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8e4d533 | 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 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 | import OAI.Analysis.MatrixScaling.Expansion
import OAI.Geometry.ComplexBundle.Hom
namespace OAI
noncomputable section
/-! Coordinates and realization of Hermitian scaling sections. -/
section
open scoped BigOperators ComplexOrder MatrixOrder Matrix.Norms.L2Operator
open Matrix Bundle Topology
namespace Kaplansky
variable {n d r : ℕ} {Y : Type*} [TopologicalSpace Y]
variable {E F : Y → Type*}
variable [∀ y, NormedAddCommGroup (E y)] [∀ y, InnerProductSpace ℂ (E y)]
variable [∀ y, NormedAddCommGroup (F y)] [∀ y, InnerProductSpace ℂ (F y)]
variable [TopologicalSpace (TotalSpace (Fin d → ℂ) E)]
variable [TopologicalSpace (TotalSpace (Fin r → ℂ) F)]
variable [FiberBundle (Fin d → ℂ) E] [VectorBundle ℂ (Fin d → ℂ) E]
variable [FiberBundle (Fin r → ℂ) F] [VectorBundle ℂ (Fin r → ℂ) F]
/-- Coordinates of genuine fiber maps in the constructed orthonormal frames. -/
def bundleMapCoordinates (T : ∀ y, Fin n → E y →ₗ[ℂ] F y) (a y : Y) :
Fin n → Matrix (Fin r) (Fin d) ℂ :=
fun i k j => inner ℂ
(bundleOrthonormalFrame (trivializationAt (Fin r → ℂ) F a) y k)
(T y i (bundleOrthonormalFrame (trivializationAt (Fin d → ℂ) E a) y j))
theorem bundleMapCoordinates_eq (T : ∀ y, Fin n → E y →ₗ[ℂ] F y) (a y : Y)
(hy : y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) a) (i : Fin n) :
bundleMapCoordinates (d := d) (r := r) T a y i = LinearMap.toMatrix
(bundleOrthonormalBasis (trivializationAt (Fin d → ℂ) E a) y hy.1).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin r → ℂ) F a) y hy.2).toBasis (T y i) := by
ext k j
simp only [bundleMapCoordinates, LinearMap.toMatrix_apply,
OrthonormalBasis.coe_toBasis_repr_apply, OrthonormalBasis.repr_apply_apply,
OrthonormalBasis.coe_toBasis, bundleOrthonormalBasis_apply]
theorem continuousOn_bundleMapCoordinates
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(T : ∀ y, Fin n → E y →ₗ[ℂ] F y)
(hT : ∀ i, Continuous (fun z : TotalSpace (Fin d → ℂ) E =>
TotalSpace.mk' (Fin r → ℂ) z.proj (T z.proj i z.2))) (a : Y) :
ContinuousOn (bundleMapCoordinates (d := d) (r := r) T a) (hermitianChartSet (d := d) (r := r) (E := E) (F := F) a) := by
apply continuousOn_pi.mpr
intro i
apply continuousOn_pi.mpr
intro k
apply continuousOn_pi.mpr
intro j
apply hF
· exact (continuousOn_bundleOrthonormalFrame hF _ k).mono Set.inter_subset_right
· exact (hT i).comp_continuousOn
((continuousOn_bundleOrthonormalFrame hE _ j).mono Set.inter_subset_left)
theorem bundleMapCoordinates_expanding
(T : ∀ y, Fin n → E y →ₗ[ℂ] F y) (hT : ∀ y, IntrinsicExpanding (T y))
(a y : Y) (hy : y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) a) :
Expanding (bundleMapCoordinates (d := d) (r := r) T a y) := by
have he : bundleMapCoordinates (d := d) (r := r) T a y = fun i => LinearMap.toMatrix
(bundleOrthonormalBasis (trivializationAt (Fin d → ℂ) E a) y hy.1).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin r → ℂ) F a) y hy.2).toBasis (T y i) :=
funext (bundleMapCoordinates_eq T a y hy)
rw [he]
exact expanding_in_basis _ _ _ (hT y)
theorem bundleUnitaryTransition_star
{k : ℕ} {V : Y → Type*} [∀ y, NormedAddCommGroup (V y)] [∀ y, InnerProductSpace ℂ (V y)]
[TopologicalSpace (TotalSpace (Fin k → ℂ) V)]
[FiberBundle (Fin k → ℂ) V] [VectorBundle ℂ (Fin k → ℂ) V]
(a b : Trivialization (Fin k → ℂ) (π (Fin k → ℂ) V)) [a.IsLinear ℂ] [b.IsLinear ℂ]
(y : Y) : (bundleUnitaryTransition a b y)ᴴ = bundleUnitaryTransition b a y := by
apply Matrix.ext
intro i j
exact inner_conj_symm _ _
theorem bundleMapCoordinates_compatible
(T : ∀ y, Fin n → E y →ₗ[ℂ] F y) (a b y : Y)
(ha : y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) a)
(hb : y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) b) :
bundleMapCoordinates (d := d) (r := r) T b y = frameTuple (bundleMapCoordinates (d := d) (r := r) T a y)
(hermitianSourceTransition (d := d) (E := E) a b y)
(hermitianTargetTransition (r := r) (F := F) a b y) := by
funext i
dsimp only [frameTuple, hermitianSourceTransition, hermitianTargetTransition]
rw [bundleUnitaryTransition_star, bundleMapCoordinates_eq T a y ha i,
bundleMapCoordinates_eq T b y hb i,
bundleUnitaryTransition_eq _ _ y hb.1 ha.1,
bundleUnitaryTransition_eq _ _ y ha.2 hb.2]
symm
exact basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix
(bundleOrthonormalBasis (trivializationAt (Fin d → ℂ) E b) y hb.1).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin d → ℂ) E a) y ha.1).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin r → ℂ) F b) y hb.2).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin r → ℂ) F a) y ha.2).toBasis (T y i)
omit [∀ y, InnerProductSpace ℂ (E y)] [∀ y, InnerProductSpace ℂ (F y)]
[VectorBundle ℂ (Fin d → ℂ) E] [VectorBundle ℂ (Fin r → ℂ) F] in
theorem mem_hermitianChartSet (y : Y) :
y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) y :=
⟨mem_baseSet_trivializationAt (Fin d → ℂ) E y,
mem_baseSet_trivializationAt (Fin r → ℂ) F y⟩
/-- Interpret a section of the constructed core as linear maps between
its original bundles, using the core's selected chart in each fiber. -/
def realizeHomSection
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(s : ∀ y, (hermitianHomCore (n := n) hE hF).Fiber y) (y : Y) (i : Fin n) :
E y →ₗ[ℂ] F y :=
Matrix.toLin
(bundleOrthonormalBasis (trivializationAt (Fin d → ℂ) E y) y
(mem_baseSet_trivializationAt (Fin d → ℂ) E y)).toBasis
(bundleOrthonormalBasis (trivializationAt (Fin r → ℂ) F y) y
(mem_baseSet_trivializationAt (Fin r → ℂ) F y)).toBasis (s y i)
theorem realizeHomSection_at
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(s : ∀ y, (hermitianHomCore (n := n) hE hF).Fiber y) (y : Y) :
bundleMapCoordinates (d := d) (r := r) (realizeHomSection hE hF s) y y = s y := by
funext i
rw [bundleMapCoordinates_eq _ y y (mem_hermitianChartSet (d := d) (r := r) y) i]
exact LinearMap.toMatrix_toLin _ _ (s y i)
theorem realizeHomSection_coordinates
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(s : ∀ y, (hermitianHomCore (n := n) hE hF).Fiber y) (a y : Y)
(hy : y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) a) :
bundleMapCoordinates (d := d) (r := r) (realizeHomSection hE hF s) a y =
((hermitianHomCore hE hF).localTriv a ⟨y,s y⟩).2 := by
rw [bundleMapCoordinates_compatible _ y a y
(mem_hermitianChartSet (d := d) (r := r) y) hy, realizeHomSection_at]
rfl
theorem continuousOn_realizeHomSection_coordinates
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(s : ∀ y, (hermitianHomCore (n := n) hE hF).Fiber y)
(hs : Continuous (fun y => TotalSpace.mk' (Fin n → Matrix (Fin r) (Fin d) ℂ) y (s y)))
(a : Y) :
ContinuousOn (bundleMapCoordinates (d := d) (r := r) (realizeHomSection hE hF s) a)
(hermitianChartSet (d := d) (r := r) (E := E) (F := F) a) := by
have hc := ((hermitianHomCore hE hF).localTriv a).continuousOn.comp hs.continuousOn
(show Set.MapsTo (fun y => TotalSpace.mk' (Fin n → Matrix (Fin r) (Fin d) ℂ) y (s y))
(hermitianChartSet (d := d) (r := r) (E := E) (F := F) a)
((hermitianHomCore hE hF).localTriv a).source from fun y hy =>
((hermitianHomCore hE hF).localTriv a).mem_source.mpr hy)
exact hc.snd.congr (fun y hy => realizeHomSection_coordinates hE hF s a y hy)
end Kaplansky
end
open scoped BigOperators ComplexOrder MatrixOrder Matrix.Norms.L2Operator
open Matrix Bundle Topology
namespace Kaplansky
variable {n d r : ℕ} {Y : Type*} [TopologicalSpace Y] [LocallyCompactSpace Y]
variable {E F : Y → Type*}
variable [∀ y, NormedAddCommGroup (E y)] [∀ y, InnerProductSpace ℂ (E y)]
variable [∀ y, NormedAddCommGroup (F y)] [∀ y, InnerProductSpace ℂ (F y)]
variable [TopologicalSpace (TotalSpace (Fin d → ℂ) E)]
variable [TopologicalSpace (TotalSpace (Fin r → ℂ) F)]
variable [FiberBundle (Fin d → ℂ) E] [VectorBundle ℂ (Fin d → ℂ) E]
variable [FiberBundle (Fin r → ℂ) F] [VectorBundle ℂ (Fin r → ℂ) F]
/-- Normalization of intrinsic expanding maps descends in the constructed
Hom-tuple bundle. The ordinary input bundles may be nontrivial. -/
theorem exists_hermitian_scaled_section
(hE : ContinuousHermitianMetric (d := d) (E := E))
(hF : ContinuousHermitianMetric (d := r) (E := F))
(T : ∀ y, Fin n → E y →ₗ[ℂ] F y)
(hT : ∀ i, Continuous (fun z : TotalSpace (Fin d → ℂ) E =>
TotalSpace.mk' (Fin r → ℂ) z.proj (T z.proj i z.2)))
(hexp : ∀ y, IntrinsicExpanding (T y)) :
∃ s : ∀ y, (hermitianHomCore (n := n) hE hF).Fiber y,
Continuous (fun y => TotalSpace.mk' (Fin n → Matrix (Fin r) (Fin d) ℂ) y (s y)) ∧
∀ a y, y ∈ hermitianChartSet (d := d) (r := r) (E := E) (F := F) a →
let X := ((hermitianHomCore hE hF).localTriv a ⟨y,s y⟩).2
(∑ i, (X i)ᴴ * X i) = 1 ∧
((2/3 : ℂ) • (1 : Matrix (Fin r) (Fin r) ℂ) - ∑ i, X i * (X i)ᴴ).PosSemidef := by
exact exists_scaled_core_section (hermitianHomCore hE hF)
(bundleMapCoordinates (d := d) (r := r) T)
(continuousOn_bundleMapCoordinates hE hF T hT)
(bundleMapCoordinates_expanding T hexp)
(hermitianSourceTransition (d := d) (E := E))
(hermitianTargetTransition (r := r) (F := F))
(hermitianHomCore_unitary (d := d) (r := r))
(fun a b y _ _ M => hermitianHomCore_coordChange hE hF a b y M)
(bundleMapCoordinates_compatible T)
end Kaplansky
end
end OAI
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