Clifford algebra

In, a Clifford algebra is an algebra generated by a with a , and is a. As, they generalize the s, s, s and several other systems. The theory of Clifford algebras is intimately connected with the theory of s and s. Clifford algebras have important applications in a variety of fields including, and. They are named after the English mathematician.

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  that contains and is generated by a  V over a  K, where V is equipped with a  Q : V → K. The Clifford algebra Cℓ(V, Q) is the  generated by V subject to the condition
 * $$v^2 = Q(v)1\ \text{ for all } v\in V,$$

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

Clifford algebras can be identified by the label Cℓp,q(R), indicating that the algebra is constructed from p simple basis elements with ei2 = +1, q with ei2 = −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 ⊕n≥0 V ⊗ ... ⊗ V, that is, the sum of the  of n copies of V over all n, and so a Clifford algebra would be the  of this tensor algebra by the two-sided  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 of the ground field K is not 2, then one can rewrite this fundamental identity in the form
 * $$uv + vu = 2\langle u, v\rangle1\ \text{ for all }u,v \in V,$$

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

is the associated with Q, via the.

Quadratic forms and Clifford algebras in 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 s. 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 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 since it makes use of the extra information provided by Q.

The Clifford algebra is a, the is the exterior algebra.

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

Weyl algebras and Clifford algebras admit a further structure of a, and can be unified as even and odd terms of a , as discussed in.

Universal property and construction
Let V be a over a  K, and let Q : V → K be a  on V. In most cases of interest the field K is either the field of s R, or the field of s C, or a.

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


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

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

In characteristic not 2, the quadratic form Q may be replaced by a symmetric $$\langle {\cdot}, {\cdot} \rangle$$, in which case an equivalent requirement on j is


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

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 T(V), and then enforce the fundamental identity by taking a suitable. In our case we want to take the IQ in T(V) generated by all elements of the form


 * $$v\otimes v - Q(v)1$$ for all $$v\in V$$

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


 * $$\operatorname{C\ell}(V, Q) = T(V) / I_Q .$$

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. One usually drops the i and considers V as a 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 from the  of vector spaces with quadratic forms (whose s 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 s 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. An is one such that


 * $$\langle e_i, e_j \rangle = 0 $$ for $$ i\neq j$$, and $$\langle e_i, e_i \rangle = Q(e_i)$$

where $$\langle \cdot, \cdot \rangle $$ is the symmetric bilinear form associated to Q. The fundamental Clifford identity implies that for an orthogonal basis


 * $$e_ie_j = -e_je_i $$ for $$ i\neq j $$, and $$ e_i^2= Q(e_i) \,$$.

This makes manipulation of orthogonal basis vectors quite simple. Given a product $$e_{i_1}e_{i_2}\cdots e_{i_k}$$ 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 needed to do so (i.e. the  of the ordering ).

If the of V over K is n and {e1, ..., en} is an  of (V, Q), then Cℓ(V, Q) is free over K with a basis


 * $$\{e_{i_1}e_{i_2}\cdots e_{i_k} \mid 1\le i_1 < i_2 < \cdots < i_k \le n\mbox{ and } 0\le k\le n\}$$.

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


 * $$\dim \operatorname{C\ell}(V,Q) = \sum_{k=0}^n\begin{pmatrix}n\\ k\end{pmatrix} = 2^n.$$

Examples: real and complex Clifford algebras
The most important Clifford algebras are those over and  vector spaces equipped with s.

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

Real numbers
Real Clifford algebras are also sometimes referred to as s.

Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:
 * $$Q(v) = v_1^2 + \cdots + v_p^2 - v_{p+1}^2 - \cdots - v_{p+q}^2 ,$$

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

A standard {e1, ..., en} for Rp,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$0,n$(R) will therefore have p vectors that square to +1 and q vectors that square to −1.

A few low-dimensional cases are:
 * Cℓ$p,q$(R) is naturally isomorphic to R since there are no nonzero vectors.
 * Cℓ$0,0$(R) is a two-dimensional algebra generated by e1 that squares to −1, and is algebra-isomorphic to C, the field of s.
 * Cℓ$0,1$(R) is a four-dimensional algebra spanned by {1, e1, e2, e1e2}. The latter three elements all square to −1 and anticommute, and so the algebra is isomorphic to the s H.
 * Cℓ$0,2$(R) is an 8-dimensional algebra isomorphic to the H ⊕ H, the s.

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
 * $$Q(z) = z_1^2 + z_2^2 + \cdots + z_n^2$$.

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 Cn with the standard quadratic form by Cℓ$0,3$(C).

For the first few cases one finds that
 * Cℓ$n$(C) ≅ C, the s
 * Cℓ$0$(C) ≅ C ⊕ C, the s
 * Cℓ$1$(C) ≅ M2(C), the s

where Mn(C) denotes the algebra of n × n matrices over C.

Quaternions
In this section, Hamilton's s are constructed as the even sub algebra of the Clifford algebra Cℓ$2$(R).

Let the vector space V be real three-dimensional space R3, and the quadratic form Q be the negative of the usual Euclidean metric. Then, for v, w in R3 we have the bilinear form (or scalar product)
 * $$v \cdot w = v_1 w_1 + v_2 w_2 + v_3 w_3.$$

Now introduce the Clifford product of vectors v and w given by
 * $$ v w + w v = -2 (v \cdot w) .$$

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

Denote a set of orthogonal unit vectors of R3 as e1, e2, and e3, then the Clifford product yields the relations
 * $$ e_2 e_3 = -e_3 e_2, \,\,\, e_3 e_1 = -e_1 e_3,\,\,\,  e_1 e_2 = -e_2 e_1,$$

and
 * $$ e_1 ^2 = e_2^2 = e_3^2 = -1. $$

The general element of the Clifford algebra Cℓ$0,3$(R) is given by
 * $$ A = a_0 + a_1 e_1 + a_2 e_2 + a_3 e_3 + a_4 e_2 e_3 + a_5 e_3 e_1 + a_6 e_1 e_2 + a_7 e_1 e_2 e_3.$$

The linear combination of the even degree elements of Cℓ$0,3$(R) defines the even subalgebra Cℓ$0,3$(R) with the general element
 * $$ q = q_0 + q_1 e_2 e_3 + q_2 e_3 e_1 + q_3 e_1 e_2. $$

The basis elements can be identified with the quaternion basis elements i, j, k as
 * $$ i= e_2 e_3, j=  e_3 e_1, k =  e_1 e_2,$$

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

To see this, compute
 * $$ i^2 = (e_2 e_3)^2 = e_2 e_3 e_2 e_3 = - e_2 e_2 e_3 e_3 = -1,$$

and
 * $$ ij = e_2 e_3 e_3 e_1 = -e_2 e_1 = e_1 e_2 = k.$$

Finally,
 * $$ ijk = e_2 e_3 e_3 e_1 e_1 e_2 = -1.$$

Dual quaternions
In this section, s 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 R4, and let the quadratic form Q be a degenerate form derived from the Euclidean metric on R3. For v, w in R4 introduce the degenerate bilinear form
 * $$d(v, w)= v_1 w_1 + v_2 w_2 + v_3 w_3 .$$

This degenerate scalar product projects distance measurements in R4 onto the R3 hyperplane.

The Clifford product of vectors v and w is given by
 * $$ v w + w v = -2 \,d(v, w).$$

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

Denote a set of mutually orthogonal unit vectors of R4 as e1, e2, e3 and e4, then the Clifford product yields the relations
 * $$ e_m e_n = -e_n e_m, \,\,\, m \ne n,$$

and
 * $$ e_1 ^2 = e_2^2 =e_3^2 = -1, \,\, e_4^2 =0.$$

The general element of the Clifford algebra Cℓ(R$[0] 0,3$, d) has 16 components. The linear combination of the even degree elements defines the even subalgebra Cℓ$4$(R4, d) with the general element
 * $$ H = h_0 + h_1 e_2 e_3 + h_2 e_3 e_1 + h_3 e_1 e_2 + h_4 e_4 e_1 + h_5 e_4 e_2 + h_6 e_4 e_3 + h_7 e_1 e_2 e_3 e_4.$$

The basis elements can be identified with the quaternion basis elements i, j, k and the dual unit ε as
 * $$ i = e_2 e_3, j = e_3 e_1, k = e_1 e_2, \,\, \varepsilon = e_1 e_2 e_3 e_4.$$

This provides the correspondence of Cℓ$[0]$(R) with algebra.

To see this, compute
 * $$ \varepsilon ^2 = (e_1 e_2 e_3 e_4)^2 = e_1 e_2 e_3 e_4 e_1 e_2 e_3 e_4 = -e_1 e_2 e_3 (e_4 e_4 ) e_1 e_2 e_3 = 0 ,$$

and
 * $$ \varepsilon i = (e_1 e_2 e_3 e_4) e_2 e_3 = e_1 e_2 e_3 e_4 e_2 e_3 = e_2 e_3 (e_1 e_2 e_3 e_4) = i\varepsilon.$$

The exchanges of e1 and e4 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 x2 = a, so it is the étale quadratic algebra K[X] / (X2 − a).

In particular, if a = 0 (that is, Q is the zero quadratic form) then Cℓ(V, Q) is the s 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 x2 = a, y2 = b and xy = −yx.

Thus Cℓ(V, Q) is the (generalized) (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).

Relation to the exterior algebra
Given a vector space V one can construct the ⋀(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 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 {e1, ..., en} for V and extend it to a basis for Cℓ(V, Q) as described. The map Cℓ(V, Q) → ⋀(V) is determined by
 * $$e_{i_1}e_{i_2}\cdots e_{i_k} \mapsto e_{i_1}\wedge e_{i_2}\wedge \cdots \wedge e_{i_k}.$$

Note that this only works if the basis {e1, ..., en} is orthogonal. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism.

If the of K is 0, one can also establish the isomorphism by antisymmetrizing. Define functions fk: V × ... × V → Cℓ(V, Q) by
 * $$f_k(v_1, \cdots, v_k) = \frac{1}{k!}\sum_{\sigma\in S_k}{\rm sgn}(\sigma)\, v_{\sigma(1)}\cdots v_{\sigma(k)}$$

where the sum is taken over the on k elements. Since fk is it induces a unique linear map ⋀k(V) → Cℓ(V, Q). The 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 on Cℓ(V, Q). Recall that the T(V) has a natural filtration: F0 ⊂ F1 ⊂ F2 ⊂ ..., where Fk contains sums of tensors with  ≤ k. Projecting this down to the Clifford algebra gives a filtration on Cℓ(V, Q). The
 * $$Gr_F \operatorname{C\ell}(V,Q) = \bigoplus_k F^k/F^{k-1}$$

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 Fk+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 Z2-s (also known as s). Indeed, the linear map on V defined by v ↦ −v preserves the quadratic form Q and so by the universal property of Clifford algebras extends to an algebra


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

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


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

where


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

Since α is an automorphism it follows that:


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

where the bracketed superscripts are read modulo 2. This gives Cℓ(V, Q) the structure of a Z2-. The subspace Cℓ$[0] 0,3,1$(V, Q) forms a of Cℓ(V, Q), called the even subalgebra. The subspace Cℓ$[0]$(V, Q) is called the odd part of Cℓ(V, Q) (it is not a subalgebra). This Z2-grading plays an important role in the analysis and application of Clifford algebras. The automorphism α is called the main  or grade involution. Elements that are pure in this Z2-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 Z2-grading: for instance if Q(v) ≠ 0, then v ∈ Cℓ$[1]$(V, Q), but v$1$ ∈ Cℓ$2$(V, Q), not in Cℓ$0$(V, Q). Happily, the gradings are related in the natural way: Z2 ≅ N/2N ≅ Z/2Z. Further, the Clifford algebra is Z-:


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

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

The even subalgebra Cℓ$2$(V, Q) of a Clifford algebra is itself isomorphic to a Clifford algebra. If V is the 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:


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

In the negative-definite case this gives an inclusion Cℓ$[0]$(R) ⊂ Cℓ$0,n−1$(R), which extends the sequence


 * R ⊂ C ⊂ H ⊂ H ⊕ H ⊂ ...

Likewise, in the complex case, one can show that the even subalgebra of Cℓ$0,n$(C) is isomorphic to Cℓ$n$(C).

Antiautomorphisms
In addition to the automorphism α, there are two s that play an important role in the analysis of Clifford algebras. Recall that the T(V) comes with an antiautomorphism that reverses the order in all products of vectors:
 * $$v_1\otimes v_2\otimes \cdots \otimes v_k \mapsto v_k\otimes \cdots \otimes v_2\otimes v_1.$$

Since the ideal IQ is invariant under this reversal, this operation descends to an antiautomorphism of Cℓ(V, Q) called the  or reversal operation, denoted by xt. The transpose is an antiautomorphism: (xy)t = yt xt. The transpose operation makes no use of the Z2-grading so we define a second antiautomorphism by composing α and the transpose. We call this operation Clifford conjugation denoted $$\bar x$$
 * $$\bar x = \alpha(x^\mathrm{t}) = \alpha(x)^\mathrm{t}.$$

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

Note that all of these operations are. 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
 * $$\alpha(x) = \pm x \qquad x^\mathrm{t} = \pm x \qquad \bar x = \pm x$$

where the signs are given by the following table:

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

where ⟨a⟩ denotes the scalar part of a (the degree 0 part in the Z-grading). One can show that
 * $$Q(v_1v_2\cdots v_k) = Q(v_1)Q(v_2)\cdots Q(v_k)$$

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

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 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 $$a^\mathrm{t}$$ of an element $$a$$ is the of left/right Clifford multiplication by $$a$$ itself with respect to this inner product. That is,
 * $$\langle ax, y\rangle = \langle x, a^\mathrm{t} y\rangle,$$

and
 * $$\langle xa, y\rangle = \langle x, y a^\mathrm{t}\rangle.$$

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 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$n−1$(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 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)/2dV, which is the space V with its quadratic form multiplied by (−1)dim(U)/2d. Over the reals, this implies in particular that
 * $$ \operatorname{C\ell}_{p+2,q}(\mathbf{R}) = \mathrm{M}_2(\mathbf{R})\otimes \operatorname{C\ell}_{q,p}(\mathbf{R}) $$
 * $$ \operatorname{C\ell}_{p+1,q+1}(\mathbf{R}) = \mathrm{M}_2(\mathbf{R})\otimes \operatorname{C\ell}_{p,q}(\mathbf{R}) $$
 * $$ \operatorname{C\ell}_{p,q+2}(\mathbf{R}) = \mathbf{H}\otimes \operatorname{C\ell}_{q,p}(\mathbf{R}). $$

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

Notably, the 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.

Clifford group
The class of Clifford groups (a.k.a. Clifford–Lipschitz groups) was discovered by.

In this section we assume that V is finite-dimensional and the quadratic form Q is.

An action on the elements of a Clifford algebra by its may be defined in terms of a twisted conjugation: twisted conjugation by x maps y ↦ x y α(x)−1, where α is the main involution defined.

The Clifford group Γ is defined to be the set of invertible elements x that stabilize the set of vectors under this action, meaning that for all v in V we have:
 * $$\alpha(x) v x^{-1}\in V .$$

This formula also defines an action of the Clifford group on the vector space V that preserves the quadratic form Q, and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group contains all elements r of V for which Q(r) is invertible in K, and these act on V by the corresponding reflections that take v to v − 2⟨v,r⟩r/Q(r). (In characteristic 2 these are called orthogonal transvections rather than reflections.)

If V is a finite-dimensional real vector space with a quadratic form then the Clifford group maps onto the orthogonal group of V with respect to the form (by the ) and the kernel consists of the nonzero elements of the field K. This leads to exact sequences
 * $$ 1 \rightarrow K^* \rightarrow \Gamma \rightarrow \mbox{O}_V(K) \rightarrow 1,\,$$
 * $$ 1 \rightarrow K^* \rightarrow \Gamma^0 \rightarrow \mbox{SO}_V(K) \rightarrow 1.\,$$

Over other fields or with indefinite forms, the map is not in general onto, and the failure is captured by the spinor norm.

Spinor norm
In arbitrary characteristic, the Q is defined on the Clifford group by
 * $$Q(x) = x^\mathrm{t}x.$$

It is a homomorphism from the Clifford group to the group K× of non-zero elements of K. It coincides with the quadratic form Q of V when V is identified with a subspace of the Clifford algebra. Several authors define the spinor norm slightly differently, so that it differs from the one here by a factor of −1, 2, or −2 on Γ1. The difference is not very important in characteristic other than 2.

The nonzero elements of K have spinor norm in the group (K×)2 of squares of nonzero elements of the field K. So when V is finite-dimensional and non-singular we get an induced map from the orthogonal group of V to the group K×/(K×)2, also called the spinor norm. The spinor norm of the reflection about r⊥, for any vector r, has image Q(r) in K×/(K×)2, and this property uniquely defines it on the orthogonal group. This gives exact sequences:


 * $$ 1 \to \{\pm 1\} \to \mbox{Pin}_V(K) \to \mbox{O}_V(K)  \to K^{\times}/(K^{\times})^2,$$
 * $$ 1 \to \{\pm 1\} \to \mbox{Spin}_V(K) \to \mbox{SO}_V(K) \to K^{\times}/(K^{\times})^2.$$

Note that in characteristic 2 the group {±1} has just one element.

From the point of view of of s, the spinor norm is a  on cohomology. Writing μ2 for the (over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action), the short exact sequence
 * $$ 1 \to \mu_2 \rightarrow \mbox{Pin}_V \rightarrow \mbox{O}_V \rightarrow 1$$

yields a long exact sequence on cohomology, which begins
 * $$ 1 \to H^0(\mu_2;K) \to H^0(\mbox{Pin}_V;K) \to H^0(\mbox{O}_V;K) \to H^1(\mu_2;K).$$

The 0th Galois cohomology group of an algebraic group with coefficients in K is just the group of K-valued points: H0(G; K) = G(K), and H1(μ2; K) ≅ K×/(K×)2, which recovers the previous sequence
 * $$ 1 \to \{\pm 1\} \to \mbox{Pin}_V(K) \to \mbox{O}_V(K) \to K^{\times}/(K^{\times})^2,$$

where the spinor norm is the connecting homomorphism H0(OV; K) → H1(μ2; K).

Spin and Pin groups
In this section we assume that V is finite-dimensional and its bilinear form is non-singular. (If K has characteristic 2 this implies that the dimension of V is even.)

The PinV(K) is the subgroup of the Clifford group Γ of elements of spinor norm 1, and similarly the  SpinV(K) is the subgroup of elements of  0 in PinV(K). When the characteristic is not 2, these are the elements of determinant 1. The Spin group usually has index 2 in the Pin group.

Recall from the previous section that there is a homomorphism from the Clifford group onto the orthogonal group. We define the to be the image of Γ0. If K does not have characteristic 2 this is just the group of elements of the orthogonal group of determinant 1. If K does have characteristic 2, then all elements of the orthogonal group have determinant 1, and the special orthogonal group is the set of elements of Dickson invariant 0.

There is a homomorphism from the Pin group to the orthogonal group. The image consists of the elements of spinor norm 1 ∈ K×/(K×)2. The kernel consists of the elements +1 and −1, and has order 2 unless K has characteristic 2. Similarly there is a homomorphism from the Spin group to the special orthogonal group of V.

In the common case when V is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected when V has dimension at least 3. Further the kernel of this homomorphism consists of 1 and −1. So in this case the spin group, Spin(n), is a double cover of SO(n). Please note, however, that the simple connectedness of the spin group is not true in general: if V is Rp,q for p and q both at least 2 then the spin group is not simply connected. In this case the algebraic group Spinp,q is simply connected as an algebraic group, even though its group of real valued points Spinp,q(R) is not simply connected. This is a rather subtle point, which completely confused the authors of at least one standard book about spin groups.

Spinors
Clifford algebras Cℓ$[0]$(C), with p + q = 2n even, are matrix algebras which have a complex representation of dimension 2n. By restricting to the group Pinp,q(R) we get a complex representation of the Pin group of the same dimension, called the. If we restrict this to the spin group Spinp,q(R) then it splits as the sum of two half spin representations (or Weyl representations) of dimension 2n−1.

If p + q = 2n + 1 is odd then the Clifford algebra Cℓ$p,q$(C) is a sum of two matrix algebras, each of which has a representation of dimension 2n, and these are also both representations of the Pin group Pinp,q(R). On restriction to the spin group Spinp,q(R) these become isomorphic, so the spin group has a complex spinor representation of dimension 2n.

More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the : whenever a Clifford algebra has a factor that is a matrix algebra over some division algebra, we get a corresponding representation of the pin and spin groups over that division algebra. For examples over the reals see the article on s.

Real spinors
To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The, Pinp,q is the set of invertible elements in Cℓ$p,q$ that can be written as a product of unit vectors:
 * $${\mbox{Pin}}_{p,q}=\{v_1v_2\dots v_r |\,\, \forall i\, \|v_i\|=\pm 1\}.$$

Comparing with the above concrete realizations of the Clifford algebras, the Pin group corresponds to the products of arbitrarily many reflections: it is a cover of the full orthogonal group O(p, q). The consists of those elements of Pinp, q which are products of an even number of unit vectors. Thus by the Spin is a cover of the group of proper rotations SO(p, q).

Let α : Cℓ → Cℓ be the automorphism which is given by the mapping v ↦ −v acting on pure vectors. Then in particular, Spinp,q is the subgroup of Pinp,q whose elements are fixed by α. Let
 * $$\operatorname{C\ell}_{p,q}^{[0]} = \{ x\in \operatorname{C\ell}_{p,q} |\, \alpha(x)=x\}.$$

(These are precisely the elements of even degree in Cℓ$p,q$.) Then the spin group lies within Cℓ$p,q$.

The irreducible representations of Cℓ$[0] p,q$ restrict to give representations of the pin group. Conversely, since the pin group is generated by unit vectors, all of its irreducible representation are induced in this manner. Thus the two representations coincide. For the same reasons, the irreducible representations of the spin coincide with the irreducible representations of Cℓ$p,q$.

To classify the pin representations, one need only appeal to the. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above)
 * $$\operatorname{C\ell}^{[0]}_{p,q} \approx \operatorname{C\ell}_{p,q-1}, \text{ for } q > 0$$
 * $$\operatorname{C\ell}^{[0]}_{p,q} \approx \operatorname{C\ell}_{q,p-1}, \text{ for } p > 0$$

and realize a spin representation in signature (p, q) as a pin representation in either signature (p, q − 1) or (q, p − 1).

Differential geometry
One of the principal applications of the exterior algebra is in where it is used to define the  of s on a. In the case of a (-), the s come equipped with a natural quadratic form induced by the. Thus, one can define a in analogy with the. This has a number of important applications in. Perhaps more importantly is the link to a, its associated 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 γ0, ..., γ3 called which have the property that


 * $$\gamma_i\gamma_j + \gamma_j\gamma_i = 2\eta_{ij}\,$$

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$[0] p,q$(R), whose is Cℓ$1,3$(R)C which, by the, is isomorphic to the algebra of 4 × 4 complex matrices Cℓ$1,3$(C) ≈ M4(C). However, it is best to retain the notation Cℓ$4$(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


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


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

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

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

The use of Clifford algebras to describe quantum theory has been advanced among others by, by in terms of , by  and  and co-workers in form of a , and by Elio Conte et al.

Computer vision
Clifford algebras have been applied in the problem of action recognition and classification in. Rodriguez et al. propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such as. Vector-valued data is analyzed using the. 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 over any unital, associative, commutative ring.
 * Clifford algebras may be generalized to a form of degree higher than quadratic over a vector space.