Clifford Algebras

We construct the Clifford algebra of a module M over a commutative ring R, equipped with a quadratic form Q.

Notation

The Clifford algebra of the R-module M equipped with a quadratic form Q is an R-algebra denoted CliffordAlgebra Q.

Given a linear morphism f : M → A from a module M to another R-algebra A, such that cond : ∀ m, f m * f m = algebraMap _ _ (Q m), there is a (unique) lift of f to an R-algebra morphism from CliffordAlgebra Q to A, which is denoted CliffordAlgebra.lift Q f cond.

The canonical linear map M → CliffordAlgebra Q is denoted CliffordAlgebra.ι Q.

Theorems

The main theorems proved ensure that CliffordAlgebra Q satisfies the universal property of the Clifford algebra.

1. ι_comp_lift is the fact that the composition of ι Q with lift Q f cond agrees with f.
2. lift_unique ensures the uniqueness of lift Q f cond with respect to 1.

Implementation details

The Clifford algebra of M is constructed as a quotient of the tensor algebra, as follows.

1. We define a relation CliffordAlgebra.Rel Q on TensorAlgebra R M. This is the smallest relation which identifies squares of elements of M with Q m.
2. The Clifford algebra is the quotient of the tensor algebra by this relation.

This file is almost identical to Mathlib/LinearAlgebra/ExteriorAlgebra/Basic.lean.

inductive CliffordAlgebra.Rel {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) : TensorAlgebra R M → TensorAlgebra R M → Prop

Rel relates each ι m * ι m, for m : M, with Q m.

The Clifford algebra of M is defined as the quotient modulo this relation.

• of {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (m : M) : Rel Q ((TensorAlgebra.ι R) m * (TensorAlgebra.ι R) m) ((algebraMap R (TensorAlgebra R M)) (Q m))

def CliffordAlgebra {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) : Type (max u_2 u_1)

The Clifford algebra of an R-module M equipped with a quadratic_form Q.

instance CliffordAlgebra.instInhabited {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) : Inhabited (CliffordAlgebra Q)

instance CliffordAlgebra.instRing {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) : Ring (CliffordAlgebra Q)

@[instance 900]

instance CliffordAlgebra.instAlgebra' {R : Type u_3} {A : Type u_4} {M : Type u_5} [CommSemiring R] [AddCommGroup M] [CommRing A] [Algebra R A] [Module R M] [Module A M] (Q : QuadraticForm A M) [IsScalarTower R A M] : Algebra R (CliffordAlgebra Q)

instance CliffordAlgebra.instAlgebra {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) : Algebra R (CliffordAlgebra Q)

instance CliffordAlgebra.instSMulCommClass {R : Type u_3} {S : Type u_4} {A : Type u_5} {M : Type u_6} [CommSemiring R] [CommSemiring S] [AddCommGroup M] [CommRing A] [Algebra R A] [Algebra S A] [Module R M] [Module S M] [Module A M] (Q : QuadraticForm A M) [IsScalarTower R A M] [IsScalarTower S A M] : SMulCommClass R S (CliffordAlgebra Q)

instance CliffordAlgebra.instIsScalarTower {R : Type u_3} {S : Type u_4} {A : Type u_5} {M : Type u_6} [CommSemiring R] [CommSemiring S] [AddCommGroup M] [CommRing A] [SMul R S] [Algebra R A] [Algebra S A] [Module R M] [Module S M] [Module A M] [IsScalarTower R A M] [IsScalarTower S A M] [IsScalarTower R S A] (Q : QuadraticForm A M) : IsScalarTower R S (CliffordAlgebra Q)

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

The canonical linear map M →ₗ[R] CliffordAlgebra Q.

@[simp]

theorem CliffordAlgebra.ι_sq_scalar {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) (m : M) : (ι Q) m * (ι Q) m = (algebraMap R (CliffordAlgebra Q)) (Q m)

As well as being linear, ι Q squares to the quadratic form

@[simp]

theorem CliffordAlgebra.comp_ι_sq_scalar {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {A : Type u_3} [Semiring A] [Algebra R A] (g : CliffordAlgebra Q →ₐ[R] A) (m : M) : g ((ι Q) m) * g ((ι Q) m) = (algebraMap R A) (Q m)

def CliffordAlgebra.lift {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) {A : Type u_3} [Semiring A] [Algebra R A] : { f : M →ₗ[R] A // ∀ (m : M), f m * f m = (algebraMap R A) (Q m) } ≃ (CliffordAlgebra Q →ₐ[R] A)

Given a linear map f : M →ₗ[R] A into an R-algebra A, which satisfies the condition: cond : ∀ m : M, f m * f m = Q(m), this is the canonical lift of f to a morphism of R-algebras from CliffordAlgebra Q to A.

@[simp]

theorem CliffordAlgebra.lift_symm_apply {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] (Q : QuadraticForm R M) {A : Type u_3} [Semiring A] [Algebra R A] (F : CliffordAlgebra Q →ₐ[R] A) : (lift Q).symm F = ⟨F.toLinearMap ∘ₗ ι Q, ⋯⟩

@[simp]

theorem CliffordAlgebra.ι_comp_lift {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {A : Type u_3} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = (algebraMap R A) (Q m)) : ((lift Q) ⟨f, cond⟩).toLinearMap ∘ₗ ι Q = f

@[simp]

theorem CliffordAlgebra.lift_ι_apply {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {A : Type u_3} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = (algebraMap R A) (Q m)) (x : M) : ((lift Q) ⟨f, cond⟩) ((ι Q) x) = f x

@[simp]

theorem CliffordAlgebra.lift_unique {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {A : Type u_3} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = (algebraMap R A) (Q m)) (g : CliffordAlgebra Q →ₐ[R] A) : g.toLinearMap ∘ₗ ι Q = f ↔ g = (lift Q) ⟨f, cond⟩

@[simp]

theorem CliffordAlgebra.lift_comp_ι {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {A : Type u_3} [Semiring A] [Algebra R A] (g : CliffordAlgebra Q →ₐ[R] A) : (lift Q) ⟨g.toLinearMap ∘ₗ ι Q, ⋯⟩ = g

theorem CliffordAlgebra.hom_ext {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {A : Type u_4} [Semiring A] [Algebra R A] {f g : CliffordAlgebra Q →ₐ[R] A} : f.toLinearMap ∘ₗ ι Q = g.toLinearMap ∘ₗ ι Q → f = g

See note [partially-applied ext lemmas].

theorem CliffordAlgebra.hom_ext_iff {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {A : Type u_4} [Semiring A] [Algebra R A] {f g : CliffordAlgebra Q →ₐ[R] A} : f = g ↔ f.toLinearMap ∘ₗ ι Q = g.toLinearMap ∘ₗ ι Q

theorem CliffordAlgebra.induction {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {C : CliffordAlgebra Q → Prop} (algebraMap : ∀ (r : R), C ((algebraMap R (CliffordAlgebra Q)) r)) (ι : ∀ (x : M), C ((ι Q) x)) (mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)) (add : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a + b)) (a : CliffordAlgebra Q) : C a

If C holds for the algebraMap of r : R into CliffordAlgebra Q, the ι of x : M, and is preserved under addition and multiplication, then it holds for all of CliffordAlgebra Q.

See also the stronger CliffordAlgebra.left_induction and CliffordAlgebra.right_induction.

@[simp]

theorem CliffordAlgebra.adjoin_range_ι {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} : Algebra.adjoin R (Set.range ⇑(ι Q)) = ⊤

@[simp]

theorem CliffordAlgebra.range_lift {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {A : Type u_3} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = (algebraMap R A) (Q m)) : ((lift Q) ⟨f, cond⟩).range = Algebra.adjoin R (Set.range ⇑f)

theorem CliffordAlgebra.mul_add_swap_eq_polar_of_forall_mul_self_eq {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {A : Type u_4} [Ring A] [Algebra R A] (f : M →ₗ[R] A) (hf : ∀ (x : M), f x * f x = (algebraMap R A) (Q x)) (a b : M) : f a * f b + f b * f a = (algebraMap R A) (QuadraticMap.polar (⇑Q) a b)

theorem CliffordAlgebra.forall_mul_self_eq_iff {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {A : Type u_4} [Ring A] [Algebra R A] (h2 : IsUnit 2) (f : M →ₗ[R] A) : (∀ (x : M), f x * f x = (algebraMap R A) (Q x)) ↔ (LinearMap.mul R A).compl₂ f ∘ₗ f + (LinearMap.mul R A).flip.compl₂ f ∘ₗ f = LinearMap.compr₂ (QuadraticMap.polarBilin Q) (Algebra.linearMap R A)

An alternative way to provide the argument to CliffordAlgebra.lift when 2 is invertible.

To show a function squares to the quadratic form, it suffices to show that f x * f y + f y * f x = algebraMap _ _ (polar Q x y)

theorem CliffordAlgebra.ι_mul_ι_add_swap {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (a b : M) : (ι Q) a * (ι Q) b + (ι Q) b * (ι Q) a = (algebraMap R (CliffordAlgebra Q)) (QuadraticMap.polar (⇑Q) a b)

The symmetric product of vectors is a scalar

theorem CliffordAlgebra.ι_mul_ι_comm {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (a b : M) : (ι Q) a * (ι Q) b = (algebraMap R (CliffordAlgebra Q)) (QuadraticMap.polar (⇑Q) a b) - (ι Q) b * (ι Q) a

@[simp]

theorem CliffordAlgebra.ι_mul_ι_add_swap_of_isOrtho {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {a b : M} (h : QuadraticMap.IsOrtho Q a b) : (ι Q) a * (ι Q) b + (ι Q) b * (ι Q) a = 0

theorem CliffordAlgebra.ι_mul_ι_comm_of_isOrtho {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {a b : M} (h : QuadraticMap.IsOrtho Q a b) : (ι Q) a * (ι Q) b = -((ι Q) b * (ι Q) a)

theorem CliffordAlgebra.mul_ι_mul_ι_of_isOrtho {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (x : CliffordAlgebra Q) {a b : M} (h : QuadraticMap.IsOrtho Q a b) : x * (ι Q) a * (ι Q) b = -(x * (ι Q) b * (ι Q) a)

theorem CliffordAlgebra.ι_mul_ι_mul_of_isOrtho {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (x : CliffordAlgebra Q) {a b : M} (h : QuadraticMap.IsOrtho Q a b) : (ι Q) a * ((ι Q) b * x) = -((ι Q) b * ((ι Q) a * x))

theorem CliffordAlgebra.ι_mul_ι_mul_ι {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (a b : M) : (ι Q) a * (ι Q) b * (ι Q) a = (ι Q) (QuadraticMap.polar (⇑Q) a b • a - Q a • b)

$aba$ is a vector.

@[simp]

theorem CliffordAlgebra.ι_range_map_lift {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} {A : Type u_3} [Semiring A] [Algebra R A] (f : M →ₗ[R] A) (cond : ∀ (m : M), f m * f m = (algebraMap R A) (Q m)) : Submodule.map ((lift Q) ⟨f, cond⟩).toLinearMap (LinearMap.range (ι Q)) = LinearMap.range f

def CliffordAlgebra.map {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) : CliffordAlgebra Q₁ →ₐ[R] CliffordAlgebra Q₂

Any linear map that preserves the quadratic form lifts to an AlgHom between algebras.

See CliffordAlgebra.equivOfIsometry for the case when f is a QuadraticForm.IsometryEquiv.

@[simp]

theorem CliffordAlgebra.map_comp_ι {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) : (map f).toLinearMap ∘ₗ ι Q₁ = ι Q₂ ∘ₗ f.toLinearMap

@[simp]

theorem CliffordAlgebra.map_apply_ι {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) (m : M₁) : (map f) ((ι Q₁) m) = (ι Q₂) (f m)

@[simp]

theorem CliffordAlgebra.map_id {R : Type u_1} [CommRing R] {M₁ : Type u_4} [AddCommGroup M₁] [Module R M₁] (Q₁ : QuadraticForm R M₁) : map (QuadraticMap.Isometry.id Q₁) = AlgHom.id R (CliffordAlgebra Q₁)

@[simp]

theorem CliffordAlgebra.map_comp_map {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} {M₃ : Type u_6} [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [Module R M₁] [Module R M₂] [Module R M₃] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Q₃ : QuadraticForm R M₃} (f : Q₂ →qᵢ Q₃) (g : Q₁ →qᵢ Q₂) : (map f).comp (map g) = map (f.comp g)

@[simp]

theorem CliffordAlgebra.ι_range_map_map {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) : Submodule.map (map f).toLinearMap (LinearMap.range (ι Q₁)) = Submodule.map (ι Q₂) (LinearMap.range f)

theorem CliffordAlgebra.leftInverse_map_of_leftInverse {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) (g : Q₂ →qᵢ Q₁) (h : Function.LeftInverse ⇑g ⇑f) : Function.LeftInverse ⇑(map g) ⇑(map f)

If f is a linear map from M₁ to M₂ that preserves the quadratic forms, and if it has a linear retraction g that also preserves the quadratic forms, then CliffordAlgebra.map g is a retraction of CliffordAlgebra.map f.

theorem CliffordAlgebra.map_surjective {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (f : Q₁ →qᵢ Q₂) (hf : Function.Surjective ⇑f) : Function.Surjective ⇑(map f)

If a linear map preserves the quadratic forms and is surjective, then the algebra maps it induces between Clifford algebras is also surjective.

def CliffordAlgebra.equivOfIsometry {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (e : QuadraticMap.IsometryEquiv Q₁ Q₂) : CliffordAlgebra Q₁ ≃ₐ[R] CliffordAlgebra Q₂

Two CliffordAlgebras are equivalent as algebras if their quadratic forms are equivalent.

@[simp]

theorem CliffordAlgebra.equivOfIsometry_apply {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (e : QuadraticMap.IsometryEquiv Q₁ Q₂) (a : CliffordAlgebra Q₁) : (equivOfIsometry e) a = (map e.toIsometry) a

@[simp]

theorem CliffordAlgebra.equivOfIsometry_symm {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} [AddCommGroup M₁] [AddCommGroup M₂] [Module R M₁] [Module R M₂] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} (e : QuadraticMap.IsometryEquiv Q₁ Q₂) : (equivOfIsometry e).symm = equivOfIsometry e.symm

@[simp]

theorem CliffordAlgebra.equivOfIsometry_trans {R : Type u_1} [CommRing R] {M₁ : Type u_4} {M₂ : Type u_5} {M₃ : Type u_6} [AddCommGroup M₁] [AddCommGroup M₂] [AddCommGroup M₃] [Module R M₁] [Module R M₂] [Module R M₃] {Q₁ : QuadraticForm R M₁} {Q₂ : QuadraticForm R M₂} {Q₃ : QuadraticForm R M₃} (e₁₂ : QuadraticMap.IsometryEquiv Q₁ Q₂) (e₂₃ : QuadraticMap.IsometryEquiv Q₂ Q₃) : (equivOfIsometry e₁₂).trans (equivOfIsometry e₂₃) = equivOfIsometry (e₁₂.trans e₂₃)

@[simp]

theorem CliffordAlgebra.equivOfIsometry_refl {R : Type u_1} [CommRing R] {M₁ : Type u_4} [AddCommGroup M₁] [Module R M₁] {Q₁ : QuadraticForm R M₁} : equivOfIsometry (QuadraticMap.IsometryEquiv.refl Q₁) = AlgEquiv.refl

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

The canonical image of the TensorAlgebra in the CliffordAlgebra, which maps TensorAlgebra.ι R x to CliffordAlgebra.ι Q x.

@[simp]

theorem TensorAlgebra.toClifford_ι {R : Type u_1} [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {Q : QuadraticForm R M} (m : M) : toClifford ((ι R) m) = (CliffordAlgebra.ι Q) m