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} [inst : CommRing R] {M : Type u_2} [inst_1 : AddCommGroup M] [inst_2 : Module R M] {Q : QuadraticForm R M} (m : M), CliffordAlgebra.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.

Equations

• CliffordAlgebra Q = RingQuot (CliffordAlgebra.Rel 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)

Equations

• CliffordAlgebra.instInhabited Q = RingQuot.instInhabited (CliffordAlgebra.Rel 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)

Equations

• CliffordAlgebra.instRing Q = RingQuot.instRing (CliffordAlgebra.Rel Q)

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)

Equations

• CliffordAlgebra.instAlgebra' Q = RingQuot.instAlgebra (CliffordAlgebra.Rel 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)

Equations

• CliffordAlgebra.instAlgebra Q = CliffordAlgebra.instAlgebra' Q

instance CliffordAlgebra.instSMulCommClassCliffordAlgebraToSMulToSemiringInstRingInstAlgebra'ToSMulInstAlgebra' {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)

Equations

• (_ : SMulCommClass R S (CliffordAlgebra Q)) = (_ : SMulCommClass R S (RingQuot (CliffordAlgebra.Rel Q)))

instance CliffordAlgebra.instIsScalarTowerCliffordAlgebraToSMulToSemiringInstRingInstAlgebra'ToSMulInstAlgebra' {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)

Equations

• (_ : IsScalarTower R S (CliffordAlgebra Q)) = (_ : IsScalarTower R S (RingQuot (CliffordAlgebra.Rel 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.

Equations

• CliffordAlgebra.ι Q = LinearMap.comp (AlgHom.toLinearMap (RingQuot.mkAlgHom R (CliffordAlgebra.Rel Q))) (TensorAlgebra.ι R)

@[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) : (CliffordAlgebra.ι Q) m * (CliffordAlgebra.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 ((CliffordAlgebra.ι Q) m) * g ((CliffordAlgebra.ι Q) m) = (algebraMap R A) (Q m)

@[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) : (CliffordAlgebra.lift Q).symm F = { val := LinearMap.comp (AlgHom.toLinearMap F) (CliffordAlgebra.ι Q), property := (_ : ∀ (m : M), (AlgHom.toLinearMap F ∘ₗ CliffordAlgebra.ι Q) m * (AlgHom.toLinearMap F ∘ₗ CliffordAlgebra.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.

Equations

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

@[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)) : AlgHom.toLinearMap ((CliffordAlgebra.lift Q) { val := f, property := cond }) ∘ₗ CliffordAlgebra.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) : ((CliffordAlgebra.lift Q) { val := f, property := cond }) ((CliffordAlgebra.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) : AlgHom.toLinearMap g ∘ₗ CliffordAlgebra.ι Q = f ↔ g = (CliffordAlgebra.lift Q) { val := f, property := 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) : (CliffordAlgebra.lift Q) { val := AlgHom.toLinearMap g ∘ₗ CliffordAlgebra.ι Q, property := (_ : ∀ (m : M), g ((CliffordAlgebra.ι Q) m) * g ((CliffordAlgebra.ι Q) m) = (algebraMap R A) (Q m)) } = 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 : CliffordAlgebra Q →ₐ[R] A} {g : CliffordAlgebra Q →ₐ[R] A} : AlgHom.toLinearMap f ∘ₗ CliffordAlgebra.ι Q = AlgHom.toLinearMap g ∘ₗ CliffordAlgebra.ι Q → f = g

See note [partially-applied ext lemmas].

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} (h_grade0 : ∀ (r : R), C ((algebraMap R (CliffordAlgebra Q)) r)) (h_grade1 : ∀ (x : M), C ((CliffordAlgebra.ι Q) x)) (h_mul : ∀ (a b : CliffordAlgebra Q), C a → C b → C (a * b)) (h_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 muliplication, then it holds for all of CliffordAlgebra Q.

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

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 : M) (b : M) : f a * f b + f b * f a = (algebraMap R A) (QuadraticForm.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.compl₂ (LinearMap.mul R A) f ∘ₗ f + LinearMap.compl₂ (LinearMap.flip (LinearMap.mul R A)) f ∘ₗ f = LinearMap.compr₂ (BilinForm.toLin (QuadraticForm.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 : M) (b : M) : (CliffordAlgebra.ι Q) a * (CliffordAlgebra.ι Q) b + (CliffordAlgebra.ι Q) b * (CliffordAlgebra.ι Q) a = (algebraMap R (CliffordAlgebra Q)) (QuadraticForm.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 : M) (b : M) : (CliffordAlgebra.ι Q) a * (CliffordAlgebra.ι Q) b = (algebraMap R (CliffordAlgebra Q)) (QuadraticForm.polar (⇑Q) a b) - (CliffordAlgebra.ι Q) b * (CliffordAlgebra.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 : M} {b : M} (h : QuadraticForm.IsOrtho Q a b) : (CliffordAlgebra.ι Q) a * (CliffordAlgebra.ι Q) b + (CliffordAlgebra.ι Q) b * (CliffordAlgebra.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 : M} {b : M} (h : QuadraticForm.IsOrtho Q a b) : (CliffordAlgebra.ι Q) a * (CliffordAlgebra.ι Q) b = -((CliffordAlgebra.ι Q) b * (CliffordAlgebra.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 : M} {b : M} (h : QuadraticForm.IsOrtho Q a b) : x * (CliffordAlgebra.ι Q) a * (CliffordAlgebra.ι Q) b = -(x * (CliffordAlgebra.ι Q) b * (CliffordAlgebra.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 : M} {b : M} (h : QuadraticForm.IsOrtho Q a b) : (CliffordAlgebra.ι Q) a * ((CliffordAlgebra.ι Q) b * x) = -((CliffordAlgebra.ι Q) b * ((CliffordAlgebra.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 : M) (b : M) : (CliffordAlgebra.ι Q) a * (CliffordAlgebra.ι Q) b * (CliffordAlgebra.ι Q) a = (CliffordAlgebra.ι Q) (QuadraticForm.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 (AlgHom.toLinearMap ((CliffordAlgebra.lift Q) { val := f, property := cond })) (LinearMap.range (CliffordAlgebra.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.

Equations

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

@[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) : AlgHom.toLinearMap (CliffordAlgebra.map f) ∘ₗ CliffordAlgebra.ι Q₁ = CliffordAlgebra.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) : (CliffordAlgebra.map f) ((CliffordAlgebra.ι Q₁) m) = (CliffordAlgebra.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) : CliffordAlgebra.map (QuadraticForm.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) : AlgHom.comp (CliffordAlgebra.map f) (CliffordAlgebra.map g) = CliffordAlgebra.map (QuadraticForm.Isometry.comp f 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 (AlgHom.toLinearMap (CliffordAlgebra.map f)) (LinearMap.range (CliffordAlgebra.ι Q₁)) = Submodule.map (CliffordAlgebra.ι Q₂) (LinearMap.range f.toLinearMap)

@[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 : QuadraticForm.IsometryEquiv Q₁ Q₂) (a : CliffordAlgebra Q₁) : (CliffordAlgebra.equivOfIsometry e) a = (CliffordAlgebra.map (QuadraticForm.IsometryEquiv.toIsometry e)) a

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 : QuadraticForm.IsometryEquiv Q₁ Q₂) : CliffordAlgebra Q₁ ≃ₐ[R] CliffordAlgebra Q₂

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

Equations

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

@[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 : QuadraticForm.IsometryEquiv Q₁ Q₂) : AlgEquiv.symm (CliffordAlgebra.equivOfIsometry e) = CliffordAlgebra.equivOfIsometry (QuadraticForm.IsometryEquiv.symm e)

@[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 : QuadraticForm.IsometryEquiv Q₁ Q₂) (e₂₃ : QuadraticForm.IsometryEquiv Q₂ Q₃) : AlgEquiv.trans (CliffordAlgebra.equivOfIsometry e₁₂) (CliffordAlgebra.equivOfIsometry e₂₃) = CliffordAlgebra.equivOfIsometry (QuadraticForm.IsometryEquiv.trans e₁₂ 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} : CliffordAlgebra.equivOfIsometry (QuadraticForm.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.

Equations

• TensorAlgebra.toClifford = (TensorAlgebra.lift R) (CliffordAlgebra.ι Q)

@[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) : TensorAlgebra.toClifford ((TensorAlgebra.ι R) m) = (CliffordAlgebra.ι Q) m