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Merge branch 'master' into SpaceTime/LorentzGroup

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jstoobysmith 2024-05-22 09:19:20 -04:00
commit 7107ba8b80
16 changed files with 36 additions and 34 deletions
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12
HepLean/SpaceTime/LorentzGroup/Basic.lean
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@ -113,7 +113,7 @@ def lorentzGroup : Subgroup (GL (Fin 4) ℝ) where
instance : TopologicalGroup lorentzGroup :=
Subgroup.instTopologicalGroupSubtypeMem lorentzGroup
/-- The lift of a matrix perserving `ηLin` to a Lorentz Group element. -/
/-- The lift of a matrix preserving `ηLin` to a Lorentz Group element. -/
def PreservesηLin.liftLor {Λ : Matrix (Fin 4) (Fin 4) ℝ} (h : PreservesηLin Λ) :
lorentzGroup := ⟨liftGL h, h⟩
@ -127,24 +127,24 @@ def transpose (Λ : lorentzGroup) : lorentzGroup :=
PreservesηLin.liftLor ((PreservesηLin.iff_transpose Λ.1).mp Λ.2)
/-- The continuous map from `GL (Fin 4) ℝ` to `Matrix (Fin 4) (Fin 4) ℝ` whose kernal is
/-- The continuous map from `GL (Fin 4) ℝ` to `Matrix (Fin 4) (Fin 4) ℝ` whose kernel is
the Lorentz group. -/
def kernalMap : C(GL (Fin 4) ℝ, Matrix (Fin 4) (Fin 4) ℝ) where
def kernelMap : C(GL (Fin 4) ℝ, Matrix (Fin 4) (Fin 4) ℝ) where
toFun Λ := η * Λ.1ᵀ * η * Λ.1
continuous_toFun := by
apply Continuous.mul _ Units.continuous_val
apply Continuous.mul _ continuous_const
exact Continuous.mul continuous_const (Continuous.matrix_transpose (Units.continuous_val))
lemma kernalMap_kernal_eq_lorentzGroup : lorentzGroup = kernalMap ⁻¹' {1} := by
lemma kernelMap_kernel_eq_lorentzGroup : lorentzGroup = kernelMap ⁻¹' {1} := by
ext Λ
erw [mem_iff Λ, PreservesηLin.iff_matrix]
rfl
/-- The Lorentz Group is a closed subset of `GL (Fin 4) ℝ`. -/
theorem isClosed_of_GL4 : IsClosed (lorentzGroup : Set (GL (Fin 4) ℝ)) := by
rw [kernalMap_kernal_eq_lorentzGroup]
exact continuous_iff_isClosed.mp kernalMap.2 {1} isClosed_singleton
rw [kernelMap_kernel_eq_lorentzGroup]
exact continuous_iff_isClosed.mp kernelMap.2 {1} isClosed_singleton
end lorentzGroup