Documentation

Monlib.LinearAlgebra.Coalgebra.FiniteDimensional

theorem LinearMap.adjoint_id {𝕜 : Type u_3} {A : Type u_4} [RCLike 𝕜] [NormedAddCommGroup A] [InnerProductSpace 𝕜 A] [FiniteDimensional 𝕜 A] :
theorem LinearMap.rTensor_adjoint {𝕜 : Type u_3} {A : Type u_4} {B : Type u_5} {C : Type u_6} [RCLike 𝕜] [NormedAddCommGroup A] [NormedAddCommGroup B] [NormedAddCommGroup C] [InnerProductSpace 𝕜 A] [InnerProductSpace 𝕜 B] [InnerProductSpace 𝕜 C] [FiniteDimensional 𝕜 A] [FiniteDimensional 𝕜 B] [FiniteDimensional 𝕜 C] (f : A →ₗ[𝕜] B) :
theorem LinearMap.lTensor_adjoint {𝕜 : Type u_3} {A : Type u_4} {B : Type u_5} {C : Type u_6} [RCLike 𝕜] [NormedAddCommGroup A] [NormedAddCommGroup B] [NormedAddCommGroup C] [InnerProductSpace 𝕜 A] [InnerProductSpace 𝕜 B] [InnerProductSpace 𝕜 C] [FiniteDimensional 𝕜 A] [FiniteDimensional 𝕜 B] [FiniteDimensional 𝕜 C] (f : A →ₗ[𝕜] B) :
theorem Coalgebra.inner_eq_counit' {R : Type u_1} {A : Type u_2} [RCLike R] [NormedAddCommGroupOfRing A] [InnerProductSpace R A] [SMulCommClass R A A] [IsScalarTower R A A] [FiniteDimensional R A] :
(fun (x : A) => inner 1 x) = counit
theorem Coalgebra.rTensor_mul_comp_lTensor_comul {R : Type u_1} {A : Type u_2} [RCLike R] [NormedAddCommGroupOfRing A] [Star A] [InnerProductSpace R A] [SMulCommClass R A A] [IsScalarTower R A A] [FiniteDimensional R A] (h : ∃ (σ : AA), ∀ (x y z : A), inner (x * y) z = inner y (σ (star x) * z)) :
theorem Coalgebra.lTensor_mul_comp_rTensor_comul_of {R : Type u_1} {A : Type u_2} [RCLike R] [NormedAddCommGroupOfRing A] [Star A] [InnerProductSpace R A] [SMulCommClass R A A] [IsScalarTower R A A] [FiniteDimensional R A] (h : ∃ (σ : AA), ∀ (x y z : A), inner (x * y) z = inner y (σ (star x) * z)) :
noncomputable def FiniteDimensionalCoAlgebra_isFrobeniusAlgebra_of {R : Type u_1} {A : Type u_2} [RCLike R] [NormedAddCommGroupOfRing A] [Star A] [InnerProductSpace R A] [SMulCommClass R A A] [IsScalarTower R A A] [FiniteDimensional R A] (h : ∃ (σ : AA), ∀ (x y z : A), inner (x * y) z = inner y (σ (star x) * z)) :
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Instances For
    structure LinearMap.IsAlgHom {R : Type u_3} {A : Type u_4} {B : Type u_5} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (x : A →ₗ[R] B) :
    Instances For
      theorem LinearMap.isAlgHom_iff {R : Type u_3} {A : Type u_4} {B : Type u_5} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (x : A →ₗ[R] B) :
      theorem AlgHom.isAlgHom {R : Type u_3} {A : Type u_4} {B : Type u_5} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (x : A →ₐ[R] B) :
      theorem AlgEquiv.isAlgHom {R : Type u_3} {A : Type u_4} {B : Type u_5} [CommSemiring R] [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (x : A ≃ₐ[R] B) :