refactor: Change structure of SM file

HepLean/StandardModel/Representations.lean

@@ -0,0 +1,59 @@
+/-
+Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
+Released under Apache 2.0 license.
+Authors: Joseph Tooby-Smith
+-/
+import HepLean.StandardModel.Basic
+import Mathlib.Data.Complex.Exponential
+import Mathlib.Geometry.Manifold.VectorBundle.Basic
+import Mathlib.Geometry.Manifold.VectorBundle.SmoothSection
+import Mathlib.Geometry.Manifold.Instances.Real
+import Mathlib.RepresentationTheory.Basic
+import Mathlib.LinearAlgebra.Matrix.ToLin
+import Mathlib.Analysis.InnerProductSpace.Adjoint
+/-!
+# Representations appearing in the Standard Model
+
+This file defines the basic representations which appear in the Standard Model.
+
+-/
+universe v u
+namespace StandardModel
+
+open Manifold
+open Matrix
+open Complex
+open ComplexConjugate
+
+
+@[simps!]
+noncomputable def repU1Map (g : unitary ℂ) : unitaryGroup (Fin 2) ℂ :=
+  ⟨g ^ 3 • 1, by
+    rw [mem_unitaryGroup_iff, smul_one_mul, show g = ⟨g.1, g.prop⟩ from rfl]
+    simp only [SubmonoidClass.mk_pow, Submonoid.mk_smul, star_smul, star_pow, RCLike.star_def,
+      star_one]
+    rw [smul_smul, ← mul_pow]
+    erw [(unitary.mem_iff.mp g.prop).2]
+    simp only [one_pow, one_smul]⟩
+
+@[simps!]
+noncomputable def repU1 : unitary ℂ →* unitaryGroup (Fin 2) ℂ where
+  toFun g := repU1Map g
+  map_mul' g h := by
+    simp only [repU1Map, Submonoid.mk_mul_mk, mul_smul_one, smul_smul, mul_comm, ← mul_pow]
+  map_one' := by
+    simp only [repU1Map, one_pow, one_smul, Submonoid.mk_eq_one]
+
+@[simps!]
+def fundamentalSU2 : specialUnitaryGroup (Fin 2) ℂ →* unitaryGroup (Fin 2) ℂ where
+  toFun g := ⟨g.1, g.prop.1⟩
+  map_mul' _ _ := Subtype.ext rfl
+  map_one' := Subtype.ext rfl
+
+lemma repU1_fundamentalSU2_commute (u1 : unitary ℂ) (g : specialUnitaryGroup (Fin 2) ℂ) :
+    repU1 u1 * fundamentalSU2 g = fundamentalSU2 g * repU1 u1 := by
+  apply Subtype.ext
+  simp
+
+
+end StandardModel