Documentation

BrauerGroup.DoubleCentralizer

theorem centralizer_tensor_centralizer {F A A' : Type u} [Field F] [Ring A] [Algebra F A] [Ring A'] [Algebra F A'] (B : Subalgebra F A) (B' : Subalgebra F A') {ι : Type u_1} {ι' : Type u_2} (𝒜 : Basis ι F A) (𝒜' : Basis ι' F A') :
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  • One or more equations did not get rendered due to their size.
Instances For
    noncomputable instance center_field {F B : Type u} [Field F] [Ring B] [Algebra F B] [IsSimpleRing B] :
    Equations
    instance center_algebra {F B : Type u} [Field F] [Ring B] [Algebra F B] :
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    def Module.End.leftMul (F B : Type u) [Field F] [Ring B] [Algebra F B] :
    Subalgebra F (End F B)
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      @[simp]
      theorem Module.End.coe_leftMul (F B : Type u) [Field F] [Ring B] [Algebra F B] :
      def Module.End.rightMul (F B : Type u) [Field F] [Ring B] [Algebra F B] :
      Subalgebra F (End F B)
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        noncomputable def Module.End.rightMulEquiv {F B : Type u} [Field F] [Ring B] [Algebra F B] :
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        • One or more equations did not get rendered due to their size.
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          def Subalgebra.conj {F A : Type u} [Field F] [Ring A] [Algebra F A] (B : Subalgebra F A) (x : Aˣ) :
          Equations
          • B.conj x = { carrier := {y : A | bB, y = x * b * x⁻¹}, mul_mem' := , one_mem' := , add_mem' := , zero_mem' := , algebraMap_mem' := }
          Instances For
            @[simp]
            theorem Subalgebra.coe_conj {F A : Type u} [Field F] [Ring A] [Algebra F A] (B : Subalgebra F A) (x : Aˣ) :
            (B.conj x) = {y : A | bB, y = x * b * x⁻¹}
            theorem Subalgebra.mem_conj {F A : Type u} [Field F] [Ring A] [Algebra F A] {B : Subalgebra F A} {x : Aˣ} {y : A} :
            y B.conj x bB, y = x * b * x⁻¹
            def Subalgebra.toConj {F A : Type u} [Field F] [Ring A] [Algebra F A] (B : Subalgebra F A) (x : Aˣ) :
            B →ₐ[F] (B.conj x)
            Equations
            • B.toConj x = { toFun := fun (b : B) => x * b * x⁻¹, , map_one' := , map_mul' := , map_zero' := , map_add' := , commutes' := }
            Instances For
              @[simp]
              theorem Subalgebra.toConj_apply_coe {F A : Type u} [Field F] [Ring A] [Algebra F A] (B : Subalgebra F A) (x : Aˣ) (b : B) :
              ((B.toConj x) b) = x * b * x⁻¹
              def Subalgebra.fromConj {F A : Type u} [Field F] [Ring A] [Algebra F A] (B : Subalgebra F A) (x : Aˣ) :
              (B.conj x) →ₐ[F] B
              Equations
              • B.fromConj x = { toFun := fun (b : (B.conj x)) => x⁻¹ * b * x, , map_one' := , map_mul' := , map_zero' := , map_add' := , commutes' := }
              Instances For
                @[simp]
                theorem Subalgebra.fromConj_apply_coe {F A : Type u} [Field F] [Ring A] [Algebra F A] (B : Subalgebra F A) (x : Aˣ) (b : (B.conj x)) :
                ((B.fromConj x) b) = x⁻¹ * b * x
                def Subalgebra.conjEquiv {F A : Type u} [Field F] [Ring A] [Algebra F A] (B : Subalgebra F A) (x : Aˣ) :
                B ≃ₐ[F] (B.conj x)
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                Instances For
                  theorem Subalgebra.finrank_conj {F A : Type u} [Field F] [Ring A] [Algebra F A] (B : Subalgebra F A) (x : Aˣ) :
                  theorem Subalgebra.conj_centralizer {F A : Type u} [Field F] [Ring A] [Algebra F A] (B : Subalgebra F A) {x : Aˣ} :
                  centralizer F (B.conj x) = (centralizer F B).conj x
                  theorem Subalgebra.conj_centralizer' {F A : Type u} [Field F] [Ring A] [Algebra F A] (B : Subalgebra F A) {x : Aˣ} :
                  theorem centralizer_isSimple.aux.step1 {F A : Type u} [Field F] [Ring A] [Algebra F A] [FiniteDimensional F A] [Algebra.IsCentral F A] [IsSimpleRing A] (B : Subalgebra F A) [IsSimpleRing B] {ι : Type u_1} (ℬ : Basis ι F (Module.End F B)) :
                  theorem centralizer_isSimple {F A : Type u} [Field F] [Ring A] [Algebra F A] [FiniteDimensional F A] [Algebra.IsCentral F A] [IsSimpleRing A] (B : Subalgebra F A) [IsSimpleRing B] {ι : Type u_1} (ℬ : Basis ι F (Module.End F B)) :