Inmathematics, aClifford algebra^[a] is analgebra generated by avector space with aquadratic form, and is aunitalassociative algebra with the additional structure of a distinguished subspace. AsK-algebras, they generalize thereal numbers,complex numbers,quaternions and several otherhypercomplex number systems.^[1][2] The theory of Clifford algebras is intimately connected with the theory ofquadratic forms andorthogonal transformations. Clifford algebras have important applications in a variety of fields includinggeometry,theoretical physics anddigital image processing. They are named after the English mathematicianWilliam Kingdon Clifford (1845–1879).

The most familiar Clifford algebras, theorthogonal Clifford algebras, are also referred to as (pseudo-)Riemannian Clifford algebras, as distinct fromsymplectic Clifford algebras.^[b]

A Clifford algebra is aunitalassociative algebra that contains and is generated by avector spaceV over afieldK, whereV is equipped with aquadratic formQ :V →K. The Clifford algebraCl(V,Q) is the "freest" unital associative algebra generated byV subject to the condition^[c]$${\displaystyle v^2=Q(v)1\mathrm{\forall }v\in V,}$$where the product on the left is that of the algebra, and the1 on the right is the algebra'smultiplicative identity (not to be confused with the multiplicative identity ofK). The idea of being the "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of auniversal property, as donebelow.

WhenV is a finite-dimensional real vector space andQ isnondegenerate,Cl(V,Q) may be identified by the labelCl_p,q(R), indicating thatV has an orthogonal basis withp elements withe_i² = +1,q withe_i² = −1, and whereR indicates that this is a Clifford algebra over the reals; i.e. coefficients of elements of the algebra are real numbers. Such a basis may be found byorthogonal diagonalization.

Thefree algebra generated byV may be written as thetensor algebra⨁_n≥0V ⊗ ⋯ ⊗V, that is, thedirect sum of thetensor product ofn copies ofV over alln. Therefore one obtains a Clifford algebra as thequotient of this tensor algebra by the two-sidedideal generated by elements of the formv ⊗v −Q(v)1 for all elementsv ∈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 distinguishedsubspace V, being theimage of theembedding map. Such a subspace cannot in general be uniquely determined given only aK-algebra that isisomorphic to the Clifford algebra.

If2 isinvertible in the ground fieldK, then one can rewrite the fundamental identity above in the form$${\displaystyle uv+vu=2⟨u,v⟩1\text{ for all }u,v\in V,}$$where$${\displaystyle ⟨u,v⟩=\frac{1}{2}\left(Q(u+v)-Q(u)-Q(v)\right)}$$is thesymmetric bilinear form associated withQ, via thepolarization identity.

Quadratic forms and Clifford algebras incharacteristic2 form an exceptional case in this respect. In particular, ifchar(K) = 2 it is not true that a quadratic form necessarily or uniquely determines a symmetric bilinear form that satisfiesQ(v) =⟨v,v⟩,^[3] Many of the statements in this article include the condition that the characteristic is not2, and are false if this condition is removed.

Clifford algebras are closely related toexterior algebras. Indeed, ifQ = 0 then the Clifford algebraCl(V,Q) is just the exterior algebra⋀V. Whenever2 is invertible in the ground field K, there exists a canonicallinear isomorphism between⋀V andCl(V,Q). That is, they arenaturally 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 theexterior product since it makes use of the extra information provided by Q.

The Clifford algebra is afiltered algebra; theassociated graded algebra is the exterior algebra.

More precisely, Clifford algebras may be thought of asquantizations (cf.quantum group) of the exterior algebra, in the same way that theWeyl algebra is a quantization of thesymmetric algebra.

Weyl algebras and Clifford algebras admit a further structure of a*-algebra, and can be unified as even and odd terms of asuperalgebra, as discussed inCCR and CAR algebras.

LetV be avector space over afield K, and letQ :V →K be aquadratic form onV. In most cases of interest the fieldK is either the field ofreal numbers R, or the field ofcomplex numbers C, or afinite field.

A Clifford algebraCl(V,Q) is a pair(B,i),^[d][4] whereB is aunitalassociative algebra overK andi is alinear mapi :V →B that satisfiesi(v)² =Q(v)1_B for allv inV, defined by the followinguniversal property: given any unital associative algebraA overK and any linear mapj :V →A such that$${\displaystyle j(v{)}^2=Q(v)1_A\text{ for all }v\in V}$$(where1_A denotes the multiplicative identity ofA), there is a uniquealgebra homomorphismf :B →A such that the following diagramcommutes (i.e. such thatf ∘i =j):

The quadratic formQ may be replaced by a (not necessarily symmetric^[5])bilinear form⟨⋅,⋅⟩ that has the property⟨v,v⟩ =Q(v),v ∈V, in which case an equivalent requirement onj is$${\displaystyle j(v)j(v)=⟨v,v⟩1_A\text{ for all }v\in V.}$$

When the characteristic of the field is not2, this may be replaced by what is then an equivalent requirement,$${\displaystyle j(v)j(w)+j(w)j(v)=(⟨v,w⟩+⟨w,v⟩)1_A\text{ for all }v,w\in V,}$$where the bilinear form may additionally be restricted to being symmetric without loss of generality.

A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that containsV, namely thetensor algebraT(V), and then enforce the fundamental identity by taking a suitablequotient. In our case we want to take the two-sidedidealI_Q inT(V) generated by all elements of the form$${\displaystyle v\otimes v-Q(v)1}$$ for all${\displaystyle v\in V}$and defineCl(V,Q) as the quotient algebra$${\displaystyle \mathrm{Cl}(V,Q)=T(V)/I_Q.}$$

Thering product inherited by this quotient is sometimes referred to as theClifford product^[6] to distinguish it from the exterior product and the scalar product.

It is then straightforward to show thatCl(V,Q) containsV and satisfies the above universal property, so thatCl is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebraCl(V,Q). It also follows from this construction thati isinjective. One usually drops the i and considersV as alinear subspace ofCl(V,Q).

The universal characterization of the Clifford algebra shows that the construction ofCl(V,Q) isfunctorial in nature. Namely,Cl can be considered as afunctor from thecategory of vector spaces with quadratic forms (whosemorphisms are linear maps that preserve the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (that preserve the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras.

SinceV comes equipped with a quadratic form Q, in characteristic not equal to2 there existbases forV that areorthogonal. Anorthogonal basis is one such that for a symmetric bilinear form$${\displaystyle ⟨e_i,e_j⟩=0}$$ for${\displaystyle i\ne j}$, and$${\displaystyle ⟨e_i,e_i⟩=Q(e_i).}$$

The fundamental Clifford identity implies that for an orthogonal basis$${\displaystyle e_i{e}_j=-e_j{e}_i}$$ for${\displaystyle i\ne j}$, and$${\displaystyle e_i^2=Q(e_i).}$$

This makes manipulation of orthogonal basis vectors quite simple. Given a product${\displaystyle e_{i_1}{e}_{i_2}\cdots e_{i_k}}$ ofdistinct orthogonal basis vectors ofV, one can put them into a standard order while including an overall sign determined by the number ofpairwise swaps needed to do so (i.e. thesignature of the orderingpermutation).

If thedimension ofV overK isn and{e₁, ...,e_n} is an orthogonal basis of(V,Q), thenCl(V,Q) isfree overK with a basis

The empty product (k = 0) is defined as being the multiplicativeidentity element. For each value ofk there aren choosek basis elements, so the total dimension of the Clifford algebra is$${\displaystyle \dim \mathrm{Cl}(V,Q)=\sum _{k=0}^n(\genfrac{}{}{0ex}{}{n}{k})=2^n.}$$

The most important Clifford algebras are those overreal andcomplex vector spaces equipped withnondegenerate quadratic forms.

Each of the algebrasCl_p,q(R) andCl_n(C) is isomorphic toA orA ⊕A, whereA is afull matrix ring with entries fromR,C, or H. For a complete classification of these algebras seeClassification of Clifford algebras.

Clifford algebras are also sometimes referred to asgeometric algebras, most often over the real numbers.

Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form:$${\displaystyle Q(v)=v_1^2+\cdots +v_p^2-v_{p+1}^2-\cdots -v_{p+q}^2,}$$wheren =p +q is the dimension of the vector space. The pair of integers(p,q) is called thesignature of the quadratic form. The real vector space with this quadratic form is often denotedR^p,q. The Clifford algebra onR^p,q is denotedCl_p,q(R). The symbolCl_n(R) means eitherCl_n,0(R) orCl_0,n(R), depending on whether the author prefers positive-definite or negative-definite spaces.

A standardbasis{e₁, ...,e_n} forR^p,q consists ofn =p +q mutually orthogonal vectors,p of which square to+1 andq of which square to −1. Of such a basis, the algebraCl_p,q(R) will therefore havep vectors that square to+1 andq vectors that square to −1.

A few low-dimensional cases are:

• Cl_0,0(R) is naturally isomorphic toR since there are no nonzero vectors.
• Cl_0,1(R) is a two-dimensional algebra generated bye₁ that squares to−1, and is algebra-isomorphic toC, the field ofcomplex numbers.
• Cl_1,0(R) is a two-dimensional algebra generated bye₁ that squares to1, and is algebra-isomorphic to thesplit-complex numbers.
• Cl_0,2(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 thequaternions H.
• Cl_2,0(R) ≅ Cl_1,1(R) is isomorphic to the algebra ofsplit-quaternions.
• Cl_0,3(R) is an 8-dimensional algebra isomorphic to thedirect sumH ⊕H, thesplit-biquaternions.
• Cl_3,0(R) ≅ Cl_1,2(R), also called thePauli algebra,^[7][8] is isomorphic to the algebra ofbiquaternions.

One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space of dimensionn is equivalent to the standard diagonal form$${\displaystyle Q(z)=z_1^2+z_2^2+\cdots +z_n^2.}$$Thus, for each dimensionn, 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 onCⁿ with the standard quadratic form byCl_n(C).

For the first few cases one finds that

• Cl₀(C) ≅C, thecomplex numbers
• Cl₁(C) ≅C ⊕C, thebicomplex numbers
• Cl₂(C) ≅ M₂(C), thebiquaternions

whereM_n(C) denotes the algebra ofn ×n matrices overC.

In this section, Hamilton'squaternions are constructed as the even subalgebra of the Clifford algebraCl_3,0(R).

Let the vector spaceV be real three-dimensional space R³, and the quadratic form be the usual quadratic form. Then, forv,w inR³ we have the bilinear form (or scalar product)$${\displaystyle v\cdot w=v_1{w}_1+v_2{w}_2+v_3{w}_3.}$$Now introduce the Clifford product of vectorsv andw given by$${\displaystyle vw+wv=2(v\cdot w).}$$

Denote a set of orthogonal unit vectors ofR³ as{e₁,e₂,e₃}, then the Clifford product yields the relations$${\displaystyle e_2{e}_3=-e_3{e}_2,e_1{e}_3=-e_3{e}_1,e_1{e}_2=-e_2{e}_1,}$$and$${\displaystyle e_1^2=e_2^2=e_3^2=1.}$$The general element of the Clifford algebraCl_3,0(R) is given by$${\displaystyle A=a_0+a_1{e}_1+a_2{e}_2+a_3{e}_3+a_4{e}_2{e}_3+a_5{e}_1{e}_3+a_6{e}_1{e}_2+a_7{e}_1{e}_2{e}_3.}$$

The linear combination of the even degree elements ofCl_3,0(R) defines the even subalgebraCl^[0]
_3,0(R) with the general element$${\displaystyle q=q_0+q_1{e}_2{e}_3+q_2{e}_1{e}_3+q_3{e}_1{e}_2.}$$The basis elements can be identified with the quaternion basis elementsi,j,k as$${\displaystyle i=e_2{e}_3,j=e_1{e}_3,k=e_1{e}_2,}$$which shows that the even subalgebraCl^[0]
_3,0(R) is Hamilton's realquaternion algebra.

To see this, compute$${\displaystyle i^2=(e_2{e}_3{)}^2=e_2{e}_3{e}_2{e}_3=-e_2{e}_2{e}_3{e}_3=-1,}$$and$${\displaystyle ij=e_2{e}_3{e}_1{e}_3=-e_2{e}_3{e}_3{e}_1=-e_2{e}_1=e_1{e}_2=k.}$$Finally,$${\displaystyle ijk=e_2{e}_3{e}_1{e}_3{e}_1{e}_2=-1.}$$

In this section,dual quaternions are constructed as the even subalgebra of a Clifford algebra of real four-dimensional space with a degenerate quadratic form.^[9][10]

Let the vector spaceV be real four-dimensional spaceR⁴, and let the quadratic formQ be a degenerate form derived from the Euclidean metric onR³. Forv,w inR⁴ introduce the degenerate bilinear form$${\displaystyle d(v,w)=v_1{w}_1+v_2{w}_2+v_3{w}_3.}$$This degenerate scalar product projects distance measurements inR⁴ onto theR³ hyperplane.

The Clifford product of vectorsv andw is given by$${\displaystyle vw+wv=-2d(v,w).}$$Note the negative sign is introduced to simplify the correspondence with quaternions.

Denote a set of mutually orthogonal unit vectors ofR⁴ as{e₁,e₂,e₃,e₄}, then the Clifford product yields the relations$${\displaystyle e_m{e}_n=-e_n{e}_m,m\ne n,}$$and$${\displaystyle e_1^2=e_2^2=e_3^2=-1,e_4^2=0.}$$

The general element of the Clifford algebraCl(R⁴,d) has 16 components. The linear combination of the even degree elements defines the even subalgebraCl^[0](R⁴,d) with the general element$${\displaystyle H=h_0+h_1{e}_2{e}_3+h_2{e}_3{e}_1+h_3{e}_1{e}_2+h_4{e}_4{e}_1+h_5{e}_4{e}_2+h_6{e}_4{e}_3+h_7{e}_1{e}_2{e}_3{e}_4.}$$

The basis elements can be identified with the quaternion basis elementsi,j,k and the dual unitε as$${\displaystyle i=e_2{e}_3,j=e_3{e}_1,k=e_1{e}_2,\epsilon =e_1{e}_2{e}_3{e}_4.}$$This provides the correspondence ofCl^[0]
_0,3,1(R) withdual quaternion algebra.

To see this, compute$${\displaystyle {\epsilon }^2=(e_1{e}_2{e}_3{e}_4{)}^2=e_1{e}_2{e}_3{e}_4{e}_1{e}_2{e}_3{e}_4=-e_1{e}_2{e}_3(e_4{e}_4)e_1{e}_2{e}_3=0,}$$and$${\displaystyle \epsilon i=(e_1{e}_2{e}_3{e}_4)e_2{e}_3=e_1{e}_2{e}_3{e}_4{e}_2{e}_3=e_2{e}_3(e_1{e}_2{e}_3{e}_4)=i\epsilon .}$$The exchanges ofe₁ ande₄ alternate signs an even number of times, and show the dual unitε commutes with the quaternion basis elementsi,j,k.

LetK be any field of characteristic not2.

FordimV = 1, ifQ has diagonalizationdiag(a), that is there is a non-zero vectorx such thatQ(x) =a, thenCl(V,Q) is algebra-isomorphic to aK-algebra generated by an elementx that satisfiesx² =a, the quadratic algebraK[X] / (X² −a).

In particular, ifa = 0 (that is,Q is the zero quadratic form) thenCl(V,Q) is algebra-isomorphic to thedual numbers algebra overK.

Ifa is a non-zero square inK, thenCl(V,Q) ≃K ⊕K.

Otherwise,Cl(V,Q) is isomorphic to the quadratic field extensionK(√a) ofK.

FordimV = 2, ifQ has diagonalizationdiag(a,b) with non-zeroa andb (which always exists ifQ is non-degenerate), thenCl(V,Q) is isomorphic to aK-algebra generated by elementsx andy that satisfiesx² =a,y² =b andxy = −yx.

ThusCl(V,Q) is isomorphic to the (generalized)quaternion algebra(a,b)_K. We retrieve Hamilton's quaternions whena =b = −1, sinceH = (−1, −1)_R.

As a special case, if somex inV satisfiesQ(x) = 1, thenCl(V,Q) ≃ M₂(K).

Given a vector space V, one can construct theexterior algebra⋀V, whose definition is independent of any quadratic form onV. It turns out that ifK does not have characteristic2 then there is anatural isomorphism between⋀V andCl(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 ifQ = 0. One can thus consider the Clifford algebraCl(V,Q) as an enrichment (or more precisely, a quantization, cf. the Introduction) of the exterior algebra onV 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 anorthogonal basis{e₁, ...,e_n} forV and extend it to a basis forCl(V,Q) as describedabove. The mapCl(V,Q) → ⋀V is determined by$${\displaystyle e_{i_1}{e}_{i_2}\cdots e_{i_k}\mapsto e_{i_1}\wedge e_{i_2}\wedge \cdots \wedge e_{i_k}.}$$Note that this works only if the basis{e₁, ...,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 thecharacteristic ofK is0, one can also establish the isomorphism by antisymmetrizing. Define functionsf_k :V × ⋯ ×V → Cl(V,Q) by$${\displaystyle f_k(v_1,\dots ,v_k)=\frac{1}{k!}\sum _{\sigma \in {\mathrm{S}}_k}\mathrm{sgn}(\sigma )v_{\sigma (1)}\cdots v_{\sigma (k)}}$$where the sum is taken over thesymmetric group onk elements,S_k. Sincef_k isalternating, it induces a unique linear map⋀^kV → Cl(V,Q). Thedirect sum of these maps gives a linear map between⋀V andCl(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 afiltration onCl(V,Q). Recall that thetensor algebraT(V) has a natural filtration:F⁰ ⊂F¹ ⊂F² ⊂ ⋯, whereF^k contains sums of tensors withorder≤k. Projecting this down to the Clifford algebra gives a filtration onCl(V,Q). Theassociated graded algebra$${\displaystyle {\mathrm{Gr}}_F\mathrm{Cl}(V,Q)=\underset{k}{⨁}{F}^k/F^{k-1}}$$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 ofF^k inF^k+1 for all k), this provides an isomorphism (although not a natural one) in any characteristic, even two.

In the following, assume that the characteristic is not 2.^[e]

Clifford algebras areZ₂-graded algebras (also known assuperalgebras). Indeed, the linear map onV defined byv ↦ −v (reflection through the origin) preserves the quadratic formQ and so by the universal property of Clifford algebras extends to an algebraautomorphism$${\displaystyle \alpha :\mathrm{Cl}(V,Q)\to \mathrm{Cl}(V,Q).}$$

Sinceα is aninvolution (i.e. it squares to theidentity) one can decomposeCl(V,Q) into positive and negative eigenspaces of α$${\displaystyle \mathrm{Cl}(V,Q)={\mathrm{Cl}}^{[0]}(V,Q)\oplus {\mathrm{Cl}}^{[1]}(V,Q)}$$where$${\displaystyle {\mathrm{Cl}}^{[i]}(V,Q)=\left\{x\in \mathrm{Cl}(V,Q)\mid \alpha (x)=(-1{)}^{i}x\right\}.}$$

Sinceα is an automorphism it follows that:$${\displaystyle {\mathrm{Cl}}^{[i]}(V,Q){\mathrm{Cl}}^{[j]}(V,Q)={\mathrm{Cl}}^{[i+j]}(V,Q)}$$where the bracketed superscripts are read modulo 2. This givesCl(V,Q) the structure of aZ₂-graded algebra. The subspaceCl^[0](V,Q) forms asubalgebra ofCl(V,Q), called theeven subalgebra. The subspaceCl^[1](V,Q) is called theodd part ofCl(V,Q) (it is not a subalgebra).ThisZ₂-grading plays an important role in the analysis and application of Clifford algebras. The automorphismα is called themaininvolution orgrade involution. Elements that are pure in thisZ₂-grading are simply said to be even or odd.

Remark. The Clifford algebra is not aZ-graded algebra, but isZ-filtered, whereCl^≤i(V,Q) is the subspace spanned by all products of at mosti elements of V.$${\displaystyle {\mathrm{Cl}}^{⩽i}(V,Q)\cdot {\mathrm{Cl}}^{⩽j}(V,Q)\subset {\mathrm{Cl}}^{⩽i+j}(V,Q).}$$

Thedegree of a Clifford number usually refers to the degree in theZ-grading.

The even subalgebraCl^[0](V,Q) of a Clifford algebra is itself isomorphic to a Clifford algebra.^[f][g] IfV is theorthogonal direct sum of a vectora of nonzero normQ(a) and a subspaceU, thenCl^[0](V,Q) is isomorphic toCl(U, −Q(a)Q|_U), whereQ|_U is the formQ restricted toU. In particular over the reals this implies that:

In the negative-definite case this gives an inclusionCl_{0,n − 1}(R) ⊂ Cl_0,n(R), which extends the sequence

Likewise, in the complex case, one can show that the even subalgebra ofCl_n(C) is isomorphic toCl_n−1(C).

In addition to the automorphismα, there are twoantiautomorphisms that play an important role in the analysis of Clifford algebras. Recall that thetensor algebraT(V) comes with an antiautomorphism that reverses the order in all products of vectors:$${\displaystyle v_1\otimes v_2\otimes \cdots \otimes v_k\mapsto v_k\otimes \cdots \otimes v_2\otimes v_1.}$$Since the idealI_Q is invariant under this reversal, this operation descends to an antiautomorphism ofCl(V,Q) called thetranspose orreversal operation, denoted byx^t. The transpose is an antiautomorphism:(xy)^t =y^tx^t. The transpose operation makes no use of theZ₂-grading so we define a second antiautomorphism by composingα and the transpose. We call this operationClifford conjugation denoted${\displaystyle \overline{x}}$$${\displaystyle \overline{x}=\alpha (x^{\mathrm{t}})=\alpha (x{)}^{\mathrm{t}}.}$$Of the two antiautomorphisms, the transpose is the more fundamental.^[h]

Note that all of these operations areinvolutions. One can show that they act as±1 on elements that are pure in theZ-grading. In fact, all three operations depend on only the degree modulo 4. That is, ifx is pure with degreek then$${\displaystyle \alpha (x)=\pm x{x}^{\mathrm{t}}=\pm x\overline{x}=\pm x}$$where the signs are given by the following table:

k mod 4	0	1	2	3	…
${\displaystyle \alpha (x)}$	+	−	+	−	(−1)k
${\displaystyle x^{\mathrm{t}}}$	+	+	−	−	(−1)k(k−1)/2
${\displaystyle \overline{x}}$	+	−	−	+	(−1)k(k+1)/2

When the characteristic is not2, the quadratic formQ onV can be extended to a quadratic form on all ofCl(V,Q) (which we also denoted byQ). A basis-independent definition of one such extension is$${\displaystyle Q(x)={⟨x^{\mathrm{t}}x⟩}_0}$$where⟨a⟩₀ denotes the scalar part ofa (the degree-0 part in theZ-grading). One can show that$${\displaystyle Q(v_1{v}_2\cdots v_k)=Q(v_1)Q(v_2)\cdots Q(v_k)}$$where thev_i are elements ofV – this identity isnot true for arbitrary elements ofCl(V,Q).

The associated symmetric bilinear form onCl(V,Q) is given by$${\displaystyle ⟨x,y⟩={⟨x^{\mathrm{t}}y⟩}_0.}$$One can check that this reduces to the original bilinear form when restricted toV. The bilinear form on all ofCl(V,Q) isnondegenerate if and only if it is nondegenerate onV.

The operator of left (respectively right) Clifford multiplication by the transposea^t of an elementa is theadjoint of left (respectively right) Clifford multiplication bya with respect to this inner product. That is,$${\displaystyle ⟨ax,y⟩=⟨x,a^{\mathrm{t}}y⟩,}$$and$${\displaystyle ⟨xa,y⟩=⟨x,y{a}^{\mathrm{t}}⟩.}$$

In this section we assume that characteristic is not2, the vector spaceV is finite-dimensional and that the associated symmetric bilinear form ofQ is nondegenerate.

Acentral simple algebra overK is a matrix algebra over a (finite-dimensional) division algebra with centerK. For example, the central simple algebras over the reals are matrix algebras over either the reals or the quaternions.

• IfV has even dimension thenCl(V,Q) is a central simple algebra over K.
• IfV has even dimension then the even subalgebraCl^[0](V,Q) is a central simple algebra over a quadratic extension ofK or a sum of two isomorphic central simple algebras over K.
• IfV has odd dimension thenCl(V,Q) is a central simple algebra over a quadratic extension ofK or a sum of two isomorphic central simple algebras over K.
• IfV has odd dimension then the even subalgebraCl^[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 thatU has even dimension and a non-singular bilinear form withdiscriminantd, and suppose thatV is another vector space with a quadratic form. The Clifford algebra ofU +V is isomorphic to the tensor product of the Clifford algebras ofU and(−1)^dim(U)/2dV, which is the spaceV with its quadratic form multiplied by(−1)^dim(U)/2d. Over the reals, this implies in particular that$${\displaystyle {\mathrm{Cl}}_{p+2,q}(\mathbf{R})={\mathrm{M}}_2(\mathbf{R})\otimes {\mathrm{Cl}}_{q,p}(\mathbf{R})}$$$${\displaystyle {\mathrm{Cl}}_{p+1,q+1}(\mathbf{R})={\mathrm{M}}_2(\mathbf{R})\otimes {\mathrm{Cl}}_{p,q}(\mathbf{R})}$$$${\displaystyle {\mathrm{Cl}}_{p,q+2}(\mathbf{R})=\mathbf{H}\otimes {\mathrm{Cl}}_{q,p}(\mathbf{R}).}$$These formulas can be used to find the structure of all real Clifford algebras and all complex Clifford algebras; see theclassification of Clifford algebras.

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

The class of Lipschitz groups (a.k.a.[11]Clifford groups or Clifford–Lipschitz groups) was discovered byRudolf Lipschitz.^[12]

In this section we assume thatV is finite-dimensional and the quadratic formQ isnondegenerate.

An action on the elements of a Clifford algebra by itsgroup of units may be defined in terms of a twisted conjugation: twisted conjugation byx mapsy ↦α(x)yx⁻¹, whereα is themain involution definedabove.

The Lipschitz groupΓ is defined to be the set of invertible elementsx thatstabilize the set of vectors under this action,^[13] meaning that for allv inV we have:$${\displaystyle \alpha (x)v{x}^{-1}\in V.}$$

This formula also defines an action of the Lipschitz group on the vector spaceV that preserves the quadratic formQ, and so gives a homomorphism from the Lipschitz group to the orthogonal group. The Lipschitz group contains all elementsr ofV for whichQ(r) is invertible inK, and these act onV by the corresponding reflections that takev tov − (⟨r,v⟩ +⟨v,r⟩)r‍/‍Q(r). (In characteristic2 these are called orthogonal transvections rather than reflections.)

IfV is a finite-dimensional real vector space with anon-degenerate quadratic form then the Lipschitz group maps onto the orthogonal group ofV with respect to the form (by theCartan–Dieudonné theorem) and the kernel consists of the nonzero elements of the field K. This leads to exact sequences$${\displaystyle 1\to K^{\times }\to \mathrm{\Gamma }\to {\mathrm{O}}_V(K)\to 1,}$$$${\displaystyle 1\to K^{\times }\to {\mathrm{\Gamma }}^0\to {\mathrm{SO}}_V(K)\to 1.}$$

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

In arbitrary characteristic, thespinor normQ is defined on the Lipschitz group by$${\displaystyle Q(x)=x^{\mathrm{t}}x.}$$It is a homomorphism from the Lipschitz group to the groupK^× of non-zero elements ofK. It coincides with the quadratic formQ ofV whenV 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 ofK have spinor norm in the group (K^×)² of squares of nonzero elements of the fieldK. So whenV is finite-dimensional and non-singular we get an induced map from the orthogonal group ofV to the groupK^×‍/‍(K^×)², also called the spinor norm. The spinor norm of the reflection aboutr^⊥, for any vectorr, has imageQ(r) inK^×‍/‍(K^×)², and this property uniquely defines it on the orthogonal group. This gives exact sequences:

Note that in characteristic2 the group{±1} has just one element.

From the point of view ofGalois cohomology ofalgebraic groups, the spinor norm is aconnecting homomorphism on cohomology. Writingμ₂ for thealgebraic group of square roots of 1 (over a field of characteristic not2 it is roughly the same as a two-element group with trivial Galois action), the short exact sequence$${\displaystyle 1\to {\mu }_2\to {\mathrm{Pin}}_V\to {\mathrm{O}}_V\to 1}$$yields a long exact sequence on cohomology, which begins$${\displaystyle 1\to H^0({\mu }_2;K)\to H^0({\mathrm{Pin}}_V;K)\to H^0({\mathrm{O}}_V;K)\to H^1({\mu }_2;K).}$$

The 0th Galois cohomology group of an algebraic group with coefficients inK is just the group ofK-valued points:H⁰(G;K) =G(K), andH¹(μ₂;K) ≅K^×‍/‍(K^×)², which recovers the previous sequence$${\displaystyle 1\to \{\pm 1\}\to {\mathrm{Pin}}_V(K)\to {\mathrm{O}}_V(K)\to K^{\times }/{\left(K^{\times }\right)}^2,}$$where the spinor norm is the connecting homomorphismH⁰(O_V;K) →H¹(μ₂;K).

In this section we assume thatV is finite-dimensional and its bilinear form is non-singular.

Thepin groupPin_V(K) is the subgroup of the Lipschitz groupΓ of elements of spinor norm1, and similarly thespin groupSpin_V(K) is the subgroup of elements ofDickson invariant0 inPin_V(K). When the characteristic is not2, these are the elements of determinant1. The spin group usually has index2 in the pin group.

Recall from the previous section that there is a homomorphism from the Lipschitz group onto the orthogonal group. We define thespecial orthogonal group to be the image ofΓ⁰. IfK does not have characteristic2 this is just the group of elements of the orthogonal group of determinant1. IfK does have characteristic2, then all elements of the orthogonal group have determinant1, and the special orthogonal group is the set of elements of Dickson invariant0.

There is a homomorphism from the pin group to the orthogonal group. The image consists of the elements of spinor norm1 ∈K^×‍/‍(K^×)². The kernel consists of the elements+1 and−1, and has order2 unlessK has characteristic2. Similarly there is a homomorphism from the Spin group to the special orthogonal group of V.

In the common case whenV is a positive or negative definite space over the reals, the spin group maps onto the special orthogonal group, and is simply connected whenV has dimension at least3. Further the kernel of this homomorphism consists of1 and−1. So in this case the spin group,Spin(n), is a double cover ofSO(n). Note, however, that the simple connectedness of the spin group is not true in general: ifV isR^p,q forp andq both at least2 then the spin group is not simply connected. In this case the algebraic groupSpin_p,q is simply connected as an algebraic group, even though its group of real valued pointsSpin_p,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.^[which?]

Clifford algebrasCl_p,q(C), withp +q = 2n even, are matrix algebras that have a complex representation of dimension2ⁿ. By restricting to the groupPin_p,q(R) we get a complex representation of the Pin group of the same dimension, called thespin representation. If we restrict this to the spin groupSpin_p,q(R) then it splits as the sum of twohalf spin representations (orWeyl representations) of dimension 2ⁿ⁻¹.

Ifp +q = 2n + 1 is odd then the Clifford algebraCl_p,q(C) is a sum of two matrix algebras, each of which has a representation of dimension2ⁿ, and these are also both representations of the pin groupPin_p,q(R). On restriction to the spin groupSpin_p,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 thestructure of the corresponding Clifford algebras: 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 onspinors.

To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. Thepin group,Pin_p,q is the set of invertible elements inCl_p,q that can be written as a product of unit vectors:$${\displaystyle {\mathrm{P}\mathrm{i}\mathrm{n}}_{p,q}=\left\{v_1{v}_2\cdots v_r\mid \mathrm{\forall }i\Vert v_i\Vert =\pm 1\right\}.}$$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 groupO(p,q). Thespin group consists of those elements ofPin_p,q that are products of an even number of unit vectors. Thus by theCartan–Dieudonné theorem Spin is a cover of the group of proper rotationsSO(p,q).

Letα : Cl → Cl be the automorphism that is given by the mappingv ↦ −v acting on pure vectors. Then in particular,Spin_p,q is the subgroup ofPin_p,q whose elements are fixed byα. Let$${\displaystyle {\mathrm{Cl}}_{p,q}^{[0]}=\{x\in {\mathrm{Cl}}_{p,q}\mid \alpha (x)=x\}.}$$(These are precisely the elements of even degree inCl_p,q.) Then the spin group lies withinCl^[0]
_p,q.

The irreducible representations ofCl_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 ofCl^[0]
_p,q.

To classify the pin representations, one need only appeal to theclassification of Clifford algebras. To find the spin representations (which are representations of the even subalgebra), one can first make use of either of the isomorphisms (see above)$${\displaystyle {\mathrm{Cl}}_{p,q}^{[0]}\approx {\mathrm{Cl}}_{p,q-1},\text{ for }q>0}$$$${\displaystyle {\mathrm{Cl}}_{p,q}^{[0]}\approx {\mathrm{Cl}}_{q,p-1},\text{ for }p>0}$$and realize a spin representation in signature(p,q) as a pin representation in either signature(p,q − 1) or(q,p − 1).

One of the principal applications of the exterior algebra is indifferential geometry where it is used to define thebundle ofdifferential forms on asmooth manifold. In the case of a (pseudo-)Riemannian manifold, thetangent spaces come equipped with a natural quadratic form induced by themetric. Thus, one can define aClifford bundle in analogy with theexterior bundle. This has a number of important applications inRiemannian geometry. Perhaps more important is the link to aspin manifold, its associatedspinor bundle andspin^c manifolds.

Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra that has a basis that is generated by the matricesγ₀, ...,γ₃, calledDirac matrices, which have the property that$${\displaystyle {\gamma }_i{\gamma }_j+{\gamma }_j{\gamma }_i=2{\eta }_{ij},}$$whereη 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 algebraCl
_1,3(R), whosecomplexification isCl
_1,3(R)_C, which, by theclassification of Clifford algebras, is isomorphic to the algebra of4 × 4 complex matricesCl₄(C) ≈ M₄(C). However, it is best to retain the notationCl
_1,3(R)_C, since any transformation that takes the bilinear form to the canonical form isnot a Lorentz transformation of the underlying spacetime.

The Clifford algebra of spacetime used in physics thus has more structure thanCl₄(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 algebraso(1, 3) sitting inside the Clifford algebra conventionally requires a complex Clifford algebra. For reference, the spin Lie algebra is given by

This is in the(3, 1) convention, hence fits inCl
_3,1(R)_C.^[14]

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

The use of Clifford algebras to describe quantum theory has been advanced among others byMario Schönberg,^[i] byDavid Hestenes in terms ofgeometric calculus, byDavid Bohm andBasil Hiley and co-workers in form of ahierarchy of Clifford algebras, and by Elio Conte et al.^[15][16]

Clifford algebras have been applied in the problem of action recognition and classification incomputer vision. Rodriguez et al^[17] propose a Clifford embedding to generalize traditional MACH filters to video (3D spatiotemporal volume), and vector-valued data such asoptical flow. Vector-valued data is analyzed using theClifford 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.

• While this article focuses on a Clifford algebra of a vector space over a field, the definition extends without change to amodule over any unital, associative, commutative ring.^[j]
• Clifford algebras may be generalized to a form of degree higher than quadratic over a vector space.^[18]

This section is empty. You can help byadding to it.(April 2025)

• "Clifford algebra",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
• Clifford algebra atPlanetMath.
• A history of Clifford algebras (unverified)
• John Baez on Clifford algebras
• Clifford Algebra: A Visual Introduction
• Clifford Algebra Explorer : A Pedagogical Tool

• Clifford algebras
• Ring theory
• Quadratic forms

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