| import OAI.Algebra.Drinfeld.RealBraid |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace RealDeligneDrinfeld.FreeCutoffCategory |
| open DeligneDrinfeld.AssociativeElimination DeligneDrinfeld.WordGrading |
| open DeligneDrinfeld.TruncatedWords |
| attribute [local instance] LieRing.ofAssociativeRing |
| abbrev P : Set (DeligneDrinfeld.AssociativeElimination.A ℝ Bool) := ∅ |
| abbrev Q (N : ℕ) := DeligneDrinfeld.TruncatedWords.Q P N |
| abbrev F (N n : ℕ) := filtration P N n |
|
|
| def j (N : ℕ) : L →ₗ⁅ℝ⁆ Q N := (quotient P N).toLieHom.comp embed |
| @[simp] theorem j_of (N : ℕ) (a : Bool) : j N (FreeLieAlgebra.of ℝ a)=DeligneDrinfeld.TruncatedDerivation.gen P N a := by |
| change quotient P N (embed (FreeLieAlgebra.of ℝ a))=_ |
| rw [embed_of]; rfl |
| theorem j_mem {N n : ℕ} {q : L} (hq : q∈Ln n) : j N q∈F N n := by |
| refine ⟨embed q,?_,rfl⟩ |
| exact space_mono (by intro w hw; exact le_of_eq hw.symm) (WeightBridge.embed_homogeneous hq) |
| theorem lie_mem {N r s : ℕ} {a b : Q N} (ha : a∈F N r) (hb : b∈F N s) : ⁅a,b⁆∈F N (r+s) := |
| (F N (r+s)).sub_mem (filtration_mul P N r s ha hb) |
| (by simpa only [Nat.add_comm s r] using filtration_mul P N s r hb ha) |
|
|
| def velocity (N : ℕ) (q : L) (a : Bool) : Q N := j N (RealDeligneDrinfeld.derivation q (FreeLieAlgebra.of ℝ a)) |
| theorem velocity_mem {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) (a : Bool) : velocity N q a∈F N (n+1) := by |
| cases a |
| · change j N (RealDeligneDrinfeld.derivation q x)∈_ |
| simp only [derivation_apply,D_x,map_zero,Submodule.zero_mem] |
| · change j N (RealDeligneDrinfeld.derivation q y)∈_ |
| rw [derivation_apply,D_y,LieHom.map_lie] |
| simpa only [Nat.add_comm 1 n] using lie_mem (j_mem (N:=N) WeightBridge.y_mem) (j_mem (N:=N) hq) |
|
|
| def d {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) : Module.End ℝ (Q N) := |
| DeligneDrinfeld.TruncatedDerivation.derivation P N n (velocity N q) (velocity_mem N hq) (by intros p hp; exact False.elim hp) |
|
|
| theorem d_one {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) : d N hq 1=0 := |
| DeligneDrinfeld.TruncatedDerivation.derivation_one P N n _ _ _ |
| theorem d_mul {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) (a b : Q N) : |
| d N hq (a*b)=d N hq a*b+a*d N hq b := |
| DeligneDrinfeld.TruncatedDerivation.derivation_mul P N n _ _ _ a b |
| theorem d_mem {n m : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) {a : Q N} (ha : a∈F N m) : |
| d N hq a∈F N (m+n) := DeligneDrinfeld.TruncatedDerivation.derivation_mem P N n _ _ _ ha |
| theorem d_j {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) (p : L) : |
| d N hq (j N p)=j N (RealDeligneDrinfeld.derivation q p) := |
| DeligneDrinfeld.PresentedCutoff.derivation_embed P N n _ _ _ (RealDeligneDrinfeld.derivation q) |
| (LieHom.id) (j N) (j_of N) (fun _ => rfl) p |
|
|
| def h (N : ℕ) (q : L) (p : Fin 3) : Q N := j N ((ThreeSolutionCategory.data q).h p) |
| theorem h_mem {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) (p : Fin 3) : h N q p∈F N n := by |
| have hr : CategoricalValues.rotateLie q∈Ln n := |
| WeightBridge.eval_mem hq WeightBridge.y_mem ((Ln 1).sub_mem ((Ln 1).neg_mem WeightBridge.x_mem) WeightBridge.y_mem) |
| unfold h ThreeSolutionCategory.data |
| fin_cases p |
| · change j N 0∈F N n; rw [map_zero]; exact Submodule.zero_mem _ |
| · exact j_mem hq |
| · exact j_mem ((Ln n).add_mem hq hr) |
|
|
| def data {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) : DeligneDrinfeld.CategoryExp.Data ℝ (Q N) (Fin 3) := |
| DeligneDrinfeld.CategoryExp.gauge (d N hq) (d_mul N hq) (d_one N hq) (h N q) |
|
|
| theorem loop_j {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) (p : Fin 3) (a : L) : |
| (data N hq).D p p (j N a)=j N (DeligneDrinfeld.CategoryGauge.loop (ThreeSolutionCategory.data q) p a) := by |
| change d N hq (j N a)+h N q p*j N a-j N a*h N q p=_ |
| rw [d_j,DeligneDrinfeld.CategoryGauge.loop_apply,map_add,LieHom.map_lie,Ring.lie_def] |
| change _=j N (RealDeligneDrinfeld.derivation q a)+(h N q p*j N a-j N a*h N q p) |
| abel |
| theorem unit_all {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) (p s : Fin 3) : |
| (data N hq).D p s 1=j N (DeligneDrinfeld.CategoryGauge.unit (ThreeSolutionCategory.data q) p s) := by |
| change d N hq 1+h N q s*1-1*h N q p=_ |
| rw [d_one,zero_add,mul_one,one_mul,DeligneDrinfeld.CategoryGauge.unit,map_sub] |
| rfl |
|
|
| theorem raises {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) (p s : Fin 3) |
| (m : ℕ) (a : Q N) (ha : a∈F N m) : (data N hq).D p s a∈F N (m+n) := |
| DeligneDrinfeld.CutoffCategory.raises P N (d N hq) (d_mul N hq) (d_one N hq) (h N q) n |
| (fun _ _ ha => d_mem N hq ha) (h_mem N hq) p s m a ha |
|
|
| theorem nilpotent {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) (hn : 0<n) (p s : Fin 3) : |
| IsNilpotent ((data N hq).D p s) := |
| DeligneDrinfeld.CutoffCategory.nilpotent P N (d N hq) (d_mul N hq) (d_one N hq) (h N q) n hn |
| (fun _ _ ha => d_mem N hq ha) (h_mem N hq) p s |
|
|
| theorem unit {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) : (data N hq).D 0 1 1=j N q := by |
| change (d N hq 1+h N q 1*1)-1*h N q 0=_ |
| rw [d_one] |
| simp [h,ThreeSolutionCategory.data] |
|
|
| theorem leading {n : ℕ} (N : ℕ) {q : L} (hq : q∈Ln n) (hn : 0<n) |
| {a : Q N} (ha : a-1∈F N 1) : |
| DeligneDrinfeld.CategoryExp.act (data N hq) 0 1 a-a-j N q∈F N (n+1) := by |
| rw [←unit N hq] |
| exact DeligneDrinfeld.CategoryExp.unit_leading (DeligneDrinfeld.CutoffCategory.filtration P N) |
| _ n (raises N hq 0 1) hn (filtration_one P N) ha |
|
|
| end RealDeligneDrinfeld.FreeCutoffCategory |
|
|
| namespace RealDeligneDrinfeld.BraidCutoff |
| open DeligneDrinfeld.AssociativeElimination DeligneDrinfeld.WordGrading |
| open DeligneDrinfeld.TruncatedWords |
| attribute [local instance] LieRing.ofAssociativeRing |
|
|
| def presentation : Set (DeligneDrinfeld.AssociativeElimination.A ℝ Edge) := |
| DeligneDrinfeld.AssociativeElimination.embed '' braidRelations |
| abbrev Q (N : ℕ) := DeligneDrinfeld.TruncatedWords.Q presentation N |
| abbrev F (N n : ℕ) := filtration presentation N n |
|
|
| def freeMap (N : ℕ) : BraidFree →ₗ⁅ℝ⁆ Q N := |
| (quotient presentation N).toLieHom.comp DeligneDrinfeld.AssociativeElimination.embed |
|
|
| theorem freeMap_relation (N : ℕ) {p : BraidFree} (hp : p∈braidRelations) : freeMap N p=0 := |
| quotient_relation presentation N ⟨p,hp,rfl⟩ |
|
|
| def lieMap (N : ℕ) : T4 →ₗ⁅ℝ⁆ Q N := by |
| have hk : braidIdeal ≤ (freeMap N).ker := by |
| rw [braidIdeal,LieSubmodule.lieSpan_le] |
| intro p hp |
| exact freeMap_relation N hp |
| exact { braidIdeal.toSubmodule.liftQ (freeMap N).toLinearMap hk with |
| map_lie' := by |
| intro a b |
| induction a using Submodule.Quotient.induction_on with | _ a => |
| induction b using Submodule.Quotient.induction_on with | _ b => |
| exact (freeMap N).map_lie a b } |
|
|
| def projection : BraidFree →ₗ⁅ℝ⁆ T4 where |
| toLinearMap := braidIdeal.toSubmodule.mkQ |
| map_lie' := rfl |
|
|
| @[simp] theorem lieMap_projection (N : ℕ) (p : BraidFree) : lieMap N (projection p)=freeMap N p := rfl |
| @[simp] theorem projection_of (a : Edge) : projection (FreeLieAlgebra.of ℝ a)=t a.val.1 a.val.2 := by |
| simp only [t,chord,dite_eq_left a.property]; rfl |
| theorem lieMap_of (N : ℕ) (a : Edge) : lieMap N (projection (FreeLieAlgebra.of ℝ a))= |
| DeligneDrinfeld.TruncatedDerivation.gen presentation N a := by |
| rw [lieMap_projection] |
| change quotient presentation N (DeligneDrinfeld.AssociativeElimination.embed (FreeLieAlgebra.of ℝ a))=_ |
| rw [DeligneDrinfeld.AssociativeElimination.embed_of]; rfl |
|
|
| theorem projection_relation {p : BraidFree} (hp : p∈braidRelations) : projection p=0 := |
| (Submodule.Quotient.mk_eq_zero _).mpr (LieSubmodule.subset_lieSpan hp) |
|
|
| theorem lie_mem {N r s : ℕ} {a b : Q N} (ha : a∈F N r) (hb : b∈F N s) : ⁅a,b⁆∈F N (r+s) := by |
| exact (F N (r+s)).sub_mem (filtration_mul presentation N r s ha hb) |
| (by simpa only [Nat.add_comm s r] using filtration_mul presentation N s r hb ha) |
| theorem t_mem (N : ℕ) (i j : Fin 4) : lieMap N (t i j)∈F N 1 := by |
| unfold t chord |
| split_ifs with hij hji |
| · change lieMap N (projection (FreeLieAlgebra.of ℝ ⟨(i,j),hij⟩))∈_ |
| rw [lieMap_of] |
| exact DeligneDrinfeld.TruncatedDerivation.gen_mem presentation N _ |
| · change lieMap N (projection (FreeLieAlgebra.of ℝ ⟨(j,i),hji⟩))∈_ |
| rw [lieMap_of] |
| exact DeligneDrinfeld.TruncatedDerivation.gen_mem presentation N _ |
| · change lieMap N 0∈_; rw [map_zero]; exact Submodule.zero_mem _ |
|
|
| theorem weight_mem {α : Type*} (N : ℕ) (e : FreeLieAlgebra ℝ α →ₗ⁅ℝ⁆ Q N) |
| (he : ∀ a,e (FreeLieAlgebra.of ℝ a)∈F N 1) {n : ℕ} {q : FreeLieAlgebra ℝ α} |
| (hq : q∈weightPiece (fun _ : α => 1) n) : e q∈F N n := by |
| have hw : ∀ n p,WeightedWord (fun _ : α => 1) n p → e p∈F N n := by |
| intro n p hp |
| induction hp with |
| | generator a => exact he a |
| | bracket hp hr ihp ihr => rw [LieHom.map_lie]; exact lie_mem ihp ihr |
| apply Submodule.span_induction (p:=fun q _ => e q∈F N n) ?_ ?_ ?_ ?_ hq |
| · intro p hp |
| exact hw n p hp |
| · rw [map_zero]; exact Submodule.zero_mem _ |
| · intro p r _ _ hp hr; rw [map_add]; exact Submodule.add_mem _ hp hr |
| · intro s p _ hp; rw [map_smul]; exact Submodule.smul_mem _ s hp |
|
|
| theorem eval_mem {N n : ℕ} {q : L} (hq : q∈Ln n) {a b : T4} |
| (ha : lieMap N a∈F N 1) (hb : lieMap N b∈F N 1) : lieMap N (eval a b q)∈F N n := by |
| apply weight_mem N ((lieMap N).comp (eval a b)) (fun i => ?_) hq |
| cases i <;> simpa [eval] using (by assumption : lieMap N _∈F N 1) |
|
|
| open T4Basic T4SolutionData |
| variable {n : ℕ} {q : L} |
| theorem A_mem (N : ℕ) (hq : q∈Ln n) : lieMap N (A q)∈F N n := |
| eval_mem hq (t_mem N 0 1) (t_mem N 1 2) |
| theorem D_mem (N : ℕ) (hq : q∈Ln n) : lieMap N (T4SolutionData.D q)∈F N n := |
| eval_mem hq (by rw [map_add]; exact Submodule.add_mem _ (t_mem N 0 2) (t_mem N 1 2)) (t_mem N 2 3) |
| theorem E_mem (N : ℕ) (hq : q∈Ln n) : lieMap N (E q)∈F N n := |
| eval_mem hq (t_mem N 0 1) (by rw [map_add]; exact Submodule.add_mem _ (t_mem N 1 2) (t_mem N 1 3)) |
|
|
| def velocity (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) (a : Edge) : Q N := |
| lieMap N (T4Derivation.derivation hq hqn hn (projection (FreeLieAlgebra.of ℝ a))) |
|
|
| theorem velocity_mem (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) (ij : Edge) : |
| velocity N hq hqn hn ij∈F N (n+1) := by |
| have hA := A_mem N hqn |
| have hD := D_mem N hqn |
| have hE := E_mem N hqn |
| have hb : lieMap N b∈F N 1 := t_mem N 1 2 |
| have hc : lieMap N c∈F N 1 := t_mem N 2 3 |
| have he : lieMap N e∈F N 1 := t_mem N 1 3 |
| have hbA : ⁅lieMap N b,lieMap N (A q)⁆∈F N (n+1) := by simpa [Nat.add_comm] using lie_mem hb hA |
| have hcD : ⁅lieMap N c,lieMap N (T4SolutionData.D q)⁆∈F N (n+1) := by simpa [Nat.add_comm] using lie_mem hc hD |
| have heDE : ⁅lieMap N (b+e),lieMap N (T4SolutionData.D q+E q)⁆∈F N (n+1) := by |
| rw [map_add,map_add] |
| simpa [Nat.add_comm] using lie_mem (Submodule.add_mem _ hb he) (Submodule.add_mem _ hD hE) |
| have hall (i j : Fin 4) : lieMap N (T4Derivation.derivation hq hqn hn (t i j))∈F N (n+1) := by |
| have ha : lieMap N (T4Derivation.derivation hq hqn hn a)∈F N (n+1) := by simp |
| have hb' : lieMap N (T4Derivation.derivation hq hqn hn b)∈F N (n+1) := by |
| rw [T4Derivation.derivation_b,LieHom.map_lie]; exact hbA |
| have hc' : lieMap N (T4Derivation.derivation hq hqn hn c)∈F N (n+1) := by |
| rw [T4Derivation.derivation_c,LieHom.map_lie]; exact hcD |
| have hd' : lieMap N (T4Derivation.derivation hq hqn hn d)∈F N (n+1) := by |
| rw [T4Derivation.derivation_d,map_neg,LieHom.map_lie]; exact Submodule.neg_mem _ hbA |
| have he' : lieMap N (T4Derivation.derivation hq hqn hn e)∈F N (n+1) := by |
| rw [T4Derivation.derivation_e,map_sub,LieHom.map_lie,LieHom.map_lie] |
| exact Submodule.sub_mem _ heDE hbA |
| have hf' : lieMap N (T4Derivation.derivation hq hqn hn f)∈F N (n+1) := by |
| rw [T4Derivation.derivation_f,map_sub,map_neg,LieHom.map_lie,map_sub,LieHom.map_lie,LieHom.map_lie] |
| exact Submodule.sub_mem _ (Submodule.neg_mem _ hcD) (Submodule.sub_mem _ heDE hbA) |
| fin_cases i <;> fin_cases j |
| all_goals first | simpa only [t_self,map_zero] using (Submodule.zero_mem (F N (n+1))) | exact ha | exact hb' | exact hc' | exact hd' | exact he' | exact hf' |
| unfold velocity |
| rw [projection_of] |
| exact hall _ _ |
|
|
| theorem relations_killed (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) |
| (p : DeligneDrinfeld.AssociativeElimination.A ℝ Edge) (hp : p∈presentation) : |
| DeligneDrinfeld.TruncatedDerivation.raw presentation N (velocity N hq hqn hn) p=0 := by |
| obtain ⟨p,hp,rfl⟩ := hp |
| unfold velocity |
| rw [DeligneDrinfeld.PresentedCutoff.raw_embed presentation N (T4Derivation.derivation hq hqn hn) |
| projection (lieMap N) (lieMap_of N) p,projection_relation hp,map_zero,map_zero] |
|
|
| def derivation (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) : Module.End ℝ (Q N) := |
| DeligneDrinfeld.TruncatedDerivation.derivation presentation N n (velocity N hq hqn hn) |
| (velocity_mem N hq hqn hn) (relations_killed N hq hqn hn) |
|
|
| theorem derivation_lieMap (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) (p : T4) : |
| derivation N hq hqn hn (lieMap N p)=lieMap N (T4Derivation.derivation hq hqn hn p) := by |
| induction p using Submodule.Quotient.induction_on with | _ p => |
| exact DeligneDrinfeld.PresentedCutoff.derivation_embed presentation N n _ _ _ _ projection |
| (lieMap N) (lieMap_of N) (fun _ => rfl) p |
|
|
| end RealDeligneDrinfeld.BraidCutoff |
|
|
| namespace RealDeligneDrinfeld.BraidCategoryExp |
| open DeligneDrinfeld.AssociativeElimination DeligneDrinfeld.WordGrading |
| open DeligneDrinfeld.TruncatedWords BraidCutoff T4Basic T4SolutionData |
| attribute [local instance] LieRing.ofAssociativeRing |
| variable {n : ℕ} {q : L} |
| theorem B_mem (N : ℕ) (hq : q∈Ln n) : lieMap N (B q)∈BraidCutoff.F N n := |
| eval_mem hq (by rw [map_add]; exact Submodule.add_mem _ (t_mem N 0 1) (t_mem N 0 2)) |
| (by rw [map_add]; exact Submodule.add_mem _ (t_mem N 1 3) (t_mem N 2 3)) |
|
|
| def h (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) (p : Fin 5) : BraidCutoff.Q N := |
| lieMap N ((FourSolutionCategory.data hq hqn hn).h p) |
| theorem h_mem (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) (p : Fin 5) : h N hq hqn hn p∈BraidCutoff.F N n := by |
| unfold h FourSolutionCategory.data |
| fin_cases p |
| · change lieMap N 0∈BraidCutoff.F N n; rw [map_zero]; exact Submodule.zero_mem _ |
| · exact A_mem N hqn |
| · change lieMap N (A q+B q)∈_; rw [map_add] |
| exact Submodule.add_mem _ (A_mem N hqn) (B_mem N hqn) |
| · exact D_mem N hqn |
| · change lieMap N (T4SolutionData.D q+E q)∈_; rw [map_add] |
| exact Submodule.add_mem _ (D_mem N hqn) (E_mem N hqn) |
|
|
| theorem d_one (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) : BraidCutoff.derivation N hq hqn hn 1=0 := |
| DeligneDrinfeld.TruncatedDerivation.derivation_one presentation N n _ _ _ |
| theorem d_mul (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) (a b : BraidCutoff.Q N) : |
| BraidCutoff.derivation N hq hqn hn (a*b)=BraidCutoff.derivation N hq hqn hn a*b+a*BraidCutoff.derivation N hq hqn hn b := |
| DeligneDrinfeld.TruncatedDerivation.derivation_mul presentation N n _ _ _ a b |
| theorem d_mem (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) {m : ℕ} {a : BraidCutoff.Q N} (ha : a∈BraidCutoff.F N m) : |
| BraidCutoff.derivation N hq hqn hn a∈BraidCutoff.F N (m+n) := |
| DeligneDrinfeld.TruncatedDerivation.derivation_mem presentation N n _ _ _ ha |
|
|
| def data (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) : DeligneDrinfeld.CategoryExp.Data ℝ (BraidCutoff.Q N) (Fin 5) := |
| DeligneDrinfeld.CategoryExp.gauge (BraidCutoff.derivation N hq hqn hn) (d_mul N hq hqn hn) (d_one N hq hqn hn) (h N hq hqn hn) |
|
|
| theorem raises (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) (p s : Fin 5) (m : ℕ) (a : BraidCutoff.Q N) |
| (ha : a∈BraidCutoff.F N m) : (data N hq hqn hn).D p s a∈BraidCutoff.F N (m+n) := |
| DeligneDrinfeld.CutoffCategory.raises presentation N _ _ _ _ n |
| (fun _ _ ha => d_mem N hq hqn hn ha) (h_mem N hq hqn hn) p s m a ha |
|
|
| theorem nilpotent (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) (p s : Fin 5) : |
| IsNilpotent ((data N hq hqn hn).D p s) := |
| DeligneDrinfeld.CutoffCategory.nilpotent presentation N _ _ _ _ n (by omega) |
| (fun _ _ ha => d_mem N hq hqn hn ha) (h_mem N hq hqn hn) p s |
|
|
| theorem loop_lieMap (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) (p : Fin 5) (a : T4) : |
| (data N hq hqn hn).D p p (lieMap N a)= |
| lieMap N (DeligneDrinfeld.CategoryGauge.loop (FourSolutionCategory.data hq hqn hn) p a) := by |
| change BraidCutoff.derivation N hq hqn hn (lieMap N a)+h N hq hqn hn p*lieMap N a-lieMap N a*h N hq hqn hn p=_ |
| rw [derivation_lieMap] |
| rw [DeligneDrinfeld.CategoryGauge.loop_apply,map_add,LieHom.map_lie,Ring.lie_def] |
| change _= _+(h N hq hqn hn p*lieMap N a-lieMap N a*h N hq hqn hn p) |
| change lieMap N (T4Derivation.derivation hq hqn hn a)+h N hq hqn hn p*lieMap N a-lieMap N a*h N hq hqn hn p= |
| lieMap N (T4Derivation.derivation hq hqn hn a)+(h N hq hqn hn p*lieMap N a-lieMap N a*h N hq hqn hn p) |
| abel |
|
|
| theorem unit (N : ℕ) (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) (p s : Fin 5) : |
| (data N hq hqn hn).D p s 1=lieMap N (DeligneDrinfeld.CategoryGauge.unit (FourSolutionCategory.data hq hqn hn) p s) := by |
| change BraidCutoff.derivation N hq hqn hn 1+h N hq hqn hn s*1-1*h N hq hqn hn p=_ |
| rw [d_one,zero_add,mul_one,one_mul,DeligneDrinfeld.CategoryGauge.unit,map_sub] |
| rfl |
| end RealDeligneDrinfeld.BraidCategoryExp |
|
|
| namespace RealDeligneDrinfeld.BraidInsertionExp |
| open DeligneDrinfeld.AssociativeElimination DeligneDrinfeld.WordGrading |
| open DeligneDrinfeld.TruncatedWords DeligneDrinfeld.CategoryGauge |
| attribute [local instance] LieRing.ofAssociativeRing |
| variable (N : ℕ) (X Y : T4) |
| (hX : BraidCutoff.lieMap N X∈BraidCutoff.F N 1) (hY : BraidCutoff.lieMap N Y∈BraidCutoff.F N 1) |
| def family (a : Bool) : BraidCutoff.Q N := if a then BraidCutoff.lieMap N Y else BraidCutoff.lieMap N X |
| include hX hY in |
| theorem family_mem (a : Bool) : family N X Y a∈BraidCutoff.F N 1 := by |
| cases a <;> assumption |
| def insert : FreeCutoffCategory.Q N →ₐ[ℝ] BraidCutoff.Q N := |
| DeligneDrinfeld.PresentedSubstitution.subst BraidCutoff.presentation N (family N X Y) (family_mem N X Y hX hY) |
| theorem insert_gen (a : Bool) : insert N X Y hX hY (DeligneDrinfeld.TruncatedDerivation.gen ∅ N a)=family N X Y a := |
| DeligneDrinfeld.PresentedSubstitution.subst_gen _ _ _ _ a |
| theorem insert_j (p : L) : insert N X Y hX hY (FreeCutoffCategory.j N p)=BraidCutoff.lieMap N (eval X Y p) := by |
| have he : (insert N X Y hX hY).toLieHom.comp (FreeCutoffCategory.j N)= |
| (BraidCutoff.lieMap N).comp (eval X Y) := by |
| apply FreeLieAlgebra.hom_ext |
| intro a |
| change insert N X Y hX hY (FreeCutoffCategory.j N (FreeLieAlgebra.of ℝ a))=_ |
| rw [FreeCutoffCategory.j_of,insert_gen] |
| cases a <;> simp [family,eval] |
| exact LieHom.congr_fun he p |
|
|
| variable {n : ℕ} {q : L} (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) |
| theorem edge (s t : Fin 5) |
| (he : EdgeCompatible (eval X Y) 0 1 s t (ThreeSolutionCategory.data q) (FourSolutionCategory.data hq hqn hn)) |
| (a : FreeCutoffCategory.Q N) : |
| DeligneDrinfeld.CategoryExp.act (BraidCategoryExp.data N hq hqn hn) s t (insert N X Y hX hY a)= |
| insert N X Y hX hY (DeligneDrinfeld.CategoryExp.act (FreeCutoffCategory.data N hqn) 0 1 a) := by |
| apply DeligneDrinfeld.CutoffCategory.exp_edge FreeCutoffCategory.P N |
| (FreeCutoffCategory.data N hqn) (BraidCategoryExp.data N hq hqn hn) (insert N X Y hX hY) 0 s |
| (FreeCutoffCategory.nilpotent N hqn (by omega)) (BraidCategoryExp.nilpotent N hq hqn hn) 1 t |
| · rw [BraidCategoryExp.unit,FreeCutoffCategory.unit_all,insert_j,he.1] |
| · intro b |
| rw [←FreeCutoffCategory.j_of,insert_j,BraidCategoryExp.loop_lieMap,FreeCutoffCategory.loop_j,insert_j,he.2] |
|
|
| open T4Basic |
| abbrev IAB := insert N a b (BraidCutoff.t_mem N 0 1) (BraidCutoff.t_mem N 1 2) |
| abbrev IAD := insert N (d+b) c (by rw [map_add]; exact Submodule.add_mem _ (BraidCutoff.t_mem N 0 2) (BraidCutoff.t_mem N 1 2)) (BraidCutoff.t_mem N 2 3) |
| abbrev IDE := insert N a (b+e) (BraidCutoff.t_mem N 0 1) (by rw [map_add]; exact Submodule.add_mem _ (BraidCutoff.t_mem N 1 2) (BraidCutoff.t_mem N 1 3)) |
| abbrev IBC := insert N (a+d) (e+c) (by rw [map_add]; exact Submodule.add_mem _ (BraidCutoff.t_mem N 0 1) (BraidCutoff.t_mem N 0 2)) |
| (by rw [map_add]; exact Submodule.add_mem _ (BraidCutoff.t_mem N 1 3) (BraidCutoff.t_mem N 2 3)) |
| abbrev ICE := insert N b c (BraidCutoff.t_mem N 1 2) (BraidCutoff.t_mem N 2 3) |
|
|
| include hq hn in |
| theorem pentagon_preserved (a : FreeCutoffCategory.Q N) |
| (ha : IDE N a*IAD N a=ICE N a*IBC N a*IAB N a) : |
| let a' := DeligneDrinfeld.CategoryExp.act (FreeCutoffCategory.data N hqn) 0 1 a |
| IDE N a'*IAD N a'=ICE N a'*IBC N a'*IAB N a' := by |
| let F := DeligneDrinfeld.CategoryExp.act (BraidCategoryExp.data N hq hqn hn) |
| have hh := congrArg (F 0 4) ha |
| have hn' := BraidCategoryExp.nilpotent N hq hqn hn |
| rw [DeligneDrinfeld.CategoryExp.composition _ hn' 0 3 4, |
| DeligneDrinfeld.CategoryExp.composition _ hn' 0 1 4, |
| DeligneDrinfeld.CategoryExp.composition _ hn' 1 2 4] at hh |
| let a' := DeligneDrinfeld.CategoryExp.act (FreeCutoffCategory.data N hqn) 0 1 a |
| have hDE : F 3 4 (IDE N a)=IDE N a' := edge N _ _ _ _ hq hqn hn 3 4 (FourSolutionCategory.DE hq hqn hn) a |
| have hAD : F 0 3 (IAD N a)=IAD N a' := edge N _ _ _ _ hq hqn hn 0 3 (FourSolutionCategory.AD hq hqn hn) a |
| have hCE : F 2 4 (ICE N a)=ICE N a' := edge N _ _ _ _ hq hqn hn 2 4 (FourSolutionCategory.CE hq hqn hn) a |
| have hBC : F 1 2 (IBC N a)=IBC N a' := edge N _ _ _ _ hq hqn hn 1 2 (FourSolutionCategory.BC hq hqn hn) a |
| have hAB : F 0 1 (IAB N a)=IAB N a' := edge N _ _ _ _ hq hqn hn 0 1 (FourSolutionCategory.AB hq hqn hn) a |
| rw [hDE,hAD,hCE,hBC,hAB] at hh |
| exact hh |
| end RealDeligneDrinfeld.BraidInsertionExp |
|
|
| namespace DeligneDrinfeld.CategoryExp |
| variable {R A B O P I : Type*} [CommRing R] [Ring A] [Algebra R A] |
| [Ring B] [Algebra R B] |
| def sum (s : Finset I) (g : I → Data R A O) : Data R A O where |
| D p q := ∑ i∈s,(g i).D p q |
| comp p q r a b := by |
| simp only [LinearMap.sum_apply] |
| rw [Finset.sum_mul,Finset.mul_sum,←Finset.sum_add_distrib] |
| apply Finset.sum_congr rfl |
| intro i _hi |
| exact (g i).comp p q r a b |
| one p := by simp only [LinearMap.sum_apply,Data.one,Finset.sum_const_zero] |
| theorem sum_apply (s : Finset I) (g : I → Data R A O) (p q : O) (a : A) : |
| (sum s g).D p q a=∑ i∈s,(g i).D p q a := LinearMap.sum_apply _ _ _ |
| theorem sum_raises {N : ℕ} (F : FilteredComparison.Filtration R A N) |
| (s : Finset I) (g : I → Data R A O) |
| (hg : ∀ i∈s,∀ p q m a,a∈F.step m → (g i).D p q a∈F.step (m+1)) |
| (p q : O) (m : ℕ) (a : A) (ha : a∈F.step m) : |
| (sum s g).D p q a∈F.step (m+1) := by |
| rw [sum_apply] |
| exact Submodule.sum_mem _ (fun i hi => hg i hi p q m a ha) |
| theorem sum_nilpotent {N : ℕ} (F : FilteredComparison.Filtration R A N) |
| (s : Finset I) (g : I → Data R A O) |
| (hg : ∀ i∈s,∀ p q m a,a∈F.step m → (g i).D p q a∈F.step (m+1)) |
| (p q : O) : IsNilpotent ((sum s g).D p q) := |
| nilpotent F _ (sum_raises F s g hg p q) |
| theorem sum_natural (s : Finset I) (g : I → Data R A O) (k : I → Data R B P) |
| (f : A →ₗ[R] B) (p q : O) (r t : P) |
| (hf : ∀ i∈s,∀ a,(k i).D r t (f a)=f ((g i).D p q a)) (a : A) : |
| (sum s k).D r t (f a)=f ((sum s g).D p q a) := by |
| rw [sum_apply,sum_apply,map_sum] |
| exact Finset.sum_congr rfl (fun i hi => hf i hi a) |
| theorem exp_sum_natural [Module ℚ A] [Module ℚ B] |
| (s : Finset I) (g : I → Data R A O) (k : I → Data R B P) |
| (f : A →ₗ[R] B) (p q : O) (r t : P) |
| (hg : IsNilpotent ((sum s g).D p q)) (hk : IsNilpotent ((sum s k).D r t)) |
| (hf : ∀ i∈s,∀ a,(k i).D r t (f a)=f ((g i).D p q a)) (a : A) : |
| act (sum s k) r t (f a)=f (act (sum s g) p q a) := |
| OperatorExp.natural _ _ hg hk f (sum_natural s g k f p q r t hf) a |
| end DeligneDrinfeld.CategoryExp |
|
|
| section |
| namespace RealDeligneDrinfeld.BraidInsertionExp |
| open DeligneDrinfeld.AssociativeElimination DeligneDrinfeld.WordGrading |
| open DeligneDrinfeld.TruncatedWords DeligneDrinfeld.CategoryGauge |
| variable (N : ℕ) (X Y : T4) |
| (hX : BraidCutoff.lieMap N X∈BraidCutoff.F N 1) (hY : BraidCutoff.lieMap N Y∈BraidCutoff.F N 1) |
| {n : ℕ} {q : L} (hq : q∈W) (hqn : q∈Ln n) (hn : 2<n) |
| theorem edge_derivative (s t : Fin 5) |
| (he : EdgeCompatible (eval X Y) 0 1 s t (ThreeSolutionCategory.data q) (FourSolutionCategory.data hq hqn hn)) |
| (a : FreeCutoffCategory.Q N) : |
| (BraidCategoryExp.data N hq hqn hn).D s t (insert N X Y hX hY a)= |
| insert N X Y hX hY ((FreeCutoffCategory.data N hqn).D 0 1 a) := by |
| apply DeligneDrinfeld.CutoffCategory.edge_natural FreeCutoffCategory.P N |
| (FreeCutoffCategory.data N hqn) (BraidCategoryExp.data N hq hqn hn) (insert N X Y hX hY) 0 s 1 t |
| · rw [BraidCategoryExp.unit,FreeCutoffCategory.unit_all,insert_j,he.1] |
| · intro b |
| rw [←FreeCutoffCategory.j_of,insert_j,BraidCategoryExp.loop_lieMap,FreeCutoffCategory.loop_j,insert_j,he.2] |
| end RealDeligneDrinfeld.BraidInsertionExp |
|
|
| namespace RealDeligneDrinfeld.BraidInsertionSum |
| open DeligneDrinfeld.CategoryExp |
| open BraidInsertionExp |
| variable {I : Type*} (N : ℕ) (s : Finset I) (q : I → L) (n : I → ℕ) |
| (hq : ∀ i,q i∈W) (hqn : ∀ i,q i∈Ln (n i)) (hn : ∀ i,2<n i) |
| def free : Data ℝ (FreeCutoffCategory.Q N) (Fin 3) := sum s (fun i => FreeCutoffCategory.data N (hqn i)) |
| def braid : Data ℝ (BraidCutoff.Q N) (Fin 5) := sum s (fun i => BraidCategoryExp.data N (hq i) (hqn i) (hn i)) |
| include hn in |
| theorem free_raises (p r : Fin 3) (m : ℕ) (a : FreeCutoffCategory.Q N) (ha : a∈FreeCutoffCategory.F N m) : |
| (free N s q n hqn).D p r a∈FreeCutoffCategory.F N (m+1) := by |
| apply sum_raises (DeligneDrinfeld.CutoffCategory.filtration ∅ N) s _ _ p r m a ha |
| intro i _hi p r m a ha |
| exact DeligneDrinfeld.TruncatedWords.filtration_antitone ∅ N (by have := hn i; omega) |
| (FreeCutoffCategory.raises N (hqn i) p r m a ha) |
| theorem braid_raises (p r : Fin 5) (m : ℕ) (a : BraidCutoff.Q N) (ha : a∈BraidCutoff.F N m) : |
| (braid N s q n hq hqn hn).D p r a∈BraidCutoff.F N (m+1) := by |
| apply sum_raises (DeligneDrinfeld.CutoffCategory.filtration BraidCutoff.presentation N) s _ _ p r m a ha |
| intro i _hi p r m a ha |
| exact DeligneDrinfeld.TruncatedWords.filtration_antitone _ N (by have := hn i; omega) |
| (BraidCategoryExp.raises N (hq i) (hqn i) (hn i) p r m a ha) |
| include hn in |
| theorem free_nilpotent (p r : Fin 3) : IsNilpotent ((free N s q n hqn).D p r) := |
| nilpotent (DeligneDrinfeld.CutoffCategory.filtration ∅ N) _ (free_raises N s q n hqn hn p r) |
| theorem braid_nilpotent (p r : Fin 5) : IsNilpotent ((braid N s q n hq hqn hn).D p r) := |
| nilpotent (DeligneDrinfeld.CutoffCategory.filtration BraidCutoff.presentation N) _ (braid_raises N s q n hq hqn hn p r) |
| theorem edge (X Y : T4) |
| (hX : BraidCutoff.lieMap N X∈BraidCutoff.F N 1) (hY : BraidCutoff.lieMap N Y∈BraidCutoff.F N 1) |
| (p r : Fin 5) |
| (he : ∀ i,DeligneDrinfeld.CategoryGauge.EdgeCompatible (eval X Y) 0 1 p r |
| (ThreeSolutionCategory.data (q i)) (FourSolutionCategory.data (hq i) (hqn i) (hn i))) |
| (a : FreeCutoffCategory.Q N) : |
| act (braid N s q n hq hqn hn) p r (insert N X Y hX hY a)= |
| insert N X Y hX hY (act (free N s q n hqn) 0 1 a) := by |
| exact exp_sum_natural s _ _ (insert N X Y hX hY).toLinearMap 0 1 p r |
| (free_nilpotent N s q n hqn hn 0 1) (braid_nilpotent N s q n hq hqn hn p r) |
| (fun i _hi a => edge_derivative N X Y hX hY (hq i) (hqn i) (hn i) p r (he i) a) a |
| include hq hn in |
| theorem pentagon (a : FreeCutoffCategory.Q N) |
| (ha : IDE N a*IAD N a=ICE N a*IBC N a*IAB N a) : |
| let a' := act (free N s q n hqn) 0 1 a |
| IDE N a'*IAD N a'=ICE N a'*IBC N a'*IAB N a' := by |
| let F := act (braid N s q n hq hqn hn) |
| let a' := act (free N s q n hqn) 0 1 a |
| have hh := congrArg (F 0 4) ha |
| have hn' := braid_nilpotent N s q n hq hqn hn |
| rw [composition _ hn' 0 3 4,composition _ hn' 0 1 4,composition _ hn' 1 2 4] at hh |
| have hDE : F 3 4 (IDE N a)=IDE N a' := edge N s q n hq hqn hn _ _ _ _ 3 4 (fun i => FourSolutionCategory.DE (hq i) (hqn i) (hn i)) a |
| have hAD : F 0 3 (IAD N a)=IAD N a' := edge N s q n hq hqn hn _ _ _ _ 0 3 (fun i => FourSolutionCategory.AD (hq i) (hqn i) (hn i)) a |
| have hCE : F 2 4 (ICE N a)=ICE N a' := edge N s q n hq hqn hn _ _ _ _ 2 4 (fun i => FourSolutionCategory.CE (hq i) (hqn i) (hn i)) a |
| have hBC : F 1 2 (IBC N a)=IBC N a' := edge N s q n hq hqn hn _ _ _ _ 1 2 (fun i => FourSolutionCategory.BC (hq i) (hqn i) (hn i)) a |
| have hAB : F 0 1 (IAB N a)=IAB N a' := edge N s q n hq hqn hn _ _ _ _ 0 1 (fun i => FourSolutionCategory.AB (hq i) (hqn i) (hn i)) a |
| rw [hDE,hAD,hCE,hBC,hAB] at hh |
| exact hh |
| theorem unit : (free N s q n hqn).D 0 1 1=FreeCutoffCategory.j N (∑ i∈s,q i) := by |
| rw [free,DeligneDrinfeld.CategoryExp.sum_apply,map_sum] |
| exact Finset.sum_congr rfl (fun i _hi => FreeCutoffCategory.unit N (hqn i)) |
| theorem edge_formula (a : FreeCutoffCategory.Q N) : |
| (free N s q n hqn).D 0 1 a=(free N s q n hqn).D 0 0 a+FreeCutoffCategory.j N (∑ i∈s,q i)*a := by |
| have hh := (free N s q n hqn).comp 0 0 1 a 1 |
| rw [one_mul,one_mul,unit] at hh |
| rw [hh,add_comm] |
| end RealDeligneDrinfeld.BraidInsertionSum |
|
|
| end |
|
|
| namespace DeligneDrinfeld.CategoryExp |
| open IsNilpotent |
| variable {R V : Type*} [CommRing R] [Algebra ℚ R] [AddCommGroup V] |
| [Module R V] [Module ℚ V] [IsScalarTower ℚ R V] |
| variable {N : ℕ} (F : FilteredComparison.Filtration R V N) |
| (A B : Module.End R V) (k : ℕ) |
| (hA : ∀ m a,a∈F.step m → A a∈F.step (m+1)) |
| (hB : ∀ m a,a∈F.step m → B a∈F.step (m+1)) |
| (hD : ∀ m a,a∈F.step m → A a-B a∈F.step (m+k)) |
| include hA hB hD |
| omit [Algebra ℚ R] [Module ℚ V] [IsScalarTower ℚ R V] in |
| theorem pow_difference (i m : ℕ) {a : V} (ha : a∈F.step m) : |
| (A^(i+1)) a-(B^(i+1)) a∈F.step (m+k+i) := by |
| induction i with |
| | zero => simpa only [zero_add,pow_one,Nat.add_zero] using hD m a ha |
| | succ i ih => |
| have h₁ := hA (m+k+i) _ ih |
| rw [map_sub] at h₁ |
| have hp := pow_mem_shift F B 1 hB (i+1) m ha |
| simp only [Nat.mul_one] at hp |
| have h₂ := hD (m+(i+1)) _ hp |
| have hh := (F.step (m+k+i+1)).add_mem h₁ (by convert h₂ using 1; congr 1; omega) |
| rw [pow_succ' A (i+1),pow_succ' B (i+1),Module.End.mul_apply,Module.End.mul_apply] |
| convert hh using 1; abel |
|
|
| theorem exp_difference (m : ℕ) {a : V} (ha : a∈F.step m) : |
| exp A a-exp B a-(A a-B a)∈F.step (m+k+1) := by |
| have hnil (d : Module.End R V) (hd : ∀ m a,a∈F.step m → d a∈F.step (m+1)) : d^(N+1)=0 := by |
| apply LinearMap.ext |
| intro x |
| have hh := pow_mem_shift F d 1 hd (N+1) 0 (a:=x) (by rw [F.initial]; trivial) |
| simpa only [Nat.mul_one,Nat.zero_add,F.terminal,Submodule.mem_bot,LinearMap.zero_apply] using hh |
| let r := N+1 |
| let s := N+1 |
| have hr : A^r=0 := hnil A hA |
| have hs : B^s=0 := hnil B hB |
| have hr' : A^(2+(r+s))=0 := pow_eq_zero_of_le (by omega) hr |
| have hs' : B^(2+(r+s))=0 := pow_eq_zero_of_le (by omega) hs |
| rw [exp_eq_sum hr',exp_eq_sum hs',LinearMap.sum_apply,LinearMap.sum_apply, |
| ←Finset.sum_sub_distrib,Finset.sum_range_add] |
| have ht : (∑ i∈Finset.range 2,(((↑i.factorial:ℚ)⁻¹ • A^i) a-((↑i.factorial:ℚ)⁻¹ • B^i) a))=A a-B a := by |
| simp [Finset.sum_range_succ] |
| rw [ht,add_sub_cancel_left] |
| apply Submodule.sum_mem |
| intro i hi |
| simp only [LinearMap.smul_apply,←smul_sub] |
| apply ((F.step (m+k+1)).restrictScalars ℚ).smul_mem |
| have hh := pow_difference F A B k hA hB hD (i+1) m ha |
| have he : 2+i=i+1+1 := by omega |
| rw [he] |
| exact F.decreasing (by omega) hh |
|
|
| theorem exp_difference_unit {a u : V} (ha : a-u∈F.step 1) : |
| exp A a-exp B a-(A u-B u)∈F.step (k+1) := by |
| have h₁ := exp_difference F A B k hA hB hD 0 (a:=a) (by rw [F.initial]; trivial) |
| have h₂ := hD 1 (a-u) ha |
| rw [map_sub,map_sub] at h₂ |
| have h₁' : exp A a-exp B a-(A a-B a)∈F.step (k+1) := by simpa only [Nat.zero_add] using h₁ |
| have h₂' : A a-A u-(B a-B u)∈F.step (k+1) := by simpa only [Nat.add_comm 1 k] using h₂ |
| convert (F.step (k+1)).add_mem h₁' h₂' using 1; abel |
| end DeligneDrinfeld.CategoryExp |
|
|
| end |
|
|
| end OAI |
|
|
