Clifford algebra

Before trying to read this article you should first read the article Multivectors.

In mathematics, a **Clifford algebra** is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As *K*-algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercomplex number systems. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry, theoretical physics and digital image processing. They are named after the English mathematician William Kingdon Clifford.

The most familiar Clifford algebras, the **orthogonal Clifford algebras**, are also referred to as (*pseudo-*)*Riemannian Clifford algebras*, as distinct from *symplectic Clifford algebras*.

## Introduction and basic properties

A Clifford algebra is a unital associative algebra that contains and is generated by a vector space *V* over a field *K*, where *V* is equipped with a quadratic form *Q* : *V* → *K*. The Clifford algebra Cℓ(*V*, *Q*) is the "freest" algebra generated by *V* subject to the condition

v
2

=
Q
(
v
)
1

 for all
v
∈
V
,

{\displaystyle v^2 = Q(v)1\ \text{ for all } v\in V,}

where the product on the left is that of the algebra, and the 1 is its multiplicative identity. The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a universal property, as done below.

Clifford algebras can be identified by the label *C*ℓ*p*,*q*(**R**), indicating that the algebra is constructed from *p* simple basis elements with *ei*2 = +1, *q* with *ei*2 = −1, and where **R** indicates that this is to be a Clifford algebra over the reals—i.e. coefficients of elements of the algebra are to be real numbers.

The free algebra generated by *V* may be written as the tensor algebra ⊕*n*≥0 *V* ⊗ ... ⊗ *V*, that is, the sum of the tensor product of *n* copies of *V* over all *n*, and so a Clifford algebra would be the quotient of this tensor algebra by the two-sided ideal generated by elements of the form *v* ⊗ *v* − *Q*(*v*)1 for all elements *v* ∈ *V*. The product induced by the tensor product in the quotient algebra is written using juxtaposition (e.g. *uv*). Its associativity follows from the associativity of the tensor product.

The Clifford algebra has a distinguished subspace *V*, being the image of the embedding map. Such a subspace cannot in general be uniquely determined given only a *K*-algebra isomorphic to the Clifford algebra.

If the characteristic of the ground field *K* is not 2, then one can rewrite this fundamental identity in the form

u
v
v
# u
2
⟨
u
,
v
⟩
1

 for all
u
,
v
∈
V
,

{\displaystyle uv + vu = 2\langle u, v\rangle1\ \text{ for all }u,v \in V,}

where

⟨
u
,
v
# ⟩

1
2

(

Q
(
u
v
)
−
Q
(
u
)
−
Q
(
v
)

)

{\displaystyle \langle u , v \rangle = \frac{1}{2} \left( Q(u+v) - Q(u) - Q(v) \right) }

is the symmetric bilinear form associated with *Q*, via the polarization identity.

Quadratic forms and Clifford algebras in characteristic 2 form an exceptional case. In particular, if char(*K*) = 2 it is not true that a quadratic form uniquely determines a symmetric bilinear form satisfying *Q*(*v*) = ⟨*v*, *v*⟩, nor that every quadratic form admits an orthogonal basis. Many of the statements in this article include the condition that the characteristic is not 2, and are false if this condition is removed.

### As a quantization of the exterior algebra

Clifford algebras are closely related to exterior algebras. Indeed, if *Q* = 0 then the Clifford algebra Cℓ(*V*, *Q*) is just the exterior algebra ⋀(*V*). For nonzero *Q* there exists a canonical *linear* isomorphism between ⋀(*V*) and Cℓ(*V*, *Q*) whenever the ground field *K* does not have characteristic two. That is, they are naturally isomorphic as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication together with the distinguished subspace is strictly richer than the exterior product since it makes use of the extra information provided by *Q*.

The Clifford algebra is a filtered algebra, the associated graded algebra is the exterior algebra.

More precisely, Clifford algebras may be thought of as *quantizations* (cf. Quantum group) of the exterior algebra, in the same way that the Weyl algebra is a quantization of the symmetric algebra.

Weyl algebras and Clifford algebras admit a further structure of a \*-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras.

## Universal property and construction

Let *V* be a vector space over a field *K*, and let *Q* : *V* → *K* be a quadratic form on *V*. In most cases of interest the field *K* is either the field of real numbers **R**, or the field of complex numbers **C**, or a finite field.

A Clifford algebra Cℓ(*V*, *Q*) is a unital associative algebra over *K* together with a linear map *i* : *V* → Cℓ(*V*, *Q*) satisfying *i*(*v*)2 = *Q*(*v*)1 for all *v* ∈ *V*, defined by the following universal property: given any unital associative algebra *A* over *K* and any linear map *j* : *V* → *A* such that

j
(
v

)
2

=
Q
(
v
)

1
A

 for all
v
∈
V

{\displaystyle j(v)^2 = Q(v)1\_A \text{ for all } v \in V}

(where 1*A* denotes the multiplicative identity of *A*), there is a unique algebra homomorphism *f* : Cℓ(*V*, *Q*) → *A* such that the following diagram commutes (i.e. such that *f* ∘ *i* = *j*):

In characteristic not 2, the quadratic form *Q* may be replaced by a symmetric bilinear form

⟨

⋅

,

⋅

⟩

{\displaystyle \langle {\cdot}, {\cdot} \rangle}

, in which case an equivalent requirement on *j* is

j
(
v
)
j
(
w
)
j
(
w
)
j
(
v
# )
2
⟨
v
,
w
⟩

1
A

 for all
v
,
w
∈
V

.

{\displaystyle j(v)j(w) + j(w)j(v) = 2 \langle v, w \rangle 1\_A \quad \text{ for all } v,w \in V \ . }

A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains *V*, namely the tensor algebra *T*(*V*), and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal *IQ* in *T*(*V*) generated by all elements of the form

v
⊗
v
−
Q
(
v
)
1

{\displaystyle v\otimes v - Q(v)1}

 for all

v
∈
V

{\displaystyle v \in V}

and define Cℓ(*V*, *Q*) as the quotient algebra

C
ℓ

⁡
(
V
,
Q
# )
T
(
V
)

/

I
Q

.

{\displaystyle \operatorname{C\ell}(V, Q) = T(V) / I\_Q .}

The ring product inherited by this quotient is sometimes referred to as the **Clifford product** to distinguish it from the exterior product and the scalar product.

It is then straightforward to show that Cℓ(*V*, *Q*) contains *V* and satisfies the above universal property, so that Cℓ is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra Cℓ(*V*, *Q*). It also follows from this construction that *i* is injective. One usually drops the *i* and considers *V* as a linear subspace of Cℓ(*V*, *Q*).

The universal characterization of the Clifford algebra shows that the construction of Cℓ(*V*, *Q*) is *functorial* in nature. Namely, Cℓ can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (preserving the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.

## Basis and dimension

Since *V* comes equipped with a quadratic form, in characteristic not 2 there is a set of privileged bases for *V*: those that are orthogonal. An orthogonal basis is one such that

⟨

e
i

,

e
j

# ⟩
0

{\displaystyle \langle e\_i, e\_j \rangle = 0 }

 for

i
≠
j

{\displaystyle i\ne j}

, and

⟨

e
i

,

e
i

# ⟩
Q
(

e
i

)

{\displaystyle \langle e\_i, e\_i \rangle = Q(e\_i)}

where

⟨
⋅
,
⋅
⟩

{\displaystyle \langle\cdot , \cdot\rangle}

 is the symmetric bilinear form associated to *Q*. The fundamental Clifford identity implies that for an orthogonal basis

e
i

e
j

=
−

e
j

e
i

{\displaystyle e\_ie\_j = -e\_je\_i }

 for

i
≠
j

{\displaystyle i\ne j}

, and

e
i
2

=
Q
(

e
i

)

{\displaystyle e\_i^2= Q(e\_i) \,}

.
This makes manipulation of orthogonal basis vectors quite simple. Given a product

e

i
1

e

i
2

⋯

e

i
k

{\displaystyle e\_{i\_1}e\_{i\_2}\cdots e\_{i\_k}}

 of *distinct* orthogonal basis vectors of *V*, one can put them into a standard order while including an overall sign determined by the number of pairwise swaps needed to do so (i.e. the signature of the ordering permutation).

If the dimension of *V* over *K* is *n* and {*e*1, ..., *e**n*} is an orthogonal basis of (*V*, *Q*), then Cℓ(*V*, *Q*) is free over *K* with a basis

{

e

i
1

e

i
2

⋯

e

i
k

∣
1
≤

i
1

<

i
2

<
⋯
<

i
k

≤
n

 and

0
≤
k
≤
n
}

{\displaystyle \{e\_{i\_1}e\_{i\_2}\cdots e\_{i\_k} \mid 1\le i\_1 < i\_2 < \cdots < i\_k \le n\mbox{ and } 0\le k\le n\}}

.
The empty product (*k* = 0) is defined as the multiplicative identity element. For each value of *k* there are *n* choose *k* basis elements, so the total dimension of the Clifford algebra is

dim
⁡

C
ℓ

⁡
(
V
,
Q

# k
0

n

(

n

k

)

=

2
n

.

{\displaystyle \dim \operatorname{C\ell}(V,Q) = \sum\_{k=0}^n\begin{pmatrix}n\\ k\end{pmatrix} = 2^n.}

## Examples: real and complex Clifford algebras

The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms.

Each of the algebras Cℓ*p*,*q*(**R**) and Cℓ*n*(**C**) is isomorphic to *A* or *A* ⊕ *A*, where *A* is a full matrix ring with entries from **R**, **C**, or **H**. For a complete classification of these algebras see classification of Clifford algebras.

### Real numbers

Main articles: Geometric algebra
Real Clifford algebras are also sometimes referred to as geometric algebras.

Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:

Q
(
v
# )

v
1
2

⋯

v
p
2

−

v

p
1

2

−
⋯
−

v

p
q

2

,

{\displaystyle Q(v) = v\_1^2 + \cdots + v\_p^2 - v\_{p+1}^2 - \cdots - v\_{p+q}^2 ,}

where *n* = *p* + *q* is the dimension of the vector space. The pair of integers (*p*, *q*) is called the signature of the quadratic form. The real vector space with this quadratic form is often denoted **R***p*,*q*. The Clifford algebra on **R***p*,*q* is denoted Cℓ*p*,*q*(**R**). The symbol Cℓ*n*(**R**) means either Cℓ*n*,0(**R**) or Cℓ0,*n*(**R**) depending on whether the author prefers positive-definite or negative-definite spaces.

A standard basis {*e*1, ..., *e**n*}  for **R***p*,*q* consists of *n* = *p* + *q* mutually orthogonal vectors, *p* of which square to +1 and *q* of which square to −1. Of such a basis, the algebra Cℓ*p*,*q*(**R**) will therefore have *p* vectors that square to +1 and *q* vectors that square to −1.

A few low-dimensional cases are:

Cℓ0,0(**R**) is naturally isomorphic to **R** since there are no nonzero vectors.
Cℓ0,1(**R**) is a two-dimensional algebra generated by *e*1 that squares to −1, and is algebra-isomorphic to **C**, the field of complex numbers.
Cℓ0,2(**R**) is a four-dimensional algebra spanned by {1, *e*1, *e*2, *e*1*e*2}. The latter three elements all square to −1 and anticommute, and so the algebra is isomorphic to the quaternions **H**.
Cℓ0,3(**R**) is an 8-dimensional algebra isomorphic to the direct sum **H** ⊕ **H**, the split-biquaternions.
### Complex numbers

One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimension *n* is equivalent to the standard diagonal form

Q
(
z
# )

z
1
2

z
2
2

⋯

z
n
2

{\displaystyle Q(z) = z\_1^2 + z\_2^2 + \cdots + z\_n^2}

.
Thus, for each dimension *n*, up to isomorphism there is only one Clifford algebra of a complex vector space with a nondegenerate quadratic form. We will denote the Clifford algebra on **C***n* with the standard quadratic form by Cℓ*n*(**C**).

For the first few cases one finds that

Cℓ0(**C**) ≅ **C**, the complex numbers
Cℓ1(**C**) ≅ **C** ⊕ **C**, the bicomplex numbers
Cℓ2(**C**) ≅ M2(**C**), the biquaternions
where M*n*(**C**) denotes the algebra of *n* × *n* matrices over **C**.

## Examples: constructing quaternions and dual quaternions

### Quaternions

In this section, Hamilton's quaternions are constructed as the even sub algebra of the Clifford algebra Cℓ0,3(**R**).

Let the vector space *V* be real three-dimensional space **R**3, and the quadratic form *Q* be the negative of the usual Euclidean metric. Then, for *v*, *w* in **R**3 we have the bilinear form (or scalar product)

v
⋅
# w

v
1

w
1

v
2

w
2

v
3

w
3

.

{\displaystyle v \cdot w = v\_1 w\_1 + v\_2 w\_2 + v\_3 w\_3.}

Now introduce the Clifford product of vectors *v* and *w* given by

v
w
w
# v
−
2
(
v
⋅
w
)
.

{\displaystyle v w + w v = -2 (v \cdot w) .}

This formulation uses the negative sign so the correspondence with quaternions is easily shown.

Denote a set of orthogonal unit vectors of **R**3 as *e*1, *e*2, and *e*3, then the Clifford product yields the relations

e
2

e
3

=
−

e
3

e
2

,

e
3

e
1

=
−

e
1

e
3

,

e
1

e
2

=
−

e
2

e
1

,

{\displaystyle e\_2 e\_3 = -e\_3 e\_2, \,\,\, e\_3 e\_1 = -e\_1 e\_3,\,\,\, e\_1 e\_2 = -e\_2 e\_1,}

and

e
1
2

=

e
2
2

=

e
3
2

=
−

{\displaystyle e\_1 ^2 = e\_2^2 = e\_3^2 = -1. }

The general element of the Clifford algebra Cℓ0,3(**R**) is given by

# A

a
0

a
1

e
1

a
2

e
2

a
3

e
3

a
4

e
2

e
3

a
5

e
3

e
1

a
6

e
1

e
2

a
7

e
1

e
2

e
3

.

{\displaystyle A = a\_0 + a\_1 e\_1 + a\_2 e\_2 + a\_3 e\_3 + a\_4 e\_2 e\_3 + a\_5 e\_3 e\_1 + a\_6 e\_1 e\_2 + a\_7 e\_1 e\_2 e\_3.}

The linear combination of the even degree elements of Cℓ0,3(**R**) defines the even subalgebra Cℓ[0]
0,3(**R**) with the general element

# q

q
0

q
1

e
2

e
3

q
2

e
3

e
1

q
3

e
1

e
2

.

{\displaystyle q = q\_0 + q\_1 e\_2 e\_3 + q\_2 e\_3 e\_1 + q\_3 e\_1 e\_2. }

The basis elements can be identified with the quaternion basis elements *i*, *j*, *k* as

# i

e
2

e
3

,
# j

e
3

e
1

,
# k

e
1

e
2

,

{\displaystyle i= e\_2 e\_3, j= e\_3 e\_1, k = e\_1 e\_2,}

which shows that the even subalgebra Cℓ[0]
0,3(**R**) is Hamilton's real quaternion algebra.

To see this, compute

i
2

=
(

e
2

e
3

)
2

=

e
2

e
3

e
2

e
3

=
−

e
2

e
2

e
3

e
3

=
−
1
,

{\displaystyle i^2 = (e\_2 e\_3)^2 = e\_2 e\_3 e\_2 e\_3 = - e\_2 e\_2 e\_3 e\_3 = -1,}

and

i
# j

e
2

e
3

e
3

e
1

=
−

e
2

e
1

=

e
1

e
2

=
k
.

{\displaystyle ij = e\_2 e\_3 e\_3 e\_1 = -e\_2 e\_1 = e\_1 e\_2 = k.}

Finally,

i
j
# k

e
2

e
3

e
3

e
1

e
1

e
2

=
−

{\displaystyle ijk = e\_2 e\_3 e\_3 e\_1 e\_1 e\_2 = -1.}

### Dual quaternions

In this section, dual quaternions are constructed as the even Clifford algebra of real four-dimensional space with a degenerate quadratic form.

Let the vector space *V* be real four-dimensional space **R**4, and let the quadratic form *Q* be a degenerate form derived from the Euclidean metric on **R**3. For *v*, *w* in **R**4 introduce the degenerate bilinear form

d
(
v
,
w
# )

v
1

w
1

v
2

w
2

v
3

w
3

.

{\displaystyle d(v, w)= v\_1 w\_1 + v\_2 w\_2 + v\_3 w\_3 .}

This degenerate scalar product projects distance measurements in **R**4 onto the **R**3 hyperplane.

The Clifford product of vectors *v* and *w* is given by

v
w
w
# v
−
2

d
(
v
,
w
)
.

{\displaystyle v w + w v = -2 \,d(v, w).}

Note the negative sign is introduced to simplify the correspondence with quaternions.

Denote a set of mutually orthogonal unit vectors of **R**4 as *e*1, *e*2, *e*3 and *e*4, then the Clifford product yields the relations

e
m

e
n

=
−

e
n

e
m

,

m
≠
n
,

{\displaystyle e\_m e\_n = -e\_n e\_m, \,\,\, m \ne n,}

and

e
1
2

=

e
2
2

=

e
3
2

=
−
1
,

e
4
2

=

{\displaystyle e\_1 ^2 = e\_2^2 =e\_3^2 = -1, \,\, e\_4^2 =0.}

The general element of the Clifford algebra Cℓ(**R**4, *d*) has 16 components. The linear combination of the even degree elements defines the even subalgebra Cℓ[0]
(**R**4, *d*) with the general element

# H

h
0

h
1

e
2

e
3

h
2

e
3

e
1

h
3

e
1

e
2

h
4

e
4

e
1

h
5

e
4

e
2

h
6

e
4

e
3

h
7

e
1

e
2

e
3

e
4

.

{\displaystyle H = h\_0 + h\_1 e\_2 e\_3 + h\_2 e\_3 e\_1 + h\_3 e\_1 e\_2 + h\_4 e\_4 e\_1 + h\_5 e\_4 e\_2 + h\_6 e\_4 e\_3 + h\_7 e\_1 e\_2 e\_3 e\_4.}

The basis elements can be identified with the quaternion basis elements *i*, *j*, *k* and the dual unit *ε* as

# i

e
2

e
3

,
# j

e
3

e
1

,
# k

e
1

e
2

,

# ε

e
1

e
2

e
3

e
4

.

{\displaystyle i = e\_2 e\_3, j = e\_3 e\_1, k = e\_1 e\_2, \,\, \varepsilon = e\_1 e\_2 e\_3 e\_4.}

This provides the correspondence of Cℓ[0]
0,3,1(**R**) with dual quaternion algebra.

To see this, compute

ε
2

=
(

e
1

e
2

e
3

e
4

)
2

=

e
1

e
2

e
3

e
4

e
1

e
2

e
3

e
4

=
−

e
1

e
2

e
3

(

e
4

e
4

)

e
1

e
2

e
3

=
0
,

{\displaystyle \varepsilon ^2 = (e\_1 e\_2 e\_3 e\_4)^2 = e\_1 e\_2 e\_3 e\_4 e\_1 e\_2 e\_3 e\_4 = -e\_1 e\_2 e\_3 (e\_4 e\_4 ) e\_1 e\_2 e\_3 = 0 ,}

and

ε
# i
(

e
1

e
2

e
3

e
4

)

e
2

e
3

=

e
1

e
2

e
3

e
4

e
2

e
3

=

e
2

e
3

(

e
1

e
2

e
3

e
4

# )
i
ε
.

{\displaystyle \varepsilon i = (e\_1 e\_2 e\_3 e\_4) e\_2 e\_3 = e\_1 e\_2 e\_3 e\_4 e\_2 e\_3 = e\_2 e\_3 (e\_1 e\_2 e\_3 e\_4) = i\varepsilon.}

The exchanges of *e*1 and *e*4 alternate signs an even number of times, and show the dual unit *ε* commutes with the quaternion basis elements *i*, *j*, and *k*.

## Examples: in small dimension

Let *K* be any field of characteristic not 2.

### Dimension 1

For dim *V* = 1, if *Q* has diagonalization diag(*a*), that is there is a non-zero vector *x* such that *Q*(*x*) = *a*, then Cℓ(*V*, *Q*) is a *K*-algebra generated by an element *x* satisfying *x*2 = *a*, so it is the étale quadratic algebra *K*[*X*] / (*X*2 − *a*).

In particular, if *a* = 0 (that is, *Q* is the zero quadratic form) then Cℓ(*V*, *Q*) is the dual numbers algebra over *K*.

If *a* is a non-zero square in *K*, then Cℓ(*V*, *Q*) ≃ *K* ⊕ *K*.

Otherwise, Cℓ(*V*, *Q*) is the quadratic field extension *K*(√*a*) of *K*.

### Dimension 2

For dim *V* = 2, if *Q* has diagonalization diag(*a*, *b*) with non-zero *a* and *b* (which always exists if *Q* is non-degenerate), then Cℓ(*V*, *Q*) is a *K*-algebra generated by elements *x* and *y* satisfying *x*2 = *a*, *y*2 = *b* and *xy* = −*yx*.

Thus Cℓ(*V*, *Q*) is the (generalized) quaternion algebra (*a*, *b*)*K*. We retrieve Hamilton's quaternions when *a* = *b* = −1, since **H** = (−1, −1)**R**.

As a special case, if some *x* in *V* satisfies *Q*(*x*) = 1, then Cℓ(*V*, *Q*) = M2(*K*).

## Properties

### Relation to the exterior algebra

Given a vector space *V* one can construct the exterior algebra ⋀(*V*), whose definition is independent of any quadratic form on *V*. It turns out that if *K* does not have characteristic 2 then there is a natural isomorphism between ⋀(*V*) and Cℓ(*V*, *Q*) considered as vector spaces (and there exists an isomorphism in characteristic two, which may not be natural). This is an algebra isomorphism if and only if *Q* = 0. One can thus consider the Clifford algebra Cℓ(*V*, *Q*) as an enrichment (or more precisely, a quantization, cf. the Introduction) of the exterior algebra on *V* with a multiplication that depends on *Q* (one can still define the exterior product independently of *Q*).

The easiest way to establish the isomorphism is to choose an *orthogonal* basis {*e*1, ..., *e**n*}  for *V* and extend it to a basis for Cℓ(*V*, *Q*) as described above. The map Cℓ(*V*, *Q*) → ⋀(*V*) is determined by

e

i
1

e

i
2

⋯

e

i
k

↦

e

i
1

∧

e

i
2

∧
⋯
∧

e

i
k

.

{\displaystyle e\_{i\_1}e\_{i\_2}\cdots e\_{i\_k} \mapsto e\_{i\_1}\wedge e\_{i\_2}\wedge \cdots \wedge e\_{i\_k}.}

Note that this only works if the basis {*e*1, ..., *e**n*}  is orthogonal. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism.

If the characteristic of *K* is 0, one can also establish the isomorphism by antisymmetrizing. Define functions *fk*: *V* × ... × *V* → Cℓ(*V*, *Q*) by

f
k

(

v
1

,
⋯
,

v
k

# )

1

k
!

∑

σ
∈

S
k

s
g
n

(
σ
)

v

σ
(
1
)

⋯

v

σ
(
k
)

{\displaystyle f\_k(v\_1, \cdots, v\_k) = \frac{1}{k!}\sum\_{\sigma\in S\_k}{\rm sgn}(\sigma)\, v\_{\sigma(1)}\cdots v\_{\sigma(k)}}

where the sum is taken over the symmetric group on *k* elements. Since *fk* is alternating it induces a unique linear map ⋀*k*(*V*) → Cℓ(*V*, *Q*). The direct sum of these maps gives a linear map between ⋀(*V*) and Cℓ(*V*, *Q*). This map can be shown to be a linear isomorphism, and it is natural.

A more sophisticated way to view the relationship is to construct a filtration on Cℓ(*V*, *Q*). Recall that the tensor algebra *T*(*V*) has a natural filtration: *F*0 ⊂ *F*1 ⊂ *F*2 ⊂ ..., where *Fk* contains sums of tensors with order ≤ *k*. Projecting this down to the Clifford algebra gives a filtration on Cℓ(*V*, *Q*). The associated graded algebra

G

r
F

C
ℓ

⁡
(
V
,
Q
# )

⨁
k

F
k

/

F

k
−
1

{\displaystyle Gr\_F \operatorname{C\ell}(V,Q) = \bigoplus\_k F^k/F^{k-1}}

is naturally isomorphic to the exterior algebra ⋀(*V*). Since the associated graded algebra of a filtered algebra is always isomorphic to the filtered algebra as filtered vector spaces (by choosing complements of *Fk* in *F**k*+1 for all *k*), this provides an isomorphism (although not a natural one) in any characteristic, even two.

### Grading

In the following, assume that the characteristic is not 2.

Clifford algebras are **Z**2-graded algebras (also known as superalgebras). Indeed, the linear map on *V* defined by *v* ↦ −*v* (reflection through the origin) preserves the quadratic form *Q* and so by the universal property of Clifford algebras extends to an algebra automorphism

α
:

C
ℓ

⁡
(
V
,
Q
)
→

C
ℓ

⁡
(
V
,
Q
)
.

{\displaystyle \alpha: \operatorname{C\ell}(V,Q) \to \operatorname{C\ell}(V,Q).}

Since *α* is an involution (i.e. it squares to the identity) one can decompose Cℓ(*V*, *Q*) into positive and negative eigenspaces of *α*

C
ℓ

⁡
(
V
,
Q
# )

C
ℓ

[
0
]

⁡
(
V
,
Q
)
⊕

C
ℓ

[
1
]

⁡
(
V
,
Q
)

{\displaystyle \operatorname{C\ell}(V,Q) = \operatorname{C\ell}^{[0]}(V,Q) \oplus \operatorname{C\ell}^{[1]}(V,Q)}

where

C
ℓ

[
i
]

⁡
(
V
,
Q
# )

{

x
∈

C
ℓ

⁡
(
V
,
Q
)

α
(
x
# )
(
−
1

)
i

x

}

.

{\displaystyle \operatorname{C\ell}^{[i]}(V,Q) = \left.\left\{ x \in \operatorname{C\ell}(V,Q) \right | \alpha(x) = (-1)^i x \right \}.}

Since *α* is an automorphism it follows that:

C
ℓ

[
i
]

⁡
(
V
,
Q
)

C
ℓ

[
j
]

⁡
(
V
,
Q
# )

C
ℓ

[
i
j
]

⁡
(
V
,
Q
)

{\displaystyle \operatorname{C\ell}^{[i]}(V,Q)\operatorname{C\ell}^{[j]}(V,Q) = \operatorname{C\ell}^{[i+j]}(V,Q)}

where the bracketed superscripts are read modulo 2. This gives Cℓ(*V*, *Q*) the structure of a **Z**2-graded algebra. The subspace Cℓ[0](*V*, *Q*) forms a subalgebra of Cℓ(*V*, *Q*), called the *even subalgebra*. The subspace Cℓ[1](*V*, *Q*) is called the *odd part* of Cℓ(*V*, *Q*) (it is not a subalgebra). This **Z**2-grading plays an important role in the analysis and application of Clifford algebras. The automorphism *α* is called the *main involution* or *grade involution*. Elements that are pure in this **Z**2-grading are simply said to be even or odd.

*Remark*. In characteristic not 2 the underlying vector space of Cℓ(*V*, *Q*) inherits an **N**-grading and a **Z**-grading from the canonical isomorphism with the underlying vector space of the exterior algebra ⋀(*V*). It is important to note, however, that this is a *vector space grading only*. That is, Clifford multiplication does not respect the **N**-grading or **Z**-grading, only the **Z**2-grading: for instance if *Q*(*v*) ≠ 0, then *v* ∈ Cℓ1(*V*, *Q*), but *v*2 ∈ Cℓ0(*V*, *Q*), not in Cℓ2(*V*, *Q*). Happily, the gradings are related in the natural way: **Z**2 ≅ **N**/2**N** ≅ **Z**/2**Z**. Further, the Clifford algebra is **Z**-filtered:

C
ℓ

⩽
i

⁡
(
V
,
Q
)
⋅

C
ℓ

⩽
j

⁡
(
V
,
Q
)
⊂

C
ℓ

⩽
i
j

⁡
(
V
,
Q
)
.

{\displaystyle \operatorname{C\ell}^{\leqslant i}(V,Q) \cdot \operatorname{C\ell}^{\leqslant j}(V,Q) \subset \operatorname{C\ell}^{\leqslant i+j}(V,Q).}

The *degree* of a Clifford number usually refers to the degree in the **N**-grading.

The even subalgebra Cℓ[0](*V*, *Q*) of a Clifford algebra is itself isomorphic to a Clifford algebra. If *V* is the orthogonal direct sum of a vector *a* of nonzero norm *Q*(*a*) and a subspace *U*, then Cℓ[0](*V*, *Q*) is isomorphic to Cℓ(*U*, −*Q*(*a*)*Q*), where −*Q*(*a*)*Q* is the form *Q* restricted to *U* and multiplied by −*Q*(*a*). In particular over the reals this implies that:

C
ℓ

p
,
q

[
0
]

⁡
(

R

)
≅

{

C
ℓ

p
,
q
−
1

⁡
(

R

)

q
>
0

C
ℓ

q
,
p
−
1

⁡
(

R

)

p
>
0

{\displaystyle \operatorname{C\ell}\_{p,q}^{[0]}(\mathbf{R}) \cong \begin{cases} \operatorname{C\ell}\_{p,q-1}(\mathbf{R}) & q > 0 \\ \operatorname{C\ell}\_{q,p-1}(\mathbf{R}) & p > 0 \end{cases}}

In the negative-definite case this gives an inclusion Cℓ0,*n*−1(**R**) ⊂ Cℓ0,*n*(**R**), which extends the sequence

**R** ⊂ **C** ⊂ **H** ⊂ **H** ⊕ **H** ⊂ ...
Likewise, in the complex case, one can show that the even subalgebra of Cℓ*n*(**C**) is isomorphic to Cℓ*n*−1(**C**).

### Antiautomorphisms

In addition to the automorphism *α*, there are two antiautomorphisms that play an important role in the analysis of Clifford algebras. Recall that the tensor algebra *T*(*V*) comes with an antiautomorphism that reverses the order in all products of vectors:

v
1

⊗

v
2

⊗
⋯
⊗

v
k

↦

v
k

⊗
⋯
⊗

v
2

⊗

v
1

.

{\displaystyle v\_1\otimes v\_2\otimes \cdots \otimes v\_k \mapsto v\_k\otimes \cdots \otimes v\_2\otimes v\_1.}

Since the ideal *I**Q* is invariant under this reversal, this operation descends to an antiautomorphism of Cℓ(*V*, *Q*) called the *transpose* or *reversal* operation, denoted by *x*t. The transpose is an antiautomorphism: (*xy*)t = *y*t *x*t. The transpose operation makes no use of the **Z**2-grading so we define a second antiautomorphism by composing *α* and the transpose. We call this operation *Clifford conjugation* denoted

x
¯

{\displaystyle \bar{x}}

x
¯

=
α
(

x

t

# )
α
(
x

)

t

.

{\displaystyle \bar x = \alpha(x^\mathrm{t}) = \alpha(x)^\mathrm{t}.}

Of the two antiautomorphisms, the transpose is the more fundamental.

Note that all of these operations are involutions. One can show that they act as ±1 on elements which are pure in the **Z**-grading. In fact, all three operations depend only on the degree modulo 4. That is, if *x* is pure with degree *k* then

α
(
x
# )
±
x

x

t

=
±
x

x
¯

=
±
x

{\displaystyle \alpha(x) = \pm x \qquad x^\mathrm{t} = \pm x \qquad \bar x = \pm x}

where the signs are given by the following table:

α
(
x
)

{\displaystyle \alpha(x)\,}

x

t

{\displaystyle x^\mathrm{t}\,}

x
¯

{\displaystyle \bar{x}}

### Clifford scalar product

When the characteristic is not 2, the quadratic form *Q* on *V* can be extended to a quadratic form on all of Cℓ(*V*, *Q*) (which we also denoted by *Q*). A basis independent definition of one such extension is

Q
(
x
# )
⟨

x

t

x
⟩

{\displaystyle Q(x) = \langle x^\mathrm{t} x\rangle}

where ⟨*a*⟩ denotes the scalar part of *a* (the degree 0 part in the **Z**-grading). One can show that

Q
(

v
1

v
2

⋯

v
k

# )
Q
(

v
1

)
Q
(

v
2

)
⋯
Q
(

v
k

)

{\displaystyle Q(v\_1v\_2\cdots v\_k) = Q(v\_1)Q(v\_2)\cdots Q(v\_k)}

where the *vi* are elements of *V* – this identity is *not* true for arbitrary elements of Cℓ(*V*, *Q*).

The associated symmetric bilinear form on Cℓ(*V*, *Q*) is given by

⟨
x
,
y
# ⟩
⟨

x

t

y
⟩
.

{\displaystyle \langle x, y\rangle = \langle x^\mathrm{t} y\rangle.}

One can check that this reduces to the original bilinear form when restricted to *V*. The bilinear form on all of Cℓ(*V*, *Q*) is nondegenerate if and only if it is nondegenerate on *V*.

It is not hard to verify that the operator of left/right Clifford multiplication by the transpose

a

t

{\displaystyle a^\mathrm{t}}

 of an element

a

{\displaystyle a}

 is the adjoint of left/right Clifford multiplication by

a

{\displaystyle a}

 itself with respect to this inner product. That is,

⟨
a
x
,
y
# ⟩
⟨
x
,

a

t

y
⟩
,

{\displaystyle \langle ax, y\rangle = \langle x, a^\mathrm{t} y\rangle,}

and

⟨
x
a
,
y
# ⟩
⟨
x
,
y

a

t

⟩
.

{\displaystyle \langle xa, y\rangle = \langle x, y a^\mathrm{t}\rangle.}

## Structure of Clifford algebras

In this section we assume that characteristic is not 2, the vector space *V* is finite-dimensional and that the associated symmetric bilinear form of *Q* is non-singular. A central simple algebra over *K* is a matrix algebra over a (finite-dimensional) division algebra with center *K*. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.

* If *V* has even dimension then Cℓ(*V*, *Q*) is a central simple algebra over *K*.
* If *V* has even dimension then Cℓ[0](*V*, *Q*) is a central simple algebra over a quadratic extension of *K* or a sum of two isomorphic central simple algebras over *K*.
* If *V* has odd dimension then Cℓ(*V*, *Q*) is a central simple algebra over a quadratic extension of *K* or a sum of two isomorphic central simple algebras over *K*.
* If *V* has odd dimension then Cℓ[0](*V*, *Q*) is a central simple algebra over *K*.

The structure of Clifford algebras can be worked out explicitly using the following result. Suppose that *U* has even dimension and a non-singular bilinear form with discriminant *d*, and suppose that *V* is another vector space with a quadratic form. The Clifford algebra of *U* + *V* is isomorphic to the tensor product of the Clifford algebras of *U* and (−1)dim(*U*)/2*dV*, which is the space *V* with its quadratic form multiplied by (−1)dim(*U*)/2*d*. Over the reals, this implies in particular that

C
ℓ

p
2
,
q

⁡
(

R

# )

M

2

(

R

)
⊗

C
ℓ

q
,
p

⁡
(

R

)

{\displaystyle \operatorname{C\ell}\_{p+2,q}(\mathbf{R}) = \mathrm{M}\_2(\mathbf{R})\otimes \operatorname{C\ell}\_{q,p}(\mathbf{R}) }

C
ℓ

p
1
,
q
1

⁡
(

R

# )

M

2

(

R

)
⊗

C
ℓ

p
,
q

⁡
(

R

)

{\displaystyle \operatorname{C\ell}\_{p+1,q+1}(\mathbf{R}) = \mathrm{M}\_2(\mathbf{R})\otimes \operatorname{C\ell}\_{p,q}(\mathbf{R}) }

C
ℓ

p
,
q
2

⁡
(

R

# )

H

⊗

C
ℓ

q
,
p

⁡
(

R

)
.

{\displaystyle \operatorname{C\ell}\_{p,q+2}(\mathbf{R}) = \mathbf{H}\otimes \operatorname{C\ell}\_{q,p}(\mathbf{R}). }

These formulas can be used to find the structure of all real Clifford algebras and all complex Clifford algebras; see the classification of Clifford algebras.

Notably, the Morita equivalence class of a Clifford algebra (its representation theory: the equivalence class of the category of modules over it) depends only on the signature (*p* − *q*) mod 8. This is an algebraic form of Bott periodicity.

## Applications

### Differential geometry

One of the principal applications of the exterior algebra is in differential geometry where it is used to define the bundle of differential forms on a smooth manifold. In the case of a (pseudo-)Riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric. Thus, one can define a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry. Perhaps more importantly is the link to a spin manifold, its associated spinor bundle and spinc manifolds.

### Physics

Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra with a basis generated by the matrices *γ*0, ..., *γ*3 called Dirac matrices which have the property that

γ
i

γ
j

γ
j

γ
i

=
2

η

i
j

{\displaystyle \gamma\_i\gamma\_j + \gamma\_j\gamma\_i = 2\eta\_{ij}\,}

where *η* is the matrix of a quadratic form of signature (1, 3) (or (3, 1) corresponding to the two equivalent choices of metric signature). These are exactly the defining relations for the Clifford algebra Cℓ
1,3(**R**), whose complexification is Cℓ
1,3(**R**)**C** which, by the classification of Clifford algebras, is isomorphic to the algebra of 4 × 4 complex matrices Cℓ4(**C**) ≈ M4(**C**). However, it is best to retain the notation Cℓ
1,3(**R**)**C**, since any transformation that takes the bilinear form to the canonical form is *not* a Lorentz transformation of the underlying spacetime.

The Clifford algebra of spacetime used in physics thus has more structure than Cℓ4(**C**). It has in addition a set of preferred transformations – Lorentz transformations. Whether complexification is necessary to begin with depends in part on conventions used and in part on how much one wants to incorporate straightforwardly, but complexification is most often necessary in quantum mechanics where the spin representation of the Lie algebra **so**(1, 3) sitting inside the Clifford algebra conventionally requires a complex Clifford algebra. For reference, the spin Lie algebra is given by

σ

μ
ν

=
−

i
4

[

γ
μ

,

γ
ν

]
,

{\displaystyle \sigma^{\mu\nu} = -\frac{i}{4}[\gamma^\mu, \gamma^\nu],}

[

σ

μ
ν

,

σ

ρ
τ

# ]
i
(

η

τ
μ

σ

ρ
ν

η

ν
τ

σ

μ
ρ

−

η

ρ
μ

σ

τ
ν

−

η

ν
ρ

σ

μ
τ

)
.

{\displaystyle [\sigma^{\mu\nu},\sigma^{\rho\tau}] = i(\eta^{\tau\mu}\sigma^{\rho\nu} + \eta^{\nu\tau}\sigma^{\mu\rho} - \eta^{\rho\mu}\sigma^{\tau\nu} -\eta^{\nu\rho}\sigma^{\mu\tau}).}

This is in the (3, 1) convention, hence fits in Cℓ
3,1(**R**)**C**.

The Dirac matrices were first written down by Paul Dirac when he was trying to write a relativistic first-order wave equation for the electron, and give an explicit isomorphism from the Clifford algebra to the algebra of complex matrices. The result was used to define the Dirac equation and introduce the Dirac operator. The entire Clifford algebra shows up in quantum field theory in the form of Dirac field bilinears.

The use of Clifford algebras to describe quantum theory has been advanced among others by Mario Schönberg, by David Hestenes in terms of geometric calculus, by David Bohm and Basil Hiley and co-workers in form of a hierarchy of Clifford algebras, and by Elio Conte et al.

### Computer vision

Clifford algebras have been applied in the problem of action recognition and classification in computer vision. Rodriguez et al. propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as optical flow. Vector-valued data is analyzed using the Clifford Fourier Transform. Based on these vectors action filters are synthesized in the Clifford Fourier domain and recognition of actions is performed using Clifford Correlation. The authors demonstrate the effectiveness of the Clifford embedding by recognizing actions typically performed in classic feature films and sports broadcast television.

## Generalizations

* While this article focuses on a Clifford algebra of a vector space over a field, the definition extends without change to a module over any unital, associative, commutative ring.
* Clifford algebras may be generalized to a form of degree higher than quadratic over a vector space.

## Notes

1. ↑  make it clear that the map *i* (*γ* in this quote) is included in the structure of a Clifford algebra by defining it as "The pair (*A*, *γ*) is a Clifford algebra for the quadratic space (*V*, *g*) when *A* is generated as an algebra by {*γ*(**v**) | **v** ∈ *V*}  and {*a*1*A* | *a* ∈ **R**}, and *γ* satisfies *γ*(**v**)*γ*(**u**) + *γ*(**u**)*γ*(**v**) = 2*g*(**v**, **u**) for all **v**, **u** ∈ *V*."