46 lines
1.6 KiB
Text
46 lines
1.6 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.Basic
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/-!
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# Isomorphism between rep of color `c` and rep of dual color.
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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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noncomputable section
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namespace TensorSpecies
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variable (S : TensorSpecies)
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/-- The morphism from `S.FD.obj (Discrete.mk c)` to `S.FD.obj (Discrete.mk (S.τ c))`
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defined by contracting with the metric. -/
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def toDualRep (c : S.C) : S.FD.obj (Discrete.mk c) ⟶ S.FD.obj (Discrete.mk (S.τ c)) :=
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(ρ_ (S.FD.obj (Discrete.mk c))).inv
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≫ (S.FD.obj { as := c } ◁ (S.metric.app (Discrete.mk (S.τ c))))
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≫ (α_ (S.FD.obj (Discrete.mk c)) (S.FD.obj (Discrete.mk (S.τ c)))
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(S.FD.obj (Discrete.mk (S.τ c)))).inv
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≫ (S.contr.app (Discrete.mk c) ▷ S.FD.obj { as := S.τ c })
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≫ (λ_ (S.FD.obj (Discrete.mk (S.τ c)))).hom
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/-- The `toDualRep` for equal colors is the same, up-to conjugation by a trivial equivalence. -/
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lemma toDualRep_congr {c c' : S.C} (h : c = c') : S.toDualRep c = S.FD.map (Discrete.eqToHom h) ≫
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S.toDualRep c' ≫ S.FD.map (Discrete.eqToHom (congrArg S.τ h.symm)) := by
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subst h
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simp only [eqToHom_refl, Discrete.functor_map_id, Category.comp_id, Category.id_comp]
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/-- The morphism from `S.FD.obj (Discrete.mk (S.τ c))` to `S.FD.obj (Discrete.mk c)`
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defined by contracting with the metric. -/
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def fromDualRep (c : S.C) : S.FD.obj (Discrete.mk (S.τ c)) ⟶ S.FD.obj (Discrete.mk c) :=
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S.toDualRep (S.τ c) ≫ S.FD.map (Discrete.eqToHom (S.τ_involution c))
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end TensorSpecies
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end
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