82 lines
3.4 KiB
Text
82 lines
3.4 KiB
Text
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/-
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Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Joseph Tooby-Smith
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-/
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import HepLean.Tensors.TensorSpecies.UnitTensor
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import HepLean.Tensors.TensorSpecies.ContractLemmas
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/-!
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## Metrics in tensor trees
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-/
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open IndexNotation
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open CategoryTheory
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open MonoidalCategory
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open OverColor
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open HepLean.Fin
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open TensorProduct
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noncomputable section
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namespace TensorSpecies
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open TensorTree
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/-- The metric of a tensor species in a `PiTensorProduct`. -/
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def metricTensor (S : TensorSpecies) (c : S.C) : S.F.obj (OverColor.mk ![c, c]) :=
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(OverColor.Discrete.pairIsoSep S.FD).hom.hom ((S.metric.app (Discrete.mk c)).hom (1 : S.k))
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variable {S : TensorSpecies}
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lemma pairIsoSep_inv_metricTensor (c : S.C) :
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(Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor c) =
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(S.metric.app (Discrete.mk c)).hom (1 : S.k) := by
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simp [metricTensor]
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erw [Discrete.rep_iso_inv_hom_apply]
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/-- Contraction of a metric tensor with a metric tensor gives the unit.
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Like `S.contr_metric` but with the braiding appearing on the side of the unit. -/
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lemma contr_metric_braid_unit (c : S.C) : (((S.FD.obj (Discrete.mk c)) ◁
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(λ_ (S.FD.obj (Discrete.mk (S.τ c)))).hom).hom
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(((S.FD.obj (Discrete.mk c)) ◁ ((S.contr.app (Discrete.mk c)) ▷
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(S.FD.obj (Discrete.mk (S.τ c))))).hom
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(((S.FD.obj (Discrete.mk c)) ◁ (α_ (S.FD.obj (Discrete.mk (c)))
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(S.FD.obj (Discrete.mk (S.τ c))) (S.FD.obj (Discrete.mk (S.τ c)))).inv).hom
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((α_ (S.FD.obj (Discrete.mk (c))) (S.FD.obj (Discrete.mk (c)))
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(S.FD.obj (Discrete.mk (S.τ c)) ⊗ S.FD.obj (Discrete.mk (S.τ c)))).hom.hom
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(((OverColor.Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor c) ⊗ₜ
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(OverColor.Discrete.pairIsoSep S.FD).inv.hom (S.metricTensor (S.τ c)))))))) =
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(β_ (S.FD.obj (Discrete.mk (S.τ c))) (S.FD.obj (Discrete.mk c))).hom.hom
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((S.unit.app (Discrete.mk c)).hom (1 : S.k)) := by
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have hx : Function.Injective (β_ (S.FD.obj (Discrete.mk c)) (S.FD.obj (Discrete.mk (S.τ c))) ).hom.hom := by
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change Function.Injective (β_ (S.FD.obj (Discrete.mk c)).V (S.FD.obj (Discrete.mk (S.τ c))).V ).hom
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exact (β_ (S.FD.obj (Discrete.mk c)).V (S.FD.obj (Discrete.mk (S.τ c))).V ).toLinearEquiv.toEquiv.injective
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apply hx
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rw [pairIsoSep_inv_metricTensor, pairIsoSep_inv_metricTensor]
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rw [S.contr_metric c]
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change _ = (β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).inv.hom
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((β_ (S.FD.obj { as := S.τ c }) (S.FD.obj { as := c })).hom.hom _)
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rw [Discrete.rep_iso_inv_hom_apply]
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lemma metricTensor_contr_dual_metricTensor_perm_cond (c : S.C) : ∀ (x : Fin (Nat.succ 0).succ),
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((Sum.elim ![c, c] ![S.τ c, S.τ c] ∘ ⇑finSumFinEquiv.symm) ∘ Fin.succAbove 1 ∘ Fin.succAbove 1) x =
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(![S.τ c, c] ∘ ⇑(finMapToEquiv ![1, 0] ![1, 0]).symm) x := by
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intro x
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fin_cases x
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· rfl
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· rfl
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/-- The contraction of a metric tensor with its dual gives the unit. -/
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lemma metricTensor_contr_dual_metricTensor_eq_unit (c : S.C) :
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{S.metricTensor c | μ ν ⊗ S.metricTensor (S.τ c) | ν ρ}ᵀ.tensor =
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(perm (OverColor.equivToHomEq (finMapToEquiv ![1, 0] ![1, 0])
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(metricTensor_contr_dual_metricTensor_perm_cond c)) {S.unitTensor c | μ ρ}ᵀ).tensor := by
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rw [contr_two_two_inner, contr_metric_braid_unit, Discrete.pairIsoSep_β]
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change (S.F.map _ ≫ S.F.map _ ).hom _ = _
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rw [← S.F.map_comp]
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rfl
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end TensorSpecies
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end
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