| import Mathlib.LinearAlgebra.Prod |
| import Mathlib.Tactic.FinCases |
| import OAI.Algebra.Finitistic.Encoding.QuadraticEncoding |
|
|
| namespace OAI |
|
|
| namespace LittleFinitistic.CornerSpaces |
| open ThreeVertexAlgebra |
|
|
| 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) |
|
|
| /-- The actual two-sided corner e_i B e_j as a vector subspace. -/ |
| def corner (i j : Fin 3) : Submodule k (ThreeVertexAlgebra μ) where |
| carrier := {x | vertex μ i * x * vertex μ j = x} |
| zero_mem' := by |
| change vertex μ i * (0 : ThreeVertexAlgebra μ) * vertex μ j = 0 |
| rw [mul_zero,zero_mul] |
| add_mem' := by |
| intro x y hx hy |
| change vertex μ i * x * vertex μ j = x at hx |
| change vertex μ i * y * vertex μ j = y at hy |
| change vertex μ i * (x+y) * vertex μ j = x+y |
| rw [mul_add,add_mul,hx,hy] |
| smul_mem' := by |
| intro r x hx |
| change vertex μ i * x * vertex μ j = x at hx |
| change vertex μ i * (r • x) * vertex μ j = r • x |
| rw [mul_smul_comm,smul_mul_assoc,hx] |
|
|
| noncomputable def corner10Equiv : corner μ 1 0 ≃ₗ[k] V where |
| toFun x := x.val.2.2.2.1 |
| invFun v := ⟨(0,0,0,v,0,0),by |
| change mul μ (mul μ (vertex μ 1) (0,0,0,v,0,0)) (vertex μ 0) = _ |
| simp [vertex,mul]⟩ |
| left_inv x := by |
| apply Subtype.ext |
| have hx := x.property |
| change mul μ (mul μ (vertex μ 1) x.val) (vertex μ 0) = x.val at hx |
| simpa [vertex,mul] using hx |
| right_inv _ := rfl |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| noncomputable def corner21Equiv : corner μ 2 1 ≃ₗ[k] W where |
| toFun x := x.val.2.2.2.2.1 |
| invFun w := ⟨(0,0,0,0,w,0),by |
| change mul μ (mul μ (vertex μ 2) (0,0,0,0,w,0)) (vertex μ 1) = _ |
| simp [vertex,mul]⟩ |
| left_inv x := by |
| apply Subtype.ext |
| have hx := x.property |
| change mul μ (mul μ (vertex μ 2) x.val) (vertex μ 1) = x.val at hx |
| simpa [vertex,mul] using hx |
| right_inv _ := rfl |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| noncomputable def corner20Equiv : corner μ 2 0 ≃ₗ[k] U where |
| toFun x := x.val.2.2.2.2.2 |
| invFun u := ⟨(0,0,0,0,0,u),by |
| change mul μ (mul μ (vertex μ 2) (0,0,0,0,0,u)) (vertex μ 0) = _ |
| simp [vertex,mul]⟩ |
| left_inv x := by |
| apply Subtype.ext |
| have hx := x.property |
| change mul μ (mul μ (vertex μ 2) x.val) (vertex μ 0) = x.val at hx |
| simpa [vertex,mul] using hx |
| right_inv _ := rfl |
| map_add' _ _ := rfl |
| map_smul' _ _ := rfl |
|
|
| /-- A diagonal corner has only its scalar trivial path. -/ |
| noncomputable def diagonalEquiv (i : Fin 3) : corner μ i i ≃ₗ[k] k where |
| toFun x := if i = 0 then x.val.1 else if i = 1 then x.val.2.1 else x.val.2.2.1 |
| invFun c := ⟨c • vertex μ i,by |
| change vertex μ i * (c • vertex μ i) * vertex μ i = c • vertex μ i |
| rw [mul_smul_comm,smul_mul_assoc] |
| have hi := (directedSystem μ).idem i |
| change vertex μ i * vertex μ i = vertex μ i at hi |
| rw [hi,hi]⟩ |
| left_inv x := by |
| apply Subtype.ext |
| have hx := x.property |
| change mul μ (mul μ (vertex μ i) x.val) (vertex μ i) = x.val at hx |
| fin_cases i <;> simpa [vertex,mul,smul_eq μ] using hx |
| right_inv c := by fin_cases i <;> simp [vertex,smul_eq μ] |
| map_add' x y := by fin_cases i <;> rfl |
| map_smul' c x := by fin_cases i <;> rfl |
|
|
| /-- No path points backwards in the actual algebra. -/ |
| theorem corner_zero_of_lt (i j : Fin 3) (hij : i < j) (x : corner μ i j) : x = 0 := by |
| apply Subtype.ext |
| have hx := x.property |
| change (directedSystem μ).e i * x.val * (directedSystem μ).e j = x.val at hx |
| rw [(directedSystem μ).directed j i hij x.val] at hx |
| exact hx.symm |
| end LittleFinitistic.CornerSpaces |
|
|
| end OAI |
|
|
