# Documentation

Mathlib.LinearAlgebra.CliffordAlgebra.Equivs

# 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} [] :

Since the vector space is empty the ring is commutative.

theorem CliffordAlgebraRing.reverse_apply {R : Type u_1} [] (x : ) :
CliffordAlgebra.reverse x = x
@[simp]
theorem CliffordAlgebraRing.reverse_eq_id {R : Type u_1} [] :
CliffordAlgebra.reverse = LinearMap.id
@[simp]
theorem CliffordAlgebraRing.involute_eq_id {R : Type u_1} [] :
CliffordAlgebra.involute =
def CliffordAlgebraRing.equiv {R : Type u_1} [] :

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

Instances For

### The clifford algebra isomorphic to the complex numbers #

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

Instances For
@[simp]
theorem CliffordAlgebraComplex.Q_apply (r : ) :
= -(r * r)

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

Instances For
@[simp]
@[simp]
theorem CliffordAlgebraComplex.toComplex_involute :
CliffordAlgebraComplex.toComplex (CliffordAlgebra.involute c) = ↑(starRingEnd ((fun x => ) c)) ()

CliffordAlgebra.involute is analogous to Complex.conj.

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

Instances For
@[simp]
@[simp]

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

Instances For

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 :
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 : ) :
= CliffordAlgebra.involute ()

Complex.conj is analogous to CliffordAlgebra.involute.

### The clifford algebra isomorphic to the quaternions #

def CliffordAlgebraQuaternion.Q {R : Type u_1} [] (c₁ : R) (c₂ : 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].

Instances For
@[simp]
theorem CliffordAlgebraQuaternion.Q_apply {R : Type u_1} [] (c₁ : R) (c₂ : R) (v : R × R) :
↑() v = c₁ * (v.fst * v.fst) + c₂ * (v.snd * v.snd)
@[simp]
theorem CliffordAlgebraQuaternion.quaternionBasis_i {R : Type u_1} [] (c₁ : R) (c₂ : R) :
= ↑() (1, 0)
@[simp]
theorem CliffordAlgebraQuaternion.quaternionBasis_j {R : Type u_1} [] (c₁ : R) (c₂ : R) :
= ↑() (0, 1)
@[simp]
theorem CliffordAlgebraQuaternion.quaternionBasis_k {R : Type u_1} [] (c₁ : R) (c₂ : R) :
= ↑() (1, 0) * ↑() (0, 1)
def CliffordAlgebraQuaternion.quaternionBasis {R : Type u_1} [] (c₁ : R) (c₂ : R) :

The quaternion basis vectors within the algebra.

Instances For
def CliffordAlgebraQuaternion.toQuaternion {R : Type u_1} [] {c₁ : R} {c₂ : R} :

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

Instances For
@[simp]
theorem CliffordAlgebraQuaternion.toQuaternion_ι {R : Type u_1} [] {c₁ : R} {c₂ : R} (v : R × R) :
CliffordAlgebraQuaternion.toQuaternion (↑() v) = { re := 0, imI := v.fst, imJ := v.snd, imK := 0 }
theorem CliffordAlgebraQuaternion.toQuaternion_star {R : Type u_1} [] {c₁ : R} {c₂ : R} (c : ) :
CliffordAlgebraQuaternion.toQuaternion (star c) = star (CliffordAlgebraQuaternion.toQuaternion c)

The "clifford conjugate" maps to the quaternion conjugate.

def CliffordAlgebraQuaternion.ofQuaternion {R : Type u_1} [] {c₁ : R} {c₂ : R} :

Map a quaternion into the clifford algebra.

Instances For
@[simp]
theorem CliffordAlgebraQuaternion.ofQuaternion_mk {R : Type u_1} [] {c₁ : R} {c₂ : R} (a₁ : R) (a₂ : R) (a₃ : R) (a₄ : R) :
CliffordAlgebraQuaternion.ofQuaternion { re := a₁, imI := a₂, imJ := a₃, imK := a₄ } = ↑() a₁ + a₂ ↑() (1, 0) + a₃ ↑() (0, 1) + a₄ (↑() (1, 0) * ↑() (0, 1))
@[simp]
theorem CliffordAlgebraQuaternion.ofQuaternion_comp_toQuaternion {R : Type u_1} [] {c₁ : R} {c₂ : R} :
AlgHom.comp CliffordAlgebraQuaternion.ofQuaternion CliffordAlgebraQuaternion.toQuaternion =
@[simp]
theorem CliffordAlgebraQuaternion.ofQuaternion_toQuaternion {R : Type u_1} [] {c₁ : R} {c₂ : R} (c : ) :
CliffordAlgebraQuaternion.ofQuaternion (CliffordAlgebraQuaternion.toQuaternion c) = c
@[simp]
theorem CliffordAlgebraQuaternion.toQuaternion_comp_ofQuaternion {R : Type u_1} [] {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} [] {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} [] {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} [] {c₁ : R} {c₂ : R} (a : ) :
CliffordAlgebraQuaternion.equiv a = CliffordAlgebraQuaternion.toQuaternion a
def CliffordAlgebraQuaternion.equiv {R : Type u_1} [] {c₁ : R} {c₂ : R} :

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

Instances For
@[simp]
theorem CliffordAlgebraQuaternion.ofQuaternion_star {R : Type u_1} [] {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} [] (r₁ : R) (r₂ : R) :
↑() r₁ * ↑() r₂ = 0

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

Instances For
@[simp]
theorem CliffordAlgebraDualNumber.equiv_ι {R : Type u_1} [] (r : R) :
CliffordAlgebraDualNumber.equiv (↑() r) = r DualNumber.eps
@[simp]
theorem CliffordAlgebraDualNumber.equiv_symm_eps {R : Type u_1} [] :
↑(AlgEquiv.symm CliffordAlgebraDualNumber.equiv) DualNumber.eps = ↑() 1