File: Mathlib/Algebra/Lie/SkewAdjoint.lean

full_name: LinearMap.BilinForm.isSkewAdjoint_bracket
code: theorem LinearMap.BilinForm.isSkewAdjoint_bracket {f g : Module.End R M}
    (hf : f ∈ B.skewAdjointSubmodule) (hg : g ∈ B.skewAdjointSubmodule) :
    ⁅f, g⁆ ∈ B.skewAdjointSubmodule 
goal: ⁅f, g⁆ ∈ LinearMap.skewAdjointSubmodule B

full_name: skewAdjointLieSubalgebra
code: /-- Given an `R`-module `M`, equipped with a bilinear form, the skew-adjoint endomorphisms form a
Lie subalgebra of the Lie algebra of endomorphisms. -/
def skewAdjointLieSubalgebra : LieSubalgebra R (Module.End R M) 
goal: LieSubalgebra R (Module.End R M)

full_name: skewAdjointLieSubalgebraEquiv
code: /-- An equivalence of modules with bilinear forms gives equivalence of Lie algebras of skew-adjoint
endomorphisms. -/
def skewAdjointLieSubalgebraEquiv :
    skewAdjointLieSubalgebra (B.compl₁₂ (↑e : N →ₗ[R] M) ↑e) ≃ₗ⁅R⁆ skewAdjointLieSubalgebra B 
goal: ↥(skewAdjointLieSubalgebra (LinearMap.compl₁₂ B ↑e ↑e)) ≃ₗ⁅R⁆ ↥(skewAdjointLieSubalgebra B)

full_name: skewAdjointLieSubalgebraEquiv_apply
code: @[simp]
theorem skewAdjointLieSubalgebraEquiv_apply
    (f : skewAdjointLieSubalgebra (B.compl₁₂ (Qₗ := N) (Qₗ' := N) ↑e ↑e)) :
    ↑(skewAdjointLieSubalgebraEquiv B e f) = e.lieConj f 
goal: ↑((skewAdjointLieSubalgebraEquiv B e) f) = e.lieConj ↑f

full_name: skewAdjointLieSubalgebraEquiv_symm_apply
code: @[simp]
theorem skewAdjointLieSubalgebraEquiv_symm_apply (f : skewAdjointLieSubalgebra B) :
    ↑((skewAdjointLieSubalgebraEquiv B e).symm f) = e.symm.lieConj f 
goal: ↑((skewAdjointLieSubalgebraEquiv B e).symm f) = e.symm.lieConj ↑f

full_name: Matrix.lie_transpose
code: theorem Matrix.lie_transpose (A B : Matrix n n R) : ⁅A, B⁆ᵀ = ⁅Bᵀ, Aᵀ⁆ 
goal: ⁅A, B⁆.transpose = ⁅B.transpose, A.transpose⁆

full_name: Matrix.isSkewAdjoint_bracket
code: theorem Matrix.isSkewAdjoint_bracket {A B : Matrix n n R} (hA : A ∈ skewAdjointMatricesSubmodule J)
    (hB : B ∈ skewAdjointMatricesSubmodule J) : ⁅A, B⁆ ∈ skewAdjointMatricesSubmodule J 
goal: ⁅A, B⁆ ∈ skewAdjointMatricesSubmodule J

full_name: skewAdjointMatricesLieSubalgebra
code: /-- The Lie subalgebra of skew-adjoint square matrices corresponding to a square matrix `J`. -/
def skewAdjointMatricesLieSubalgebra : LieSubalgebra R (Matrix n n R) 
goal: LieSubalgebra R (Matrix n n R)

full_name: mem_skewAdjointMatricesLieSubalgebra
code: @[simp]
theorem mem_skewAdjointMatricesLieSubalgebra (A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra J ↔ A ∈ skewAdjointMatricesSubmodule J 
goal: A ∈ skewAdjointMatricesLieSubalgebra J ↔ A ∈ skewAdjointMatricesSubmodule J

full_name: skewAdjointMatricesLieSubalgebraEquiv
code: /-- An invertible matrix `P` gives a Lie algebra equivalence between those endomorphisms that are
skew-adjoint with respect to a square matrix `J` and those with respect to `PᵀJP`. -/
def skewAdjointMatricesLieSubalgebraEquiv (P : Matrix n n R) (h : Invertible P) :
    skewAdjointMatricesLieSubalgebra J ≃ₗ⁅R⁆ skewAdjointMatricesLieSubalgebra (Pᵀ * J * P) 
goal: ↥(skewAdjointMatricesLieSubalgebra J) ≃ₗ⁅R⁆ ↥(skewAdjointMatricesLieSubalgebra (P.transpose * J * P))

full_name: skewAdjointMatricesLieSubalgebraEquiv_apply
code: theorem skewAdjointMatricesLieSubalgebraEquiv_apply (P : Matrix n n R) (h : Invertible P)
    (A : skewAdjointMatricesLieSubalgebra J) :
    ↑(skewAdjointMatricesLieSubalgebraEquiv J P h A) = P⁻¹ * (A : Matrix n n R) * P 
goal: ↑((skewAdjointMatricesLieSubalgebraEquiv J P h) A) = P⁻¹ * ↑A * P

full_name: skewAdjointMatricesLieSubalgebraEquivTranspose
code: /-- An equivalence of matrix algebras commuting with the transpose endomorphisms restricts to an
equivalence of Lie algebras of skew-adjoint matrices. -/
def skewAdjointMatricesLieSubalgebraEquivTranspose {m : Type w} [DecidableEq m] [Fintype m]
    (e : Matrix n n R ≃ₐ[R] Matrix m m R) (h : ∀ A, (e A)ᵀ = e Aᵀ) :
    skewAdjointMatricesLieSubalgebra J ≃ₗ⁅R⁆ skewAdjointMatricesLieSubalgebra (e J) 
goal: ↥(skewAdjointMatricesLieSubalgebra J) ≃ₗ⁅R⁆ ↥(skewAdjointMatricesLieSubalgebra (e J))

full_name: skewAdjointMatricesLieSubalgebraEquivTranspose_apply
code: @[simp]
theorem skewAdjointMatricesLieSubalgebraEquivTranspose_apply {m : Type w} [DecidableEq m]
    [Fintype m] (e : Matrix n n R ≃ₐ[R] Matrix m m R) (h : ∀ A, (e A)ᵀ = e Aᵀ)
    (A : skewAdjointMatricesLieSubalgebra J) :
    (skewAdjointMatricesLieSubalgebraEquivTranspose J e h A : Matrix m m R) = e A 
goal: ↑((skewAdjointMatricesLieSubalgebraEquivTranspose J e h) A) = e ↑A

full_name: mem_skewAdjointMatricesLieSubalgebra_unit_smul
code: theorem mem_skewAdjointMatricesLieSubalgebra_unit_smul (u : Rˣ) (J A : Matrix n n R) :
    A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J 
goal: A ∈ skewAdjointMatricesLieSubalgebra (u • J) ↔ A ∈ skewAdjointMatricesLieSubalgebra J