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Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation

Conjugations #

This file defines the grade reversal and grade involution functions on multivectors, reverse and involute. Together, these operations compose to form the "Clifford conjugate", hence the name of this file.

https://en.wikipedia.org/wiki/Clifford_algebra#Antiautomorphisms

Main definitions #

Main statements #

source
def CliffordAlgebra.involute {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
CliffordAlgebra Q →ₐ[R] CliffordAlgebra Q

Grade involution, inverting the sign of each basis vector.

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    @[simp]
    theorem CliffordAlgebra.involute_ι {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (m : M) :
    CliffordAlgebra.involute ((CliffordAlgebra.ι Q) m) = -(CliffordAlgebra.ι Q) m
    source
    @[simp]
    theorem CliffordAlgebra.involute_comp_involute {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
    AlgHom.comp CliffordAlgebra.involute CliffordAlgebra.involute = AlgHom.id R (CliffordAlgebra Q)
    source
    theorem CliffordAlgebra.involute_involutive {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
    Function.Involutive ⇑CliffordAlgebra.involute
    source
    @[simp]
    theorem CliffordAlgebra.involute_involute {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (a : CliffordAlgebra Q) :
    CliffordAlgebra.involute (CliffordAlgebra.involute a) = a
    source
    @[simp]
    theorem CliffordAlgebra.involuteEquiv_symm_apply {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (a : CliffordAlgebra Q) :
    (AlgEquiv.symm CliffordAlgebra.involuteEquiv) a = CliffordAlgebra.involute a
    source
    @[simp]
    theorem CliffordAlgebra.involuteEquiv_apply {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (a : CliffordAlgebra Q) :
    CliffordAlgebra.involuteEquiv a = CliffordAlgebra.involute a
    source
    def CliffordAlgebra.involuteEquiv {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
    CliffordAlgebra Q ≃ₐ[R] CliffordAlgebra Q

    CliffordAlgebra.involute as an AlgEquiv.

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      def CliffordAlgebra.reverseOp {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
      CliffordAlgebra Q →ₐ[R] (CliffordAlgebra Q)ᵐᵒᵖ

      CliffordAlgebra.reverse as an AlgHom to the opposite algebra

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        @[simp]
        theorem CliffordAlgebra.reverseOp_ι {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (m : M) :
        CliffordAlgebra.reverseOp ((CliffordAlgebra.ι Q) m) = MulOpposite.op ((CliffordAlgebra.ι Q) m)
        source
        @[simp]
        theorem CliffordAlgebra.reverseOpEquiv_apply {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (a : CliffordAlgebra Q) :
        CliffordAlgebra.reverseOpEquiv a = CliffordAlgebra.reverseOp a
        source
        def CliffordAlgebra.reverseOpEquiv {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
        CliffordAlgebra Q ≃ₐ[R] (CliffordAlgebra Q)ᵐᵒᵖ

        CliffordAlgebra.reverseEquiv as an AlgEquiv to the opposite algebra

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          @[simp]
          theorem CliffordAlgebra.reverseOpEquiv_opComm {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
          AlgEquiv.opComm CliffordAlgebra.reverseOpEquiv = AlgEquiv.symm CliffordAlgebra.reverseOpEquiv
          source
          def CliffordAlgebra.reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
          CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q

          Grade reversion, inverting the multiplication order of basis vectors. Also called transpose in some literature.

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            @[simp]
            theorem CliffordAlgebra.unop_reverseOp {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (x : CliffordAlgebra Q) :
            MulOpposite.unop (CliffordAlgebra.reverseOp x) = CliffordAlgebra.reverse x
            source
            @[simp]
            theorem CliffordAlgebra.op_reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (x : CliffordAlgebra Q) :
            MulOpposite.op (CliffordAlgebra.reverse x) = CliffordAlgebra.reverseOp x
            source
            @[simp]
            theorem CliffordAlgebra.reverse_ι {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (m : M) :
            CliffordAlgebra.reverse ((CliffordAlgebra.ι Q) m) = (CliffordAlgebra.ι Q) m
            source
            @[simp]
            theorem CliffordAlgebra.reverse.commutes {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (r : R) :
            CliffordAlgebra.reverse ((algebraMap R (CliffordAlgebra Q)) r) = (algebraMap R (CliffordAlgebra Q)) r
            source
            @[simp]
            theorem CliffordAlgebra.reverse.map_one {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
            CliffordAlgebra.reverse 1 = 1
            source
            @[simp]
            theorem CliffordAlgebra.reverse.map_mul {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (a : CliffordAlgebra Q) (b : CliffordAlgebra Q) :
            CliffordAlgebra.reverse (a * b) = CliffordAlgebra.reverse b * CliffordAlgebra.reverse a
            source
            @[simp]
            theorem CliffordAlgebra.reverse_involutive {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
            Function.Involutive ⇑CliffordAlgebra.reverse
            source
            @[simp]
            theorem CliffordAlgebra.reverse_comp_reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
            CliffordAlgebra.reverse ∘ₗ CliffordAlgebra.reverse = LinearMap.id
            source
            @[simp]
            theorem CliffordAlgebra.reverse_reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (a : CliffordAlgebra Q) :
            CliffordAlgebra.reverse (CliffordAlgebra.reverse a) = a
            source
            @[simp]
            theorem CliffordAlgebra.reverseEquiv_apply {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
            ∀ (a : CliffordAlgebra Q), CliffordAlgebra.reverseEquiv a = CliffordAlgebra.reverse a
            source
            @[simp]
            theorem CliffordAlgebra.reverseEquiv_symm_apply {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
            ∀ (a : CliffordAlgebra Q), (LinearEquiv.symm CliffordAlgebra.reverseEquiv) a = CliffordAlgebra.reverse a
            source
            def CliffordAlgebra.reverseEquiv {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
            CliffordAlgebra Q ≃ₗ[R] CliffordAlgebra Q

            CliffordAlgebra.reverse as a LinearEquiv.

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              theorem CliffordAlgebra.reverse_comp_involute {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
              CliffordAlgebra.reverse ∘ₗ AlgHom.toLinearMap CliffordAlgebra.involute = AlgHom.toLinearMap CliffordAlgebra.involute ∘ₗ CliffordAlgebra.reverse
              source
              theorem CliffordAlgebra.reverse_involute_commute {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} :
              Function.Commute ⇑CliffordAlgebra.reverse ⇑CliffordAlgebra.involute

              CliffordAlgebra.reverse and CliffordAlgebra.involute commute. Note that the composition is sometimes referred to as the "clifford conjugate".

              source
              theorem CliffordAlgebra.reverse_involute {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (a : CliffordAlgebra Q) :
              CliffordAlgebra.reverse (CliffordAlgebra.involute a) = CliffordAlgebra.involute (CliffordAlgebra.reverse a)

              Statements about conjugations of products of lists #

              source
              theorem CliffordAlgebra.reverse_prod_map_ι {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (l : List M) :
              CliffordAlgebra.reverse (List.prod (List.map (⇑(CliffordAlgebra.ι Q)) l)) = List.prod (List.reverse (List.map (⇑(CliffordAlgebra.ι Q)) l))

              Taking the reverse of the product a list of $n$ vectors lifted via ι is equivalent to taking the product of the reverse of that list.

              source
              theorem CliffordAlgebra.involute_prod_map_ι {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (l : List M) :
              CliffordAlgebra.involute (List.prod (List.map (⇑(CliffordAlgebra.ι Q)) l)) = (-1) ^ List.length l • List.prod (List.map (⇑(CliffordAlgebra.ι Q)) l)

              Taking the involute of the product a list of $n$ vectors lifted via ι is equivalent to premultiplying by ${-1}^n$.

              Statements about Submodule.map and Submodule.comap #

              source
              theorem CliffordAlgebra.submodule_map_involute_eq_comap {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p : Submodule R (CliffordAlgebra Q)) :
              Submodule.map (AlgHom.toLinearMap CliffordAlgebra.involute) p = Submodule.comap (AlgHom.toLinearMap CliffordAlgebra.involute) p
              source
              @[simp]
              theorem CliffordAlgebra.ι_range_map_involute {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) :
              Submodule.map (AlgHom.toLinearMap CliffordAlgebra.involute) (LinearMap.range (CliffordAlgebra.ι Q)) = LinearMap.range (CliffordAlgebra.ι Q)
              source
              @[simp]
              theorem CliffordAlgebra.ι_range_comap_involute {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) :
              Submodule.comap (AlgHom.toLinearMap CliffordAlgebra.involute) (LinearMap.range (CliffordAlgebra.ι Q)) = LinearMap.range (CliffordAlgebra.ι Q)
              source
              @[simp]
              theorem CliffordAlgebra.evenOdd_map_involute {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (n : ZMod 2) :
              Submodule.map (AlgHom.toLinearMap CliffordAlgebra.involute) (CliffordAlgebra.evenOdd Q n) = CliffordAlgebra.evenOdd Q n
              source
              @[simp]
              theorem CliffordAlgebra.evenOdd_comap_involute {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (n : ZMod 2) :
              Submodule.comap (AlgHom.toLinearMap CliffordAlgebra.involute) (CliffordAlgebra.evenOdd Q n) = CliffordAlgebra.evenOdd Q n
              source
              theorem CliffordAlgebra.submodule_map_reverse_eq_comap {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p : Submodule R (CliffordAlgebra Q)) :
              Submodule.map CliffordAlgebra.reverse p = Submodule.comap CliffordAlgebra.reverse p
              source
              @[simp]
              theorem CliffordAlgebra.ι_range_map_reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) :
              Submodule.map CliffordAlgebra.reverse (LinearMap.range (CliffordAlgebra.ι Q)) = LinearMap.range (CliffordAlgebra.ι Q)
              source
              @[simp]
              theorem CliffordAlgebra.ι_range_comap_reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) :
              Submodule.comap CliffordAlgebra.reverse (LinearMap.range (CliffordAlgebra.ι Q)) = LinearMap.range (CliffordAlgebra.ι Q)
              source
              theorem CliffordAlgebra.submodule_map_mul_reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p : Submodule R (CliffordAlgebra Q)) (q : Submodule R (CliffordAlgebra Q)) :
              Submodule.map CliffordAlgebra.reverse (p * q) = Submodule.map CliffordAlgebra.reverse q * Submodule.map CliffordAlgebra.reverse p

              Like Submodule.map_mul, but with the multiplication reversed.

              source
              theorem CliffordAlgebra.submodule_comap_mul_reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p : Submodule R (CliffordAlgebra Q)) (q : Submodule R (CliffordAlgebra Q)) :
              Submodule.comap CliffordAlgebra.reverse (p * q) = Submodule.comap CliffordAlgebra.reverse q * Submodule.comap CliffordAlgebra.reverse p
              source
              theorem CliffordAlgebra.submodule_map_pow_reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p : Submodule R (CliffordAlgebra Q)) (n : ℕ) :
              Submodule.map CliffordAlgebra.reverse (p ^ n) = Submodule.map CliffordAlgebra.reverse p ^ n

              Like Submodule.map_pow

              source
              theorem CliffordAlgebra.submodule_comap_pow_reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (p : Submodule R (CliffordAlgebra Q)) (n : ℕ) :
              Submodule.comap CliffordAlgebra.reverse (p ^ n) = Submodule.comap CliffordAlgebra.reverse p ^ n
              source
              @[simp]
              theorem CliffordAlgebra.evenOdd_map_reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (n : ZMod 2) :
              Submodule.map CliffordAlgebra.reverse (CliffordAlgebra.evenOdd Q n) = CliffordAlgebra.evenOdd Q n
              source
              @[simp]
              theorem CliffordAlgebra.evenOdd_comap_reverse {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (n : ZMod 2) :
              Submodule.comap CliffordAlgebra.reverse (CliffordAlgebra.evenOdd Q n) = CliffordAlgebra.evenOdd Q n
              source
              @[simp]
              theorem CliffordAlgebra.involute_mem_evenOdd_iff {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) {x : CliffordAlgebra Q} {n : ZMod 2} :
              CliffordAlgebra.involute x ∈ CliffordAlgebra.evenOdd Q n ↔ x ∈ CliffordAlgebra.evenOdd Q n
              source
              @[simp]
              theorem CliffordAlgebra.reverse_mem_evenOdd_iff {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) {x : CliffordAlgebra Q} {n : ZMod 2} :
              CliffordAlgebra.reverse x ∈ CliffordAlgebra.evenOdd Q n ↔ x ∈ CliffordAlgebra.evenOdd Q n

              TODO: show that these are iffs when Invertible (2 : R).

              source
              theorem CliffordAlgebra.involute_eq_of_mem_even {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {x : CliffordAlgebra Q} (h : x ∈ CliffordAlgebra.evenOdd Q 0) :
              CliffordAlgebra.involute x = x
              source
              theorem CliffordAlgebra.involute_eq_of_mem_odd {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {x : CliffordAlgebra Q} (h : x ∈ CliffordAlgebra.evenOdd Q 1) :
              CliffordAlgebra.involute x = -x