_left (I := 𝓘(ℝ, E)) (a := g⁻¹) :
ContMDiff (𝓘(ℝ, E)) (𝓘(ℝ, E)) ∞ (fun x : G => g⁻¹ * x)).contMDiffAt
have hm := hbase.comp_of_eq hl (inv_mul_cancel g)
have hh := (ContinuousLinearMap.mul ℝ A (ρ g : A)).contDiff.contMDiff.contMDiffAt.comp g hm
have heq : (fun x => (ρ g : A) * (ρ (g⁻¹ * x) : A)) = fun x => (ρ x : A) := by
funext x
rw [← Units.val_mul, ← map_mul, mul_inv_cancel_left]
change ContMDiffAt (𝓘(ℝ, E)) (𝓘(ℝ, A)) ∞
(fun x => (ρ g : A) * (ρ (g⁻¹ * x) : A)) g at hh
rwa [heq] at hh
end PolycyclicRecognition
end
end OAI
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