File size: 3,397 Bytes
ae00908 | 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 | import OAI.Algebra.Finitistic.Rectification.CornerSpaces
import OAI.Algebra.Finitistic.Rectification.InducedColumns
namespace OAI
noncomputable section
namespace LittleFinitistic.CornerProjection
open ThreeVertexAlgebra CornerSpaces
universe u
variable {k V W U : Type u} [Field k]
[AddCommGroup V] [Module k V] [AddCommGroup W] [Module k W]
[AddCommGroup U] [Module k U]
variable (μ : W →ₗ[k] V →ₗ[k] U)
private theorem videm (i : Fin 3) : vertex μ i * vertex μ i = vertex μ i :=
(directedSystem μ).idem i
theorem corner_left_fixed (i j : Fin 3) (x : corner μ i j) : vertex μ i * x.val = x.val := by
have hx := x.property
change vertex μ i * x.val * vertex μ j = x.val at hx
calc
vertex μ i * x.val = vertex μ i * (vertex μ i * x.val * vertex μ j) := congrArg (fun a : ThreeVertexAlgebra μ => vertex μ i * a) hx.symm
_ = vertex μ i * x.val * vertex μ j := by simp only [← mul_assoc,videm]
_ = x.val := hx
theorem corner_right_fixed (i j : Fin 3) (x : corner μ i j) : x.val * vertex μ j = x.val := by
have hx := x.property
change vertex μ i * x.val * vertex μ j = x.val at hx
calc
x.val * vertex μ j = (vertex μ i * x.val * vertex μ j) * vertex μ j := congrArg (fun a : ThreeVertexAlgebra μ => a * vertex μ j) hx.symm
_ = vertex μ i * x.val * vertex μ j := by rw [mul_assoc,videm]
_ = x.val := hx
theorem ideal_right_fixed (j : Fin 3)
(x : ExternalTensor.leftIdeal (ThreeVertexAlgebra μ) (vertex μ j)) :
x.val * vertex μ j = x.val := by
obtain ⟨b,hb⟩ := x.property
change b * vertex μ j = x.val at hb
rw [← hb,mul_assoc,videm]
/-- Left multiplication by e_i on the actual induced-column first factor B e_j. -/
def projection (i j : Fin 3) :
ExternalTensor.leftIdeal (ThreeVertexAlgebra μ) (vertex μ j) →ₗ[k]
ExternalTensor.leftIdeal (ThreeVertexAlgebra μ) (vertex μ j) where
toFun x := ⟨vertex μ i * x.val,⟨vertex μ i * x.val,by
change (vertex μ i * x.val) * vertex μ j = vertex μ i * x.val
rw [mul_assoc,ideal_right_fixed]⟩⟩
map_add' x y := Subtype.ext (mul_add _ _ _)
map_smul' c x := Subtype.ext (mul_smul_comm _ _ _)
@[simp] theorem projection_apply (i j : Fin 3) (x) :
((projection μ i j) x).val = vertex μ i * x.val := rfl
theorem projection_idempotent (i j : Fin 3) :
(projection μ i j).comp (projection μ i j) = projection μ i j := by
apply LinearMap.ext
intro x
apply Subtype.ext
change vertex μ i * (vertex μ i * x.val) = vertex μ i * x.val
rw [← mul_assoc,videm]
/-- The image of the actual column projector is the concrete two-sided path corner. -/
def rangeEquiv (i j : Fin 3) : (projection μ i j).range ≃ₗ[k] corner μ i j where
toFun x := ⟨x.val.val,by
obtain ⟨y,hy⟩ := x.property
have h := congrArg Subtype.val hy
change vertex μ i * y.val = x.val.val at h
change vertex μ i * x.val.val * vertex μ j = x.val.val
rw [← h,← mul_assoc,videm,mul_assoc,ideal_right_fixed]⟩
invFun x := ⟨⟨x.val,⟨x.val,corner_right_fixed μ i j x⟩⟩,by
refine ⟨⟨x.val,⟨x.val,corner_right_fixed μ i j x⟩⟩,?_⟩
apply Subtype.ext
exact corner_left_fixed μ i j x⟩
left_inv x := by apply Subtype.ext; apply Subtype.ext; rfl
right_inv x := rfl
map_add' x y := rfl
map_smul' c x := rfl
end LittleFinitistic.CornerProjection
end
end OAI
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