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Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

Other constructions isomorphic to Clifford Algebras #

This file contains isomorphisms showing that other types are equivalent to some CliffordAlgebra.

Rings #

Complex numbers #

We show additionally that this equivalence sends Complex.conj to CliffordAlgebra.involute and vice-versa:

Note that in this algebra CliffordAlgebra.reverse is the identity and so the clifford conjugate is the same as CliffordAlgebra.involute.

Quaternion algebras #

We show additionally that this equivalence sends QuaternionAlgebra.conj to the clifford conjugate and vice-versa:

Dual numbers #

The clifford algebra isomorphic to a ring #

Since the vector space is empty the ring is commutative.

Equations
  • CliffordAlgebraRing.instCommRingCliffordAlgebraUnitOfNatQuadraticForm = CommRing.mk
theorem CliffordAlgebraRing.reverse_apply {R : Type u_1} [CommRing R] (x : CliffordAlgebra 0) :
CliffordAlgebra.reverse x = x
@[simp]
theorem CliffordAlgebraRing.reverse_eq_id {R : Type u_1} [CommRing R] :
CliffordAlgebra.reverse = LinearMap.id
@[simp]
theorem CliffordAlgebraRing.involute_eq_id {R : Type u_1} [CommRing R] :
CliffordAlgebra.involute = AlgHom.id R (CliffordAlgebra 0)

The clifford algebra over a 0-dimensional vector space is isomorphic to its scalars.

Equations
Instances For

    The clifford algebra isomorphic to the complex numbers #

    The quadratic form sending elements to the negation of their square.

    Equations
    Instances For
      @[simp]
      theorem CliffordAlgebraComplex.Q_apply (r : ) :
      CliffordAlgebraComplex.Q r = -(r * r)
      @[simp]
      theorem CliffordAlgebraComplex.toComplex_ι (r : ) :
      CliffordAlgebraComplex.toComplex ((CliffordAlgebra.ι CliffordAlgebraComplex.Q) r) = r Complex.I
      @[simp]
      theorem CliffordAlgebraComplex.toComplex_involute (c : CliffordAlgebra CliffordAlgebraComplex.Q) :
      CliffordAlgebraComplex.toComplex (CliffordAlgebra.involute c) = (starRingEnd ) (CliffordAlgebraComplex.toComplex c)

      CliffordAlgebra.involute is analogous to Complex.conj.

      @[simp]
      theorem CliffordAlgebraComplex.toComplex_ofComplex (c : ) :
      CliffordAlgebraComplex.toComplex (CliffordAlgebraComplex.ofComplex c) = c
      @[simp]
      theorem CliffordAlgebraComplex.ofComplex_toComplex (c : CliffordAlgebra CliffordAlgebraComplex.Q) :
      CliffordAlgebraComplex.ofComplex (CliffordAlgebraComplex.toComplex c) = c
      @[simp]
      theorem CliffordAlgebraComplex.equiv_apply (a : CliffordAlgebra CliffordAlgebraComplex.Q) :
      CliffordAlgebraComplex.equiv a = CliffordAlgebraComplex.toComplex a
      @[simp]
      theorem CliffordAlgebraComplex.equiv_symm_apply (a : ) :
      CliffordAlgebraComplex.equiv.symm a = CliffordAlgebraComplex.ofComplex a

      The clifford algebras over CliffordAlgebraComplex.Q is isomorphic as an -algebra to .

      Equations
      • One or more equations did not get rendered due to their size.
      Instances For

        The clifford algebra is commutative since it is isomorphic to the complex numbers.

        TODO: prove this is true for all CliffordAlgebras over a 1-dimensional vector space.

        Equations

        reverse is a no-op over CliffordAlgebraComplex.Q.

        @[simp]
        theorem CliffordAlgebraComplex.reverse_eq_id :
        CliffordAlgebra.reverse = LinearMap.id
        @[simp]
        theorem CliffordAlgebraComplex.ofComplex_conj (c : ) :
        CliffordAlgebraComplex.ofComplex ((starRingEnd ) c) = CliffordAlgebra.involute (CliffordAlgebraComplex.ofComplex c)

        Complex.conj is analogous to CliffordAlgebra.involute.

        The clifford algebra isomorphic to the quaternions #

        def CliffordAlgebraQuaternion.Q {R : Type u_1} [CommRing R] (c₁ : R) (c₂ : R) :

        Q c₁ c₂ is a quadratic form over R × R such that CliffordAlgebra (Q c₁ c₂) is isomorphic as an R-algebra to ℍ[R,c₁,c₂].

        Equations
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          @[simp]
          theorem CliffordAlgebraQuaternion.Q_apply {R : Type u_1} [CommRing R] (c₁ : R) (c₂ : R) (v : R × R) :
          (CliffordAlgebraQuaternion.Q c₁ c₂) v = c₁ * (v.1 * v.1) + c₂ * (v.2 * v.2)

          The quaternion basis vectors within the algebra.

          Equations
          • One or more equations did not get rendered due to their size.
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            Intermediate result of CliffordAlgebraQuaternion.equiv: clifford algebras over CliffordAlgebraQuaternion.Q can be converted to ℍ[R,c₁,c₂].

            Equations
            • One or more equations did not get rendered due to their size.
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              @[simp]
              theorem CliffordAlgebraQuaternion.toQuaternion_ι {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} (v : R × R) :
              CliffordAlgebraQuaternion.toQuaternion ((CliffordAlgebra.ι (CliffordAlgebraQuaternion.Q c₁ c₂)) v) = { re := 0, imI := v.1, imJ := v.2, imK := 0 }
              theorem CliffordAlgebraQuaternion.toQuaternion_star {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} (c : CliffordAlgebra (CliffordAlgebraQuaternion.Q c₁ c₂)) :
              CliffordAlgebraQuaternion.toQuaternion (star c) = star (CliffordAlgebraQuaternion.toQuaternion c)

              The "clifford conjugate" maps to the quaternion conjugate.

              Map a quaternion into the clifford algebra.

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                @[simp]
                theorem CliffordAlgebraQuaternion.ofQuaternion_mk {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} (a₁ : R) (a₂ : R) (a₃ : R) (a₄ : R) :
                CliffordAlgebraQuaternion.ofQuaternion { re := a₁, imI := a₂, imJ := a₃, imK := a₄ } = (algebraMap R (CliffordAlgebra (CliffordAlgebraQuaternion.Q c₁ c₂))) a₁ + a₂ (CliffordAlgebra.ι (CliffordAlgebraQuaternion.Q c₁ c₂)) (1, 0) + a₃ (CliffordAlgebra.ι (CliffordAlgebraQuaternion.Q c₁ c₂)) (0, 1) + a₄ ((CliffordAlgebra.ι (CliffordAlgebraQuaternion.Q c₁ c₂)) (1, 0) * (CliffordAlgebra.ι (CliffordAlgebraQuaternion.Q c₁ c₂)) (0, 1))
                @[simp]
                theorem CliffordAlgebraQuaternion.ofQuaternion_comp_toQuaternion {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} :
                CliffordAlgebraQuaternion.ofQuaternion.comp CliffordAlgebraQuaternion.toQuaternion = AlgHom.id R (CliffordAlgebra (CliffordAlgebraQuaternion.Q c₁ c₂))
                @[simp]
                theorem CliffordAlgebraQuaternion.ofQuaternion_toQuaternion {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} (c : CliffordAlgebra (CliffordAlgebraQuaternion.Q c₁ c₂)) :
                CliffordAlgebraQuaternion.ofQuaternion (CliffordAlgebraQuaternion.toQuaternion c) = c
                @[simp]
                theorem CliffordAlgebraQuaternion.toQuaternion_comp_ofQuaternion {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} :
                CliffordAlgebraQuaternion.toQuaternion.comp CliffordAlgebraQuaternion.ofQuaternion = AlgHom.id R (QuaternionAlgebra R c₁ c₂)
                @[simp]
                theorem CliffordAlgebraQuaternion.toQuaternion_ofQuaternion {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} (q : QuaternionAlgebra R c₁ c₂) :
                CliffordAlgebraQuaternion.toQuaternion (CliffordAlgebraQuaternion.ofQuaternion q) = q
                @[simp]
                theorem CliffordAlgebraQuaternion.equiv_apply {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} (a : CliffordAlgebra (CliffordAlgebraQuaternion.Q c₁ c₂)) :
                CliffordAlgebraQuaternion.equiv a = CliffordAlgebraQuaternion.toQuaternion a
                @[simp]
                theorem CliffordAlgebraQuaternion.equiv_symm_apply {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} (a : QuaternionAlgebra R c₁ c₂) :
                CliffordAlgebraQuaternion.equiv.symm a = CliffordAlgebraQuaternion.ofQuaternion a

                The clifford algebra over CliffordAlgebraQuaternion.Q c₁ c₂ is isomorphic as an R-algebra to ℍ[R,c₁,c₂].

                Equations
                • CliffordAlgebraQuaternion.equiv = AlgEquiv.ofAlgHom CliffordAlgebraQuaternion.toQuaternion CliffordAlgebraQuaternion.ofQuaternion
                Instances For
                  @[simp]
                  theorem CliffordAlgebraQuaternion.ofQuaternion_star {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} (q : QuaternionAlgebra R c₁ c₂) :
                  CliffordAlgebraQuaternion.ofQuaternion (star q) = star (CliffordAlgebraQuaternion.ofQuaternion q)

                  The quaternion conjugate maps to the "clifford conjugate" (aka star).

                  The clifford algebra isomorphic to the dual numbers #

                  theorem CliffordAlgebraDualNumber.ι_mul_ι {R : Type u_1} [CommRing R] (r₁ : R) (r₂ : R) :

                  The clifford algebra over a 1-dimensional vector space with 0 quadratic form is isomorphic to the dual numbers.

                  Equations
                  • One or more equations did not get rendered due to their size.
                  Instances For
                    @[simp]
                    theorem CliffordAlgebraDualNumber.equiv_ι {R : Type u_1} [CommRing R] (r : R) :
                    CliffordAlgebraDualNumber.equiv ((CliffordAlgebra.ι 0) r) = r DualNumber.eps
                    @[simp]
                    theorem CliffordAlgebraDualNumber.equiv_symm_eps {R : Type u_1} [CommRing R] :
                    CliffordAlgebraDualNumber.equiv.symm DualNumber.eps = (CliffordAlgebra.ι 0) 1