feat: Homomorphism from SL(2, C) to Lorentz Group

HepLean/SpaceTime/SL2C/Basic.lean

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+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.SpaceTime.LorentzGroup.Basic
+import Mathlib.RepresentationTheory.Basic
+/-!
+# The group SL(2, ℂ) and it's relation to the Lorentz group
+
+The aim of this file is to give the relationship between `SL(2, ℂ)` and the Lorentz group.
+
+## TODO
+
+This file is a working progress.
+
+-/
+namespace spaceTime
+
+open Matrix
+open MatrixGroups
+open Complex
+
+namespace SL2C
+
+open spaceTime
+
+noncomputable section
+
+/-- Given an element  `M ∈ SL(2, ℂ)` the linear map from `selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)` to
+ itself defined by  `A ↦ M * A * Mᴴ`. -/
+@[simps!]
+def toLinearMapSelfAdjointMatrix (M : SL(2, ℂ)) :
+    selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) →ₗ[ℝ] selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) where
+  toFun A := ⟨M.1 * A.1 * Matrix.conjTranspose M,
+    by
+      noncomm_ring [selfAdjoint.mem_iff, star_eq_conjTranspose,
+        conjTranspose_mul, conjTranspose_conjTranspose,
+        (star_eq_conjTranspose A.1).symm.trans $ selfAdjoint.mem_iff.mp A.2]⟩
+  map_add' A B := by
+    noncomm_ring [AddSubmonoid.coe_add, AddSubgroup.coe_toAddSubmonoid, AddSubmonoid.mk_add_mk,
+      Subtype.mk.injEq]
+  map_smul' r A := by
+    noncomm_ring [selfAdjoint.val_smul, Algebra.mul_smul_comm, Algebra.smul_mul_assoc,
+      RingHom.id_apply]
+
+/-- The representation of  `SL(2, ℂ)` on `selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ)` given by
+   `M A ↦ M * A * Mᴴ`. -/
+@[simps!]
+def repSelfAdjointMatrix : Representation ℝ SL(2, ℂ) $ selfAdjoint (Matrix (Fin 2) (Fin 2) ℂ) where
+  toFun := toLinearMapSelfAdjointMatrix
+  map_one' := by
+    noncomm_ring [toLinearMapSelfAdjointMatrix, SpecialLinearGroup.coe_one, one_mul,
+      conjTranspose_one, mul_one, Subtype.coe_eta]
+  map_mul' M N := by
+    ext x i j : 3
+    noncomm_ring [toLinearMapSelfAdjointMatrix, SpecialLinearGroup.coe_mul, mul_assoc,
+      conjTranspose_mul, LinearMap.coe_mk, AddHom.coe_mk, LinearMap.mul_apply]
+
+/-- The representation of  `SL(2, ℂ)` on `spaceTime` obtained from `toSelfAdjointMatrix` and
+  `repSelfAdjointMatrix`. -/
+def repSpaceTime : Representation ℝ SL(2, ℂ) spaceTime where
+  toFun M := toSelfAdjointMatrix.symm.comp ((repSelfAdjointMatrix M).comp
+    toSelfAdjointMatrix.toLinearMap)
+  map_one' := by
+    ext
+    simp
+  map_mul' M N := by
+    ext x : 3
+    simp
+
+/-- Given an element `M ∈ SL(2, ℂ)` the corresponding element of the Lorentz group. -/
+@[simps!]
+def toLorentzGroupElem (M : SL(2, ℂ)) : 𝓛 :=
+  ⟨LinearMap.toMatrix stdBasis stdBasis (repSpaceTime M) ,
+   by simp [repSpaceTime, PreservesηLin.iff_det_selfAdjoint]⟩
+
+/-- The group homomorphism from ` SL(2, ℂ)` to the Lorentz group `𝓛`. -/
+@[simps!]
+def toLorentzGroup : SL(2, ℂ) →* 𝓛 where
+  toFun := toLorentzGroupElem
+  map_one' := by
+    simp only [toLorentzGroupElem, _root_.map_one, LinearMap.toMatrix_one]
+    rfl
+  map_mul' M N := by
+    apply Subtype.eq
+    simp only [toLorentzGroupElem, _root_.map_mul, LinearMap.toMatrix_mul,
+      lorentzGroupIsGroup_mul_coe]
+
+end
+end SL2C
+
+end spaceTime