| import Mathlib.LinearAlgebra.FiniteDimensional.Basic |
| import OAI.Algebra.Finitistic.Encoding.QuadraticCoefficients |
| import OAI.Algebra.Finitistic.Localization.QuotientAction |
| import OAI.Algebra.Finitistic.Encoding.AuxiliaryQuadraticPresentation |
|
|
| namespace OAI |
|
|
| namespace LittleFinitistic.PresentationReindex |
| open QuadraticPresentation |
|
|
| variable {k : Type*} [Field k] {α β κ ι : Type*} |
|
|
| noncomputable def renameHom (e : α ≃ β) : FreeAlgebra k α →ₐ[k] FreeAlgebra k β := |
| FreeAlgebra.lift k (fun a => FreeAlgebra.ι k (e a)) |
|
|
| @[simp] theorem rename_ι (e : α ≃ β) (a : α) : |
| renameHom (k := k) e (FreeAlgebra.ι k a) = FreeAlgebra.ι k (e a) := by |
| simp [renameHom] |
|
|
| noncomputable def rename (e : α ≃ β) : FreeAlgebra k α ≃ₐ[k] FreeAlgebra k β := |
| AlgEquiv.ofAlgHom (renameHom e) (renameHom e.symm) |
| (by apply FreeAlgebra.hom_ext; funext a; simp) |
| (by apply FreeAlgebra.hom_ext; funext a; simp) |
|
|
| @[simp] theorem rename_apply (e : α ≃ β) (p : FreeAlgebra k α) : rename e p = renameHom e p := rfl |
|
|
| @[simp] theorem rename_word (e : α ≃ β) (w : List α) : |
| renameHom (k := k) e (word k α w) = word k β (w.map e) := by |
| unfold word |
| rw [map_list_prod] |
| simp only [List.map_map,Function.comp_def,rename_ι] |
|
|
| lemma rename_degree (e : α ≃ β) {p : FreeAlgebra k α} {n : ℕ} |
| (hp : p ∈ degreeLE k α n) : rename e p ∈ degreeLE k β n := by |
| induction hp using Submodule.span_induction with |
| | mem x hx => |
| rcases hx with ⟨w,hw,rfl⟩ |
| simpa using word_mem_degreeLE (R := k) (w.map e) n (by simpa using hw) |
| | zero => simp |
| | add x y hx hy ihx ihy => simpa using (degreeLE k β n).add_mem ihx ihy |
| | smul r x hx ih => simpa using (degreeLE k β n).smul_mem r ih |
|
|
| variable (e : α ≃ β) (j : κ ≃ ι) (p : κ → FreeAlgebra k α) |
|
|
| noncomputable def renamedRelation (i : ι) : FreeAlgebra k β := rename e (p (j.symm i)) |
|
|
| noncomputable def forward : Original k α p →ₐ[k] Original k β (renamedRelation e j p) := |
| (original k α p).liftₐ ((originalMap k β (renamedRelation e j p)).comp (rename e).toAlgHom) |
| (RingCon.ringConGen_le.mpr (by |
| rintro x y ⟨i,rfl,rfl⟩ |
| change originalMap k β (renamedRelation e j p) (rename e (p i)) = _ |
| have h := original_relation k β (renamedRelation e j p) (j i) |
| simpa [renamedRelation] using h)) |
|
|
| noncomputable def backward : Original k β (renamedRelation e j p) →ₐ[k] Original k α p := |
| (original k β (renamedRelation e j p)).liftₐ |
| ((originalMap k α p).comp (rename e).symm.toAlgHom) |
| (RingCon.ringConGen_le.mpr (by |
| rintro x y ⟨i,rfl,rfl⟩ |
| change originalMap k α p ((rename e).symm (rename e (p (j.symm i)))) = _ |
| simpa only [AlgEquiv.symm_apply_apply,map_zero,AlgHom.comp_apply] using original_relation k α p (j.symm i))) |
|
|
| @[simp] theorem forward_mk (f : FreeAlgebra k α) : |
| forward e j p (originalMap k α p f) = |
| originalMap k β (renamedRelation e j p) (rename e f) := rfl |
|
|
| @[simp] theorem backward_mk (f : FreeAlgebra k β) : |
| backward e j p (originalMap k β (renamedRelation e j p) f) = |
| originalMap k α p ((rename e).symm f) := rfl |
|
|
| /-- Relabeling finite generators and relations preserves the actual presented algebra. -/ |
| noncomputable def presentationEquiv : Original k α p ≃ₐ[k] Original k β (renamedRelation e j p) := |
| AlgEquiv.ofAlgHom (forward e j p) (backward e j p) |
| (by |
| apply AlgHom.ext |
| intro x |
| obtain ⟨f,rfl⟩ := (original k β (renamedRelation e j p)).mkₐ_surjective (α := k) x |
| change forward e j p (backward e j p |
| (originalMap k β (renamedRelation e j p) f)) = originalMap k β (renamedRelation e j p) f |
| rw [backward_mk,forward_mk,AlgEquiv.apply_symm_apply]) |
| (by |
| apply AlgHom.ext |
| intro x |
| obtain ⟨f,rfl⟩ := (original k α p).mkₐ_surjective (α := k) x |
| change backward e j p (forward e j p (originalMap k α p f)) = originalMap k α p f |
| rw [forward_mk,backward_mk,AlgEquiv.symm_apply_apply]) |
|
|
| /-- Every finite list of quadratic polynomials has the concrete coefficient interface |
| used by the three-vertex algebra. -/ |
| theorem exists_relations {d t : ℕ} (p : Fin t → FreeAlgebra k (Fin d)) |
| (hp : ∀ i, p i ∈ degreeLE k (Fin d) 2) : |
| ∃ P : QuadraticEncoding.Relations k d t, ∀ i, QuotientAction.polynomial P i = p i := by |
| classical |
| choose c l q h using fun i => exists_coefficients k (Fin d) (p i) (hp i) |
| refine ⟨⟨c,l,q⟩,?_⟩ |
| intro i |
| rw [← h i] |
| simp only [QuotientAction.polynomial,coefficientPolynomial,Algebra.algebraMap_eq_smul_one] |
| abel |
|
|
| /-- Finite generator and equation types may both be replaced by Fin, and all |
| coefficient choices preserve the exact quotient algebra. -/ |
| theorem finite_quadratic_encoding [Finite α] [Finite κ] |
| (p : κ → FreeAlgebra k α) (hp : ∀ i, p i ∈ degreeLE k α 2) : |
| ∃ (d t : ℕ) (P : QuadraticEncoding.Relations k d t), |
| Nonempty (Original k (Fin d) (QuotientAction.polynomial P) ≃ₐ[k] Original k α p) := by |
| classical |
| let := Fintype.ofFinite α |
| let := Fintype.ofFinite κ |
| let e := Fintype.equivFin α |
| let j := Fintype.equivFin κ |
| let p' := renamedRelation e j p |
| have hp' (i) : p' i ∈ degreeLE k (Fin (Fintype.card α)) 2 := rename_degree e (hp (j.symm i)) |
| obtain ⟨P,hP⟩ := exists_relations p' hp' |
| refine ⟨Fintype.card α,Fintype.card κ,P,?_⟩ |
| have heq : QuotientAction.polynomial P = p' := funext hP |
| rw [heq] |
| exact ⟨(presentationEquiv e j p).symm⟩ |
|
|
| /-- The entire actual complex group algebra feeds the finite directed encoding. -/ |
| theorem auxiliaryGroupAlgebra_has_encoding : |
| ∃ (d t : ℕ) (P : QuadraticEncoding.Relations ℂ d t), |
| Nonempty (Original ℂ (Fin d) (QuotientAction.polynomial P) ≃ₐ[ℂ] |
| MonoidAlgebra ℂ Presentation.AuxiliaryGroup) := by |
| obtain ⟨d,t,P,⟨E⟩⟩ := finite_quadratic_encoding auxiliaryRelation auxiliaryRelation_degree |
| exact ⟨d,t,P,⟨E.trans auxiliaryQuadraticEquiv⟩⟩ |
|
|
| end LittleFinitistic.PresentationReindex |
|
|
| end OAI |
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|
