| import Mathlib |
| import OAI.Algebra.MarkedTensor.AdjointCoordinates |
|
|
| namespace OAI |
|
|
| /-! Finite rational reference words whose adjoint actions span all endomorphisms. -/ |
|
|
| noncomputable section |
| open scoped BigOperators commutatorElement |
| open Matrix |
|
|
| namespace BoundaryOnly.Holonomy |
| open Matrix |
| open scoped BigOperators |
|
|
| |
| def referenceGenerator : Fin 3 → SpecialLinearGroup (Fin 2) ℚ := ![ |
| ⟨!![2,0;0,1/2], by norm_num [Matrix.det_fin_two]⟩, |
| ⟨!![1,1;0,1], by norm_num [Matrix.det_fin_two]⟩, |
| ⟨!![1,0;1,1], by norm_num [Matrix.det_fin_two]⟩] |
|
|
| def referenceRho : FreeGroup (Fin 3) →* SpecialLinearGroup (Fin 2) ℚ := |
| FreeGroup.lift referenceGenerator |
|
|
| def shortWords : Set (FreeGroup (Fin 3)) := {g | FreeGroup.norm g ≤ 5} |
|
|
| def spanningWord : Fin 9 → FreeGroup (Fin 3) := ![1,FreeGroup.of 0, |
| FreeGroup.of 0 * FreeGroup.of 0,FreeGroup.of 1,FreeGroup.of 2, |
| FreeGroup.of 1 * FreeGroup.of 0,FreeGroup.of 2 * FreeGroup.of 0, |
| FreeGroup.of 0 * FreeGroup.of 1,FreeGroup.of 0 * FreeGroup.of 2] |
|
|
| lemma spanningWord_mem (i : Fin 9) : spanningWord i ∈ shortWords := by |
| have hm (i j : Fin 3) : FreeGroup.norm (FreeGroup.of i * FreeGroup.of j) ≤ 5 := |
| (FreeGroup.norm_mul_le _ _).trans (by simp) |
| fin_cases i <;> simp [spanningWord, shortWords,hm] |
|
|
| def spanningMatrix : Fin 9 → Matrix (Fin 3) (Fin 3) ℚ := ![ |
| !![1,0,0; 0,1,0; 0,0,1], |
| !![4,0,0; 0,1,0; 0,0,(1/4)], |
| !![16,0,0; 0,1,0; 0,0,(1/16)], |
| !![1,-2,-1; 0,1,1; 0,0,1], |
| !![1,0,0; -1,1,0; -1,2,1], |
| !![4,-2,(-1/4); 0,1,(1/4); 0,0,(1/4)], |
| !![4,0,0; -4,1,0; -4,2,(1/4)], |
| !![4,-8,-4; 0,1,1; 0,0,(1/4)], |
| !![4,0,0; -1,1,0; (-1/4),(1/2),(1/4)]] |
|
|
| @[simp] lemma spanningMatrix_at_0 : spanningMatrix 0 = !![1,0,0; 0,1,0; 0,0,1] := rfl |
| @[simp] lemma spanningWord_at_0 : spanningWord 0 = 1 := rfl |
| @[simp] lemma spanningMatrix_at_1 : spanningMatrix 1 = !![4,0,0; 0,1,0; 0,0,(1/4)] := rfl |
| @[simp] lemma spanningWord_at_1 : spanningWord 1 = FreeGroup.of 0 := rfl |
| @[simp] lemma spanningMatrix_at_2 : spanningMatrix 2 = !![16,0,0; 0,1,0; 0,0,(1/16)] := rfl |
| @[simp] lemma spanningWord_at_2 : spanningWord 2 = FreeGroup.of 0 * FreeGroup.of 0 := rfl |
| @[simp] lemma spanningMatrix_at_3 : spanningMatrix 3 = !![1,-2,-1; 0,1,1; 0,0,1] := rfl |
| @[simp] lemma spanningWord_at_3 : spanningWord 3 = FreeGroup.of 1 := rfl |
| @[simp] lemma spanningMatrix_at_4 : spanningMatrix 4 = !![1,0,0; -1,1,0; -1,2,1] := rfl |
| @[simp] lemma spanningWord_at_4 : spanningWord 4 = FreeGroup.of 2 := rfl |
| @[simp] lemma spanningMatrix_at_5 : spanningMatrix 5 = !![4,-2,(-1/4); 0,1,(1/4); 0,0,(1/4)] := rfl |
| @[simp] lemma spanningWord_at_5 : spanningWord 5 = FreeGroup.of 1 * FreeGroup.of 0 := rfl |
| @[simp] lemma spanningMatrix_at_6 : spanningMatrix 6 = !![4,0,0; -4,1,0; -4,2,(1/4)] := rfl |
| @[simp] lemma spanningWord_at_6 : spanningWord 6 = FreeGroup.of 2 * FreeGroup.of 0 := rfl |
| @[simp] lemma spanningMatrix_at_7 : spanningMatrix 7 = !![4,-8,-4; 0,1,1; 0,0,(1/4)] := rfl |
| @[simp] lemma spanningWord_at_7 : spanningWord 7 = FreeGroup.of 0 * FreeGroup.of 1 := rfl |
| @[simp] lemma spanningMatrix_at_8 : spanningMatrix 8 = !![4,0,0; -1,1,0; (-1/4),(1/2),(1/4)] := rfl |
| @[simp] lemma spanningWord_at_8 : spanningWord 8 = FreeGroup.of 0 * FreeGroup.of 2 := rfl |
|
|
| @[simp] lemma referenceRho_of (i : Fin 3) : referenceRho (FreeGroup.of i) = referenceGenerator i := by simp [referenceRho] |
|
|
| @[simp] lemma reference_mat_0 : mat (referenceGenerator 0) = !![(2:ℚ),0;0,1/2] := rfl |
| @[simp] lemma reference_mat_1 : mat (referenceGenerator 1) = !![(1:ℚ),1;0,1] := rfl |
| @[simp] lemma reference_mat_2 : mat (referenceGenerator 2) = !![(1:ℚ),0;1,1] := rfl |
| lemma reference_mat_inv (g : SpecialLinearGroup (Fin 2) ℚ) : |
| mat g⁻¹ = Matrix.adjugate (mat g) := rfl |
| lemma spanningMatrix_0 : LinearMap.toMatrix' (adjoint (referenceRho (spanningWord 0))) = |
| spanningMatrix 0 := by |
| ext i j |
| change extract (mat (referenceRho (spanningWord 0)) * traceless (Pi.single j 1) * |
| mat ((referenceRho (spanningWord 0))⁻¹)) i = spanningMatrix 0 i j |
| simp only [spanningWord_at_0, map_one, inv_one, mat_one, one_mul, mul_one, |
| traceless_extract, spanningMatrix_at_0] |
| fin_cases i <;> fin_cases j <;> norm_num [Pi.single_apply] |
|
|
| lemma spanningMatrix_1 : LinearMap.toMatrix' (adjoint (referenceRho (spanningWord 1))) = |
| spanningMatrix 1 := by |
| ext i j |
| change extract (mat (referenceRho (spanningWord 1)) * traceless (Pi.single j 1) * |
| mat ((referenceRho (spanningWord 1))⁻¹)) i = spanningMatrix 1 i j |
| simp only [spanningWord_at_1, referenceRho_of, reference_mat_inv, reference_mat_0] |
| fin_cases i <;> fin_cases j <;> |
| norm_num [extract, traceless, Matrix.mul_apply, |
| Matrix.adjugate_fin_two, Fin.sum_univ_two, Pi.single_apply, Fin.ext_iff] |
|
|
| lemma spanningMatrix_2 : LinearMap.toMatrix' (adjoint (referenceRho (spanningWord 2))) = |
| spanningMatrix 2 := by |
| ext i j |
| change extract (mat (referenceRho (spanningWord 2)) * traceless (Pi.single j 1) * |
| mat ((referenceRho (spanningWord 2))⁻¹)) i = spanningMatrix 2 i j |
| simp only [spanningWord_at_2, map_mul, referenceRho_of, |
| reference_mat_inv, mat_mul, reference_mat_0] |
| fin_cases i <;> fin_cases j <;> |
| norm_num [extract, traceless, Matrix.mul_apply, |
| Matrix.adjugate_fin_two, Fin.sum_univ_two, Pi.single_apply, Fin.ext_iff] |
|
|
| lemma spanningMatrix_3 : LinearMap.toMatrix' (adjoint (referenceRho (spanningWord 3))) = |
| spanningMatrix 3 := by |
| ext i j |
| change extract (mat (referenceRho (spanningWord 3)) * traceless (Pi.single j 1) * |
| mat ((referenceRho (spanningWord 3))⁻¹)) i = spanningMatrix 3 i j |
| simp only [spanningWord_at_3, referenceRho_of, reference_mat_inv, reference_mat_1] |
| fin_cases i <;> fin_cases j <;> |
| norm_num [extract, traceless, Matrix.mul_apply, |
| Matrix.adjugate_fin_two, Fin.sum_univ_two, Pi.single_apply, Fin.ext_iff] |
|
|
| lemma spanningMatrix_4 : LinearMap.toMatrix' (adjoint (referenceRho (spanningWord 4))) = |
| spanningMatrix 4 := by |
| ext i j |
| change extract (mat (referenceRho (spanningWord 4)) * traceless (Pi.single j 1) * |
| mat ((referenceRho (spanningWord 4))⁻¹)) i = spanningMatrix 4 i j |
| simp only [spanningWord_at_4, referenceRho_of, reference_mat_inv, reference_mat_2] |
| fin_cases i <;> fin_cases j <;> |
| norm_num [extract, traceless, Matrix.mul_apply, |
| Matrix.adjugate_fin_two, Fin.sum_univ_two, Pi.single_apply, Fin.ext_iff] |
|
|
| lemma spanningMatrix_5 : LinearMap.toMatrix' (adjoint (referenceRho (spanningWord 5))) = |
| spanningMatrix 5 := by |
| ext i j |
| change extract (mat (referenceRho (spanningWord 5)) * traceless (Pi.single j 1) * |
| mat ((referenceRho (spanningWord 5))⁻¹)) i = spanningMatrix 5 i j |
| simp only [spanningWord_at_5, map_mul, referenceRho_of, |
| reference_mat_inv, mat_mul, reference_mat_0, reference_mat_1] |
| fin_cases i <;> fin_cases j <;> |
| norm_num [extract, traceless, Matrix.mul_apply, |
| Matrix.adjugate_fin_two, Fin.sum_univ_two, Pi.single_apply, Fin.ext_iff] |
|
|
| lemma spanningMatrix_6 : LinearMap.toMatrix' (adjoint (referenceRho (spanningWord 6))) = |
| spanningMatrix 6 := by |
| ext i j |
| change extract (mat (referenceRho (spanningWord 6)) * traceless (Pi.single j 1) * |
| mat ((referenceRho (spanningWord 6))⁻¹)) i = spanningMatrix 6 i j |
| simp only [spanningWord_at_6, map_mul, referenceRho_of, |
| reference_mat_inv, mat_mul, reference_mat_0, reference_mat_2] |
| fin_cases i <;> fin_cases j <;> |
| norm_num [extract, traceless, Matrix.mul_apply, |
| Matrix.adjugate_fin_two, Fin.sum_univ_two, Pi.single_apply, Fin.ext_iff] |
|
|
| lemma spanningMatrix_7 : LinearMap.toMatrix' (adjoint (referenceRho (spanningWord 7))) = |
| spanningMatrix 7 := by |
| ext i j |
| change extract (mat (referenceRho (spanningWord 7)) * traceless (Pi.single j 1) * |
| mat ((referenceRho (spanningWord 7))⁻¹)) i = spanningMatrix 7 i j |
| simp only [spanningWord_at_7, map_mul, referenceRho_of, |
| reference_mat_inv, mat_mul, reference_mat_0, reference_mat_1] |
| fin_cases i <;> fin_cases j <;> |
| norm_num [extract, traceless, Matrix.mul_apply, |
| Matrix.adjugate_fin_two, Fin.sum_univ_two, Pi.single_apply, Fin.ext_iff] |
|
|
| lemma spanningMatrix_8 : LinearMap.toMatrix' (adjoint (referenceRho (spanningWord 8))) = |
| spanningMatrix 8 := by |
| ext i j |
| change extract (mat (referenceRho (spanningWord 8)) * traceless (Pi.single j 1) * |
| mat ((referenceRho (spanningWord 8))⁻¹)) i = spanningMatrix 8 i j |
| simp only [spanningWord_at_8, map_mul, referenceRho_of, |
| reference_mat_inv, mat_mul, reference_mat_0, reference_mat_2] |
| fin_cases i <;> fin_cases j <;> |
| norm_num [extract, traceless, Matrix.mul_apply, |
| Matrix.adjugate_fin_two, Fin.sum_univ_two, Pi.single_apply, Fin.ext_iff] |
|
|
| lemma spanningMatrix_eq (i : Fin 9) : LinearMap.toMatrix' (adjoint (referenceRho (spanningWord i))) = |
| spanningMatrix i := by |
| fin_cases i |
| · exact spanningMatrix_0 |
| · exact spanningMatrix_1 |
| · exact spanningMatrix_2 |
| · exact spanningMatrix_3 |
| · exact spanningMatrix_4 |
| · exact spanningMatrix_5 |
| · exact spanningMatrix_6 |
| · exact spanningMatrix_7 |
| · exact spanningMatrix_8 |
|
|
| lemma matrixUnit_00 : Matrix.single 0 0 (1:ℚ) = |
| ((1/45) : ℚ) • spanningMatrix 0 + |
| ((-1/9) : ℚ) • spanningMatrix 1 + |
| ((4/45) : ℚ) • spanningMatrix 2 := by |
| ext i j |
| fin_cases i <;> fin_cases j <;> norm_num [Matrix.single_apply, Matrix.smul_apply, Matrix.add_apply] |
|
|
| lemma matrixUnit_01 : Matrix.single 0 1 (1:ℚ) = |
| ((-1/6) : ℚ) • spanningMatrix 0 + |
| ((2/3) : ℚ) • spanningMatrix 1 + |
| ((1/6) : ℚ) • spanningMatrix 3 + |
| ((-2/3) : ℚ) • spanningMatrix 5 := by |
| ext i j |
| fin_cases i <;> fin_cases j <;> norm_num [Matrix.single_apply, Matrix.smul_apply, Matrix.add_apply] |
|
|
| lemma matrixUnit_02 : Matrix.single 0 2 (1:ℚ) = |
| (-1 : ℚ) • spanningMatrix 1 + |
| ((4/3) : ℚ) • spanningMatrix 5 + |
| ((-1/3) : ℚ) • spanningMatrix 7 := by |
| ext i j |
| fin_cases i <;> fin_cases j <;> norm_num [Matrix.single_apply, Matrix.smul_apply, Matrix.add_apply] |
|
|
| lemma matrixUnit_10 : Matrix.single 1 0 (1:ℚ) = |
| ((-1/3) : ℚ) • spanningMatrix 0 + |
| ((4/3) : ℚ) • spanningMatrix 1 + |
| ((1/3) : ℚ) • spanningMatrix 4 + |
| ((-4/3) : ℚ) • spanningMatrix 8 := by |
| ext i j |
| fin_cases i <;> fin_cases j <;> norm_num [Matrix.single_apply, Matrix.smul_apply, Matrix.add_apply] |
|
|
| lemma matrixUnit_11 : Matrix.single 1 1 (1:ℚ) = |
| ((-4/9) : ℚ) • spanningMatrix 0 + |
| ((17/9) : ℚ) • spanningMatrix 1 + |
| ((-4/9) : ℚ) • spanningMatrix 2 := by |
| ext i j |
| fin_cases i <;> fin_cases j <;> norm_num [Matrix.single_apply, Matrix.smul_apply, Matrix.add_apply] |
|
|
| lemma matrixUnit_12 : Matrix.single 1 2 (1:ℚ) = |
| ((-4/3) : ℚ) • spanningMatrix 0 + |
| ((1/3) : ℚ) • spanningMatrix 1 + |
| ((4/3) : ℚ) • spanningMatrix 3 + |
| ((-1/3) : ℚ) • spanningMatrix 7 := by |
| ext i j |
| fin_cases i <;> fin_cases j <;> norm_num [Matrix.single_apply, Matrix.smul_apply, Matrix.add_apply] |
|
|
| lemma matrixUnit_20 : Matrix.single 2 0 (1:ℚ) = |
| (-1 : ℚ) • spanningMatrix 1 + |
| ((-1/3) : ℚ) • spanningMatrix 6 + |
| ((4/3) : ℚ) • spanningMatrix 8 := by |
| ext i j |
| fin_cases i <;> fin_cases j <;> norm_num [Matrix.single_apply, Matrix.smul_apply, Matrix.add_apply] |
|
|
| lemma matrixUnit_21 : Matrix.single 2 1 (1:ℚ) = |
| ((-2/3) : ℚ) • spanningMatrix 0 + |
| ((1/6) : ℚ) • spanningMatrix 1 + |
| ((2/3) : ℚ) • spanningMatrix 4 + |
| ((-1/6) : ℚ) • spanningMatrix 6 := by |
| ext i j |
| fin_cases i <;> fin_cases j <;> norm_num [Matrix.single_apply, Matrix.smul_apply, Matrix.add_apply] |
|
|
| lemma matrixUnit_22 : Matrix.single 2 2 (1:ℚ) = |
| ((64/45) : ℚ) • spanningMatrix 0 + |
| ((-16/9) : ℚ) • spanningMatrix 1 + |
| ((16/45) : ℚ) • spanningMatrix 2 := by |
| ext i j |
| fin_cases i <;> fin_cases j <;> norm_num [Matrix.single_apply, Matrix.smul_apply, Matrix.add_apply] |
|
|
| |
|
|
| lemma shortWords_finite : shortWords.Finite := by |
| let f : (Σ n : Fin 6, Fin n.val → Fin 3 × Bool) → shortWords := fun w ↦ |
| ⟨FreeGroup.mk (List.ofFn w.2), |
| FreeGroup.norm_mk_le.trans (by simpa only [List.length_ofFn] using Nat.le_of_lt_succ w.1.isLt)⟩ |
| have hf : Function.Surjective f := by |
| rintro ⟨g,hg⟩ |
| refine ⟨⟨⟨FreeGroup.norm g, Nat.lt_succ_of_le hg⟩,g.toWord.get⟩, ?_⟩ |
| apply Subtype.ext |
| change FreeGroup.mk (List.ofFn g.toWord.get) = g |
| simpa only [List.ofFn_get] using (FreeGroup.mk_toWord (x := g)) |
| let : Finite shortWords := Finite.of_surjective f hf |
| exact Set.toFinite _ |
|
|
| lemma one_mem_shortWords : (1 : FreeGroup (Fin 3)) ∈ shortWords := by |
| simp [shortWords] |
|
|
| lemma inv_mem_shortWords {g : FreeGroup (Fin 3)} (hg : g ∈ shortWords) : |
| g⁻¹ ∈ shortWords := by simpa only [shortWords, Set.mem_ofPred_eq, FreeGroup.norm_inv_eq] using hg |
|
|
| |
|
|
| def reverseWord (g : FreeGroup (Fin 3)) : FreeGroup (Fin 3) := |
| FreeGroup.mk g.toWord.reverse |
|
|
| lemma reverse_mem_shortWords {g : FreeGroup (Fin 3)} (hg : g ∈ shortWords) : |
| reverseWord g ∈ shortWords := by |
| change FreeGroup.norm (FreeGroup.mk g.toWord.reverse) ≤ 5 |
| change g.toWord.length ≤ 5 at hg |
| apply FreeGroup.norm_mk_le.trans |
| simpa only [List.length_reverse] using hg |
|
|
| lemma reference_matrix_span : Submodule.span ℚ |
| (Set.range (fun c : shortWords ↦ LinearMap.toMatrix' (adjoint (referenceRho c)))) = ⊤ := by |
| let S := Submodule.span ℚ |
| (Set.range (fun c : shortWords ↦ LinearMap.toMatrix' (adjoint (referenceRho c)))) |
| have hm (i : Fin 9) : spanningMatrix i ∈ S := by |
| rw [← spanningMatrix_eq] |
| exact Submodule.subset_span ⟨⟨spanningWord i,spanningWord_mem i⟩,rfl⟩ |
| have hs (i j : Fin 3) : Matrix.single i j (1:ℚ) ∈ S := by |
| fin_cases i <;> fin_cases j |
| · change Matrix.single 0 0 (1:ℚ) ∈ S |
| rw [matrixUnit_00] |
| repeat' first | exact hm _ | apply S.add_mem | apply S.smul_mem |
| · change Matrix.single 0 1 (1:ℚ) ∈ S |
| rw [matrixUnit_01] |
| repeat' first | exact hm _ | apply S.add_mem | apply S.smul_mem |
| · change Matrix.single 0 2 (1:ℚ) ∈ S |
| rw [matrixUnit_02] |
| repeat' first | exact hm _ | apply S.add_mem | apply S.smul_mem |
| · change Matrix.single 1 0 (1:ℚ) ∈ S |
| rw [matrixUnit_10] |
| repeat' first | exact hm _ | apply S.add_mem | apply S.smul_mem |
| · change Matrix.single 1 1 (1:ℚ) ∈ S |
| rw [matrixUnit_11] |
| repeat' first | exact hm _ | apply S.add_mem | apply S.smul_mem |
| · change Matrix.single 1 2 (1:ℚ) ∈ S |
| rw [matrixUnit_12] |
| repeat' first | exact hm _ | apply S.add_mem | apply S.smul_mem |
| · change Matrix.single 2 0 (1:ℚ) ∈ S |
| rw [matrixUnit_20] |
| repeat' first | exact hm _ | apply S.add_mem | apply S.smul_mem |
| · change Matrix.single 2 1 (1:ℚ) ∈ S |
| rw [matrixUnit_21] |
| repeat' first | exact hm _ | apply S.add_mem | apply S.smul_mem |
| · change Matrix.single 2 2 (1:ℚ) ∈ S |
| rw [matrixUnit_22] |
| repeat' first | exact hm _ | apply S.add_mem | apply S.smul_mem |
| apply top_unique |
| intro B _ |
| rw [Matrix.matrix_eq_sum_single B] |
| apply S.sum_mem; intro i _ |
| apply S.sum_mem; intro j _ |
| simpa only [Matrix.smul_single, smul_eq_mul, mul_one] using S.smul_mem (B i j) (hs i j) |
|
|
| theorem finite_reference_adjoint_span : Submodule.span ℚ |
| (Set.range (fun c : shortWords ↦ adjoint (referenceRho c))) = ⊤ := by |
| have h := congrArg (Submodule.map Matrix.toLin'.toLinearMap) reference_matrix_span |
| simpa only [Submodule.map_span, ← Set.range_comp, Submodule.map_top, |
| LinearEquiv.range, Function.comp_def, LinearEquiv.coe_coe, |
| Matrix.toLin'_toMatrix'] using h |
|
|
| end BoundaryOnly.Holonomy |
| end |
|
|
| end OAI |
|
|
