File size: 2,256 Bytes
0dd3af4 | 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 | import Mathlib
import OAI.GroupTheory.SimpleAmenable.PolygonGeometry.Sectors
namespace OAI
section
section
open scoped symmDiff
namespace SimpleAmenable
open scoped commutatorElement
open scoped commutatorElement
section EvaluationTable
variable {D H Q T : Type*} [Group D] [Group H] [Group Q] [Group T]
noncomputable def evaluationModel (q : H →* Q) (e : D →* H)
(v : D →* T) (ρ : T →* Q) (hρ : Function.Injective ρ)
(he : q.comp e = ρ.comp v) : e.range →* T :=
e.rangeRestrict.liftOfSurjective e.rangeRestrict_surjective
⟨v,by
intro w hw
apply hρ
have hh := DFunLike.congr_fun he w
change q (e w) = ρ (v w) at hh
have hw' : e w = 1 := congrArg Subtype.val hw
rw [hw',map_one] at hh
simpa only [map_one] using hh.symm⟩
theorem evaluationModel_comp (q : H →* Q) (e : D →* H)
(v : D →* T) (ρ : T →* Q) (hρ : Function.Injective ρ)
(he : q.comp e = ρ.comp v) :
(evaluationModel q e v ρ hρ he).comp e.rangeRestrict = v :=
e.rangeRestrict.liftOfRightInverse_comp _ _ _
theorem evaluationModel_projection (q : H →* Q) (e : D →* H)
(v : D →* T) (ρ : T →* Q) (hρ : Function.Injective ρ)
(he : q.comp e = ρ.comp v) :
q.comp e.range.subtype = ρ.comp (evaluationModel q e v ρ hρ he) := by
apply MonoidHom.ext
intro x
obtain ⟨w,rfl⟩ := e.rangeRestrict_surjective x
have hh := DFunLike.congr_fun (evaluationModel_comp q e v ρ hρ he) w
change evaluationModel q e v ρ hρ he (e.rangeRestrict w) = v w at hh
simp only [MonoidHom.comp_apply,hh]
exact DFunLike.congr_fun he w
noncomputable def evaluationTable {Ω E : Type*} [Group E]
(q : H →* Q) (e : D →* H) (v : D →* (Ω → E))
(ρ : (Ω → E) →* Q) (hρ : Function.Injective ρ)
(he : q.comp e = ρ.comp v) (hv : Function.Surjective v)
(h : CentralOn q e.range) : ActualLawfulTable q ρ where
carrier := e.range
law := h
model := evaluationModel q e v ρ hρ he
onto := by
intro s
obtain ⟨w,rfl⟩ := hv s
exact ⟨e.rangeRestrict w,DFunLike.congr_fun (evaluationModel_comp q e v ρ hρ he) w⟩
projection := evaluationModel_projection q e v ρ hρ he
end EvaluationTable
end SimpleAmenable
end
end
end OAI
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