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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 #

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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.

Equations
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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
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@[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} :
CliffordAlgebra.involute.comp CliffordAlgebra.involute = AlgHom.id R (CliffordAlgebra Q)
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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
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@[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
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@[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
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@[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) :
CliffordAlgebra.involuteEquiv.symm a = CliffordAlgebra.involute a
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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.

Equations
  • CliffordAlgebra.involuteEquiv = AlgEquiv.ofAlgHom CliffordAlgebra.involute CliffordAlgebra.involute ⋯ ⋯
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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)
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@[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
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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

Equations
  • CliffordAlgebra.reverseOpEquiv = AlgEquiv.ofAlgHom CliffordAlgebra.reverseOp (AlgHom.opComm CliffordAlgebra.reverseOp) ⋯ ⋯
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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 = CliffordAlgebra.reverseOpEquiv.symm
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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
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@[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
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@[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
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@[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
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@[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
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@[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
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@[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
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@[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
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@[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
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@[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
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@[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), CliffordAlgebra.reverseEquiv.symm a = CliffordAlgebra.reverse a
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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.

Equations
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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 CliffordAlgebra.involute.toLinearMap = CliffordAlgebra.involute.toLinearMap ∘ₗ CliffordAlgebra.reverse
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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".

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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 #

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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.map (⇑(CliffordAlgebra.ι Q)) l).prod = (List.map (⇑(CliffordAlgebra.ι Q)) l).reverse.prod

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.

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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.map (⇑(CliffordAlgebra.ι Q)) l).prod = (-1) ^ l.length • (List.map (⇑(CliffordAlgebra.ι Q)) l).prod

Taking the involute of the product a list of n vectors lifted via ι is equivalent to premultiplying by −1n.

Statements about Submodule.map and Submodule.comap #

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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 CliffordAlgebra.involute.toLinearMap p = Submodule.comap CliffordAlgebra.involute.toLinearMap p
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@[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 CliffordAlgebra.involute.toLinearMap (LinearMap.range (CliffordAlgebra.ι Q)) = LinearMap.range (CliffordAlgebra.ι Q)
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@[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 CliffordAlgebra.involute.toLinearMap (LinearMap.range (CliffordAlgebra.ι Q)) = LinearMap.range (CliffordAlgebra.ι Q)
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@[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 CliffordAlgebra.involute.toLinearMap (CliffordAlgebra.evenOdd Q n) = CliffordAlgebra.evenOdd Q n
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@[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 CliffordAlgebra.involute.toLinearMap (CliffordAlgebra.evenOdd Q n) = CliffordAlgebra.evenOdd Q n
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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
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@[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)
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@[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)
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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.

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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
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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

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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
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@[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
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@[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
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@[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
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@[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).

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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
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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