30 lines
940 B
Text
30 lines
940 B
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.SpaceTime.LorentzTensor.Contractions
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/-!
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# Structure for Rising and Lowering indices
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-/
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noncomputable section
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open TensorProduct
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variable {R : Type} [CommSemiring R]
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/-- Structure extending `TensorStructure` with the addition of a metric
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permitting to `rise` and `lower` indices. -/
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structure DualizeTensorStructure (R : Type) [CommSemiring R] extends TensorStructure R where
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/-- The metric for a given color. -/
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metric : Color → ColorModule (τ μ) ⊗[R] ColorModule (τ μ)
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/-- The metric contracted with its dual is the unit. -/
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metric_dual : ∀ μ,
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TensorProduct.congr (LinearEquiv.refl _ _) (toTensorStructure.colorModuleCast (τ_involutive μ))
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(toTensorStructure.contrTwoTwo (metric μ ⊗ₜ[R] metric (τ μ))) = unit μ
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end
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