Other constructions isomorphic to Clifford Algebras

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

Rings

• CliffordAlgebraRing.equiv: any ring is equivalent to a CliffordAlgebra over a zero-dimensional vector space.

Complex numbers

• CliffordAlgebraComplex.equiv: the Complex numbers are equivalent as an ℝ-algebra to a CliffordAlgebra over a one-dimensional vector space with a quadratic form that satisfies Q (ι Q 1) = -1.
• CliffordAlgebraComplex.toComplex: the forward direction of this equiv
• CliffordAlgebraComplex.ofComplex: the reverse direction of this equiv

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

• CliffordAlgebraComplex.toComplex_involute
• CliffordAlgebraComplex.ofComplex_conj

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

Quaternion algebras

• CliffordAlgebraQuaternion.equiv: a QuaternionAlgebra over R is equivalent as an R-algebra to a clifford algebra over R × R, sending i to (0, 1) and j to (1, 0).
• CliffordAlgebraQuaternion.toQuaternion: the forward direction of this equiv
• CliffordAlgebraQuaternion.ofQuaternion: the reverse direction of this equiv

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

• CliffordAlgebraQuaternion.toQuaternion_star
• CliffordAlgebraQuaternion.ofQuaternion_star

Dual numbers

• CliffordAlgebraDualNumber.equiv: R[ε] is equivalent as an R-algebra to a clifford algebra over R where Q = 0.

The clifford algebra isomorphic to a ring

@[simp]

theorem CliffordAlgebraRing.ι_eq_zero {R : Type u_1} [CommRing R] : CliffordAlgebra.ι 0 = 0

instance CliffordAlgebraRing.instCommRingCliffordAlgebraUnitAddCommGroupModuleToSemiringToCommSemiringOfNatQuadraticFormToAddCommMonoidToOrderedCancelAddCommMonoidLinearOrderedCancelAddCommMonoidToOfNat0InstZeroQuadraticForm {R : Type u_1} [CommRing R] : CommRing (CliffordAlgebra 0)

Since the vector space is empty the ring is commutative.

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)

def CliffordAlgebraRing.equiv {R : Type u_1} [CommRing R] : CliffordAlgebra 0 ≃ₐ[R] R

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

The clifford algebra isomorphic to the complex numbers

def CliffordAlgebraComplex.Q : QuadraticForm ℝ ℝ

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

@[simp]

theorem CliffordAlgebraComplex.Q_apply (r : ℝ) : ↑CliffordAlgebraComplex.Q r = -(r * r)

def CliffordAlgebraComplex.toComplex : CliffordAlgebra CliffordAlgebraComplex.Q →ₐ[ℝ] ℂ

Intermediate result for CliffordAlgebraComplex.equiv: clifford algebras over CliffordAlgebraComplex.Q above can be converted to ℂ.

@[simp]

theorem CliffordAlgebraComplex.toComplex_ι (r : 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 ((fun x => ℂ) c)) (↑CliffordAlgebraComplex.toComplex c)

CliffordAlgebra.involute is analogous to Complex.conj.

def CliffordAlgebraComplex.ofComplex : ℂ →ₐ[ℝ] CliffordAlgebra CliffordAlgebraComplex.Q

Intermediate result for CliffordAlgebraComplex.equiv: ℂ can be converted to CliffordAlgebraComplex.Q above can be converted to.

@[simp]

theorem CliffordAlgebraComplex.ofComplex_I : ↑CliffordAlgebraComplex.ofComplex Complex.I = ↑(CliffordAlgebra.ι CliffordAlgebraComplex.Q) 1

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theorem CliffordAlgebraComplex.toComplex_comp_ofComplex : AlgHom.comp CliffordAlgebraComplex.toComplex CliffordAlgebraComplex.ofComplex = AlgHom.id ℝ ℂ

@[simp]

theorem CliffordAlgebraComplex.toComplex_ofComplex (c : ℂ) : ↑CliffordAlgebraComplex.toComplex (↑CliffordAlgebraComplex.ofComplex c) = c

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theorem CliffordAlgebraComplex.ofComplex_comp_toComplex : AlgHom.comp CliffordAlgebraComplex.ofComplex CliffordAlgebraComplex.toComplex = AlgHom.id ℝ (CliffordAlgebra CliffordAlgebraComplex.Q)

@[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 : ℂ) : ↑(AlgEquiv.symm CliffordAlgebraComplex.equiv) a = ↑CliffordAlgebraComplex.ofComplex a

def CliffordAlgebraComplex.equiv : CliffordAlgebra CliffordAlgebraComplex.Q ≃ₐ[ℝ] ℂ

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

instance CliffordAlgebraComplex.instCommRingCliffordAlgebraRealCommRingInstAddCommGroupRealToModuleSemiringQ : CommRing (CliffordAlgebra CliffordAlgebraComplex.Q)

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.

theorem CliffordAlgebraComplex.reverse_apply (x : CliffordAlgebra CliffordAlgebraComplex.Q) : ↑CliffordAlgebra.reverse x = x

reverse is a no-op over CliffordAlgebraComplex.Q.

@[simp]

theorem CliffordAlgebraComplex.reverse_eq_id : CliffordAlgebra.reverse = LinearMap.id

@[simp]

theorem CliffordAlgebraComplex.ofComplex_conj (c : ℂ) : ↑CliffordAlgebraComplex.ofComplex (↑(starRingEnd ℂ) 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) : QuadraticForm R (R × 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].

@[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.fst * v.fst) + c₂ * (v.snd * v.snd)

@[simp]

theorem CliffordAlgebraQuaternion.quaternionBasis_i {R : Type u_1} [CommRing R] (c₁ : R) (c₂ : R) : (CliffordAlgebraQuaternion.quaternionBasis c₁ c₂).i = ↑(CliffordAlgebra.ι (CliffordAlgebraQuaternion.Q c₁ c₂)) (1, 0)

@[simp]

theorem CliffordAlgebraQuaternion.quaternionBasis_j {R : Type u_1} [CommRing R] (c₁ : R) (c₂ : R) : (CliffordAlgebraQuaternion.quaternionBasis c₁ c₂).j = ↑(CliffordAlgebra.ι (CliffordAlgebraQuaternion.Q c₁ c₂)) (0, 1)

@[simp]

theorem CliffordAlgebraQuaternion.quaternionBasis_k {R : Type u_1} [CommRing R] (c₁ : R) (c₂ : R) : (CliffordAlgebraQuaternion.quaternionBasis c₁ c₂).k = ↑(CliffordAlgebra.ι (CliffordAlgebraQuaternion.Q c₁ c₂)) (1, 0) * ↑(CliffordAlgebra.ι (CliffordAlgebraQuaternion.Q c₁ c₂)) (0, 1)

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

The quaternion basis vectors within the algebra.

def CliffordAlgebraQuaternion.toQuaternion {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} : CliffordAlgebra (CliffordAlgebraQuaternion.Q c₁ c₂) →ₐ[R] QuaternionAlgebra R c₁ c₂

Intermediate result of CliffordAlgebraQuaternion.equiv: clifford algebras over CliffordAlgebraQuaternion.Q can be converted to ℍ[R,c₁,c₂].

@[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.fst, imJ := v.snd, 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.

def CliffordAlgebraQuaternion.ofQuaternion {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} : QuaternionAlgebra R c₁ c₂ →ₐ[R] CliffordAlgebra (CliffordAlgebraQuaternion.Q c₁ c₂)

Map a quaternion into the clifford algebra.

@[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} : AlgHom.comp CliffordAlgebraQuaternion.ofQuaternion 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} : AlgHom.comp CliffordAlgebraQuaternion.toQuaternion 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_symm_apply {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} (a : QuaternionAlgebra R c₁ c₂) : ↑(AlgEquiv.symm CliffordAlgebraQuaternion.equiv) a = ↑CliffordAlgebraQuaternion.ofQuaternion a

@[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

def CliffordAlgebraQuaternion.equiv {R : Type u_1} [CommRing R] {c₁ : R} {c₂ : R} : CliffordAlgebra (CliffordAlgebraQuaternion.Q c₁ c₂) ≃ₐ[R] QuaternionAlgebra R c₁ c₂

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

@[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) : ↑(CliffordAlgebra.ι 0) r₁ * ↑(CliffordAlgebra.ι 0) r₂ = 0

def CliffordAlgebraDualNumber.equiv {R : Type u_1} [CommRing R] : CliffordAlgebra 0 ≃ₐ[R] DualNumber R

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

@[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] : ↑(AlgEquiv.symm CliffordAlgebraDualNumber.equiv) DualNumber.eps = ↑(CliffordAlgebra.ι 0) 1