Clifford algebra 45305 217317270 2008-06-05T14:40:56Z 142.151.171.10 /* The Clifford group &Gamma; */ In [[mathematics]], '''Clifford algebras''' are a type of [[associative algebra]]. They can be thought of as one of the possible generalizations of the [[complex number]]s and [[quaternion]]s. The theory of Clifford algebras is intimately connected with the theory of [[quadratic form]]s and [[orthogonal group|orthogonal transformation]]s. Clifford algebras have important applications in a variety of fields including [[geometry]] and [[theoretical physics]]. They are named for the English geometer [[William Kingdon Clifford]]. :''Some familiarity with the basics of [[multilinear algebra]] will be useful in reading this article.'' ==Introduction and basic properties== Specifically, a Clifford algebra is a [[unital]] associative algebra which contains and is generated by a [[vector space]] ''V'' equipped with a [[quadratic form]] ''Q''. The Clifford algebra ''C''&#x2113;(''V'',''Q'') is the "freest" algebra generated by ''V'' subject to the condition<ref>Mathematicians who work with real Clifford algebras and prefer positive definite quadratic forms (especially those working in index theory) sometimes use a different [[sign convention|choice of sign]] in the fundamental Clifford identity. That is, they take ''v''<sup>2</sup> = &minus;''Q''(''v''). One must replace ''Q'' with &minus;''Q'' in going from one convention to the other.</ref> :<math>v^2 = Q(v)\ \mbox{ for all } v\in V.</math> If the [[characteristic (algebra)|characteristic]] of the ground [[field (mathematics)|field]] ''K'' is not 2, then one can rewrite this fundamental identity in the form :<math>uv + vu = 2\lang u, v\rang \mbox{ for all }u,v \in V,</math> where <''u'',&nbsp;''v''> = ½(''Q''(''u''&nbsp;+&nbsp;''v'')&nbsp;&minus;&nbsp;''Q''(''u'')&nbsp;&minus;&nbsp;''Q''(''v'')) is the [[symmetric matrix|symmetric]] [[bilinear form]] associated to Q. This idea of "freest" or "most general" algebra subject to this identity can be formally expressed through the notion of a [[universal property]] (see below). Quadratic forms and Clifford algebras in [[characteristic (algebra)|characteristic]] 2 form an exceptional case. In particular, if char ''K'' = 2 it is not true that a quadratic form is determined by its symmetric bilinear form, or 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 quantization of exterior algebra=== Clifford algebras are closely related to [[exterior algebra]]s. In fact, if ''Q'' = 0 then the Clifford algebra ''C''&#x2113;(''V'',''Q'') is just the exterior algebra &Lambda;(''V''). For nonzero ''Q'' there exists a canonical ''linear'' isomorphism between &Lambda;(''V'') and ''C''&#x2113;(''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 is strictly richer than the [[exterior product]] since it makes use of the extra information provided by ''Q''. More precisely, they may be thought of as quantizations (cf. [[quantization (physics)]], [[Quantum group]]) of the exterior algebra, in the same way that the [[Weyl algebra]] is a quantization of the [[symmetric algebra]]. == Universal property and construction == Let ''V'' be a [[vector space]] over a [[field (mathematics)|field]] ''K'', and let ''Q'' : ''V'' &rarr; ''K'' be a [[quadratic form]] on ''V''. In most cases of interest the field ''K'' is either '''R''', '''C''' or a finite field. A Clifford algebra ''C''&#x2113;(''V'',''Q'') is a [[unital]] [[associative algebra]] over ''K'' together with a [[linear transformation|linear map]] ''i'' : ''V'' &rarr; ''C''&#x2113;(''V'',''Q'') satisfying ''i''(''v'')<sup>2</sup> = ''Q''(''v'')1 for all ''v'' &isin; ''V'', defined by the following [[universal property]]: Given any associative algebra ''A'' over ''K'' and any linear map ''j'' : ''V'' &rarr; ''A'' such that :''j''(''v'')<sup>2</sup> = ''Q''(''v'')1 for all ''v'' &isin; ''V'' (where 1 denotes the multiplicative identity of ''A''), there is a unique [[algebra homomorphism]] ''f'' : ''C''&#x2113;(''V'',''Q'') &rarr; ''A'' such that the following diagram [[commutative diagram|commutes]] (i.e. such that ''f'' o ''i'' = ''j''): <div style="text-align: center;">[[Image:CliffordAlgebra-01.png]]</div> Working with a symmetric [[bilinear form]] <&middot;,&middot;> instead of ''Q'' (in characteristic not 2), the requirement on ''j'' is :''j''(''v'')''j''(''w'') + ''j''(''w'')''j''(''v'')'' = 2<''v'',&nbsp;''w''> for all ''v'',&nbsp;''w'' &isin; ''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 ring|quotient]]. In our case we want to take the [[ideal (ring theory)|two-sided ideal]] ''I''<sub>''Q''</sub> in ''T''(''V'') generated by all elements of the form :<math>v\otimes v - Q(v)1</math> for all <math>v\in V</math> and define ''C''&#x2113;(''V'',''Q'') as the quotient :''C''&#x2113;(''V'',''Q'') = T(''V'')/''I''<sub>''Q''.</sub> It is then straightforward to show that ''C''&#x2113;(''V'',''Q'') contains ''V'' and satisfies the above universal property, so that ''C''&#x2113; is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra ''C''&#x2113;(''V'', ''Q''). It also follows from this construction that ''i'' is [[injective function|injective]]. One usually drops the ''i'' and considers ''V'' as a [[linear subspace]] of ''C''&#x2113;(''V'',''Q''). The universal characterization of the Clifford algebra shows that the construction of ''C''&#x2113;(''V'',''Q'') is ''functorial'' in nature. Namely, ''C''&#x2113; can be considered as a [[functor]] from the [[category (mathematics)|category]] of vector spaces with quadratic forms (whose [[morphism]]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 [[algebra homomorphism]]s between the associated Clifford algebras. ==Basis and dimension== If the [[dimension (linear algebra)|dimension]] of ''V'' is ''n'' and {''e''<sub>1</sub>,&hellip;,''e''<sub>''n''</sub>} is a [[basis (linear algebra)|basis]] of ''V'', then the set :<math>\{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\}</math> is a basis for ''C''&#x2113;(''V'',''Q''). The empty product (''k'' = 0) is defined as the multiplicative [[identity element]]. For each value of ''k'' there are [[Binomial coefficient|''n'' choose ''k'']] basis elements, so the total dimension of the Clifford algebra is :<math>\dim C\ell(V,Q) = \sum_{k=0}^n\begin{pmatrix}n\\ k\end{pmatrix} = 2^n.</math> Since ''V'' comes equipped with a quadratic form, there is a set of privileged bases for ''V'': the [[orthogonal]] ones. An [[orthogonal basis]] is one such that :<math>\langle e_i, e_j \rangle = 0 \qquad i\neq j. \,</math> where <&middot;,&middot;> is the symmetric bilinear form associated to ''Q''. The fundamental Clifford identity implies that for an orthogonal basis :<math>e_ie_j = -e_je_i \qquad i\neq j. \,</math> This makes manipulation of orthogonal basis vectors quite simple. Given a product <math>e_{i_1}e_{i_2}\cdots e_{i_k}</math> of ''distinct'' orthogonal basis vectors, one can put them into standard order by including an overall sign corresponding to the number of flips needed to correctly order them (i.e. the [[signature (permutation)|signature]] of the ordering [[permutation]]). If the characteristic is not 2 then an orthogonal basis for ''V'' exists, and one can easily extend the quadratic form on ''V'' to a quadratic form on all of ''C''&#x2113;(''V'',''Q'') by requiring that distinct elements <math>e_{i_1}e_{i_2}\cdots e_{i_k}</math> are orthogonal to one another whenever the {''e''<sub>''i''</sub>}'s are orthogonal. Additionally, one sets :<math>Q(e_{i_1}e_{i_2}\cdots e_{i_k}) = Q(e_{i_1})Q(e_{i_2})\cdots Q(e_{i_k})</math>. The quadratic form on a scalar is just ''Q''(&lambda;) = &lambda;<sup>2</sup>. Thus, orthogonal bases for ''V'' extend to orthogonal bases for ''C''&#x2113;(''V'',''Q''). The quadratic form defined in this way is actually independent of the orthogonal basis chosen (a basis-independent formulation will be given later). == Examples: real and complex Clifford algebras == {{main|geometric algebra}} The most important Clifford algebras are those over [[real number|real]] and [[complex number|complex]] vector spaces equipped with [[nondegenerate]] quadratic forms. Every nondegenerate quadratic form on a finite-dimensional real vector space is equivalent to the standard diagonal form: :<math>Q(v) = v_1^2 + \cdots + v_p^2 - v_{p+1}^2 - \cdots - v_{p+q}^2</math> where ''n'' = ''p'' + ''q'' is the dimension of the vector space. The pair of integers (''p'', ''q'') is called the [[metric signature|signature]] of the quadratic form. The real vector space with this quadratic form is often denoted '''R'''<sup>''p'',''q''</sup>. The Clifford algebra on '''R'''<sup>''p'',''q''</sup> is denoted ''C''&#x2113;<sub>''p'',''q''</sub>('''R'''). The symbol ''C''&#x2113;<sub>''n''</sub>('''R''') means either ''C''&#x2113;<sub>''n'',0</sub>('''R''') or ''C''&#x2113;<sub>0,''n''</sub>('''R''') depending on whether the author prefers positive definite or negative definite spaces. A standard [[orthonormal basis]] {''e''<sub>''i''</sub>} for '''R'''<sup>''p'',''q''</sup> consists of ''n'' = ''p'' + ''q'' mutually orthogonal vectors, ''p'' of which have norm +1 and ''q'' of which have norm &minus;1. The algebra ''C''&#x2113;<sub>''p'',''q''</sub>('''R''') will therefore have ''p'' vectors which square to +1 and ''q'' vectors which square to &minus;1. Note that ''C''&#x2113;<sub>0,0</sub>('''R''') is naturally isomorphic to '''R''' since there are no nonzero vectors. ''C''&#x2113;<sub>0,1</sub>('''R''') is a two-dimensional algebra generated by a single vector ''e''<sub>1</sub> which squares to &minus;1, and therefore is isomorphic to '''C''', the field of [[complex number]]s. The algebra ''C''&#x2113;<sub>0,2</sub>('''R''') is a four-dimensional algebra spanned by {1, ''e''<sub>1</sub>, ''e''<sub>2</sub>, ''e''<sub>1</sub>''e''<sub>2</sub>}. The latter three elements square to &minus;1 and all anticommute, and so the algebra is isomorphic to the [[quaternion]]s '''H'''. The next algebra in the sequence is ''C''&#x2113;<sub>0,3</sub>('''R''') is an 8-dimensional algebra isomorphic to the [[direct sum]] '''H''' &oplus; '''H''' called [[split-biquaternion]]s. One can also study Clifford algebras on complex vector spaces. Every nondegenerate quadratic form on a complex vector space is equivalent to the standard diagonal form :<math>Q(z) = z_1^2 + z_2^2 + \cdots + z_n^2</math> where ''n'' = dim ''V'', so there is essentially only one Clifford algebra in each dimension. We will denote the Clifford algebra on '''C'''<sup>''n''</sup> with the standard quadratic form by ''C''&#x2113;<sub>''n''</sub>('''C'''). One can show that the algebra ''C''&#x2113;<sub>''n''</sub>('''C''') may be obtained as the [[complexification]] of the algebra ''C''&#x2113;<sub>''p'',''q''</sub>('''R''') where ''n'' = ''p'' + ''q'': :<math>C\ell_n(\mathbb{C}) \cong C\ell_{p,q}(\mathbb{R})\otimes\mathbb{C} \cong C\ell(\mathbb{C}^{p+q},Q\otimes\mathbb{C})</math>. Here ''Q'' is the real quadratic form of signature (''p'',''q''). Note that the complexification does not depend on the signature. The first few cases are not hard to compute. One finds that :''C''&#x2113;<sub>0</sub>('''C''') = '''C''' :''C''&#x2113;<sub>1</sub>('''C''') = '''C''' &oplus; '''C''' :''C''&#x2113;<sub>2</sub>('''C''') = ''M''<sub>2</sub>('''C''') where ''M''<sub>2</sub>('''C''') denotes the algebra of 2&times;2 matrices over '''C'''. It turns out that every one of the algebras ''C''&#x2113;<sub>''p'',''q''</sub>('''R''') and ''C''&#x2113;<sub>''n''</sub>('''C''') is isomorphic to a [[matrix algebra]] over '''R''', '''C''', or '''H''' or to a direct sum of two such algebras. For a complete classification of these algebras see [[classification of Clifford algebras]]. ==Properties== ===Relation to the exterior algebra=== Given a vector space ''V'' one can construct the [[exterior algebra]] &Lambda;(''V''), whose definition is independent of any quadratic form on ''V''. It turns out that if ''F'' does not have characteristic 2 then there is a [[natural isomorphism]] between &Lambda;(''V'') and ''C''&#x2113;(''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 ''C''&#x2113;(''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 independent of ''Q''). The easiest way to establish the isomorphism is to choose an ''orthogonal'' basis {''e''<sub>''i''</sub>} for ''V'' and extend it to an orthogonal basis for ''C''&#x2113;(''V'',''Q'') as described above. The map ''C''&#x2113;(''V'',''Q'') &rarr; &Lambda;(''V'') is determined by :<math>e_{i_1}e_{i_2}\cdots e_{i_k} \mapsto e_{i_1}\wedge e_{i_2}\wedge \cdots \wedge e_{i_k}.</math> Note that this only works if the basis {''e''<sub>''i''</sub>} 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 (algebra)|characteristic]] of ''K'' is 0, one can also establish the isomorphism by antisymmetrizing. Define functions ''f''<sub>''k''</sub> : ''V'' &times; &hellip; &times; ''V'' &rarr; ''C''&#x2113;(''V'',''Q'') by :<math>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)}</math> where the sum is taken over the [[symmetric group]] on ''k'' elements. Since ''f''<sub>''k''</sub> is [[alternating form|alternating]] it induces a unique linear map &Lambda;<sup>''k''</sup>(''V'') &rarr; ''C''&#x2113;(''V'',''Q''). The [[direct sum]] of these maps gives a linear map between &Lambda;(''V'') and ''C''&#x2113;(''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 (abstract algebra)|filtration]] on ''C''&#x2113;(''V'',''Q''). Recall that the [[tensor algebra]] ''T''(''V'') has a natural filtration: ''F''<sup>0</sup> &sub; ''F''<sup>1</sup> &sub; ''F''<sup>2</sup> &sub; &hellip; where ''F''<sup>''k''</sup> contains sums of tensors with rank &le; ''k''. Projecting this down to the Clifford algebra gives a filtration on ''C''&#x2113;(''V'',''Q''). The associated [[graded algebra]] :<math>Gr_F C\ell(V,Q) = \bigoplus_k F^k/F^{k-1}</math> is naturally isomorphic to the exterior algebra &Lambda;(''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 ''F''<sup>k</sup> in ''F''<sup>k+1</sup> 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.<ref>Thus the [[group algebra]] '''K'''['''Z'''/2] is [[semisimple]] and the Clifford algebra splits into eigenspaces of the main involution.</ref> Clifford algebras are '''Z'''<sub>2</sub>-[[graded algebra]] (also known as [[superalgebra]]s). Indeed, the linear map on ''V'' defined by <math>v \mapsto -v</math> preserves the quadratic form ''Q'' and so by the universal property of Clifford algebras extends to an algebra [[automorphism]] :&alpha; : ''C''&#x2113;(''V'',''Q'') &rarr; ''C''&#x2113;(''V'',''Q''). Since &alpha; is an [[involution]] (i.e. it squares to the [[identity map|identity]]) one can decompose ''C''&#x2113;(''V'',''Q'') into positive and negative eigenspaces :<math>C\ell(V,Q) = C\ell^0(V,Q) \oplus C\ell^1(V,Q)</math> where ''C''&#x2113;<sup>''i''</sup>(''V'',''Q'') = {''x'' &isin; ''C''&#x2113;(''V'',''Q'') | &alpha;(''x'') = (&minus;1)<sup>''i''</sup>''x''}. Since &alpha; is an automorphism it follows that :<math>C\ell^{\,i}(V,Q)C\ell^{\,j}(V,Q) = C\ell^{\,i+j}(V,Q)</math> where the superscripts are read modulo 2. This gives ''C''&#x2113;(''V'',''Q'') the structure of a '''Z'''<sub>2</sub>-[[graded algebra]]. The subspace ''C''&#x2113;<sup>0</sup>(''V'',''Q'') forms a [[subalgebra]] of ''C''&#x2113;(''V'',''Q''), called the ''even subalgebra''. The subspace ''C''&#x2113;<sup>1</sup>(''V'',''Q'') is called the ''odd part'' of ''C''&#x2113;(''V'',''Q'') (it is not a subalgebra). The '''Z'''<sub>2</sub>-grading plays an important role in the analysis and application of Clifford algebras. The automorphism &alpha; is called the ''main [[involution]]'' or ''grade involution''. ''Remark''. In characteristic not 2 the underlying vector space of ''C''&#x2113;(''V'',''Q'') inherits a '''Z'''-grading from the canonical isomorphism with the underlying vector space of the exterior algebra &Lambda;(''V''). It is important to note, however, that this is a ''vector space grading only''. That is, Clifford multiplication does not respect the '''Z'''-grading, only the '''Z'''<sub>2</sub>-grading: for instance if <math>Q(v)\neq 0</math>, then <math>v\in C\ell^1(V,Q)</math>, but <math>v^2\in C\ell^0(V,Q)</math>, not in <math>C\ell^2(V,Q)</math>. Happily, the gradings are related in the natural way: '''Z'''<sub>2</sub> = '''Z'''/2'''Z'''. Further, the Clifford algebra is '''Z'''-[[filtered algebra|filtered]]: <math>C\ell^{\leq i}(V,Q) \cdot C\ell^{\leq j}(V,Q) \subset C\ell^{\leq i+j}(V,Q)</math>. The ''degree'' of a Clifford number usually refers to the degree in the '''Z'''-grading. Elements which are pure in the '''Z'''<sub>2</sub>-grading are simply said to be even or odd. The even subalgebra ''C''&#x2113;<sup>0</sup>(''V'',''Q'') of a Clifford algebra is itself a Clifford algebra<ref>We are still assuming that the characteristic is not 2.</ref>. If ''V'' is the orthogonal direct sum of a vector ''a'' of norm ''Q''(''a'') and a subspace ''U'', then ''C''&#x2113;<sup>0</sup>(''V'',''Q'') is isomorphic to ''C''&#x2113;(''U'',&minus;''Q''(''a'')''Q''), where &minus;''Q''(''a'')''Q'' is the form ''Q'' restricted to ''U'' and multiplied by &minus;''Q''(''a''). In particular over the reals this implies that :<math>C\ell_{p,q}^0(\mathbb{R}) \cong C\ell_{p,q-1}(\mathbb{R})</math> for ''q'' &gt; 0, and :<math>C\ell_{p,q}^0(\mathbb{R}) \cong C\ell_{q,p-1}(\mathbb{R})</math>for ''p'' &gt; 0. In the negative-definite case this gives an inclusion ''C''&#x2113;<sub>0,''n''&minus;1</sub>('''R''') &sub; ''C''&#x2113;<sub>0, ''n''</sub>('''R''') which extends the sequence :'''R''' &sub; '''C''' &sub; '''H''' &sub; '''H'''&oplus;'''H''' &sub; &hellip; Likewise, in the complex case, one can show that the even subalgebra of ''C''&#x2113;<sub>''n''</sub>('''C''') is isomorphic to ''C''&#x2113;<sub>''n''&minus;1</sub>('''C'''). ===Antiautomorphisms=== In addition to the automorphism &alpha;, there are two [[antiautomorphism]]s which 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: :<math>v_1\otimes v_2\otimes \cdots \otimes v_k \mapsto v_k\otimes \cdots \otimes v_2\otimes v_1</math>. Since the ideal ''I''<sub>''Q''</sub> is invariant under this reversal, this operation descends to an antiautomorphism of ''C''&#x2113;(''V'',''Q'') called the ''[[Transposition (mathematics)|transpose]]'' or ''reversal'' operation, denoted by ''x''<sup>''t''</sup>. The transpose is an antiautomorphism: <math>(xy)^t = y^t x^t</math>. The transpose operation makes no use of the '''Z'''<sub>2</sub>-grading so we define a second antiautomorphism by composing &alpha; and the transpose. We call this operation ''Clifford conjugation'' denoted <math>\bar x</math> :<math>\bar x = \alpha(x^t) = \alpha(x)^t.</math> Of the two antiautomorphisms, the transpose is the more fundamental.<ref>The opposite is true when uses the alternate (&minus;) sign convention for Clifford algebras: it is the conjugate which is more important. In general, the meanings of conjugation and transpose are interchanged when passing from one sign convention to the other. For example, in the convention used here the inverse of a vector is given by <math>v^{-1} = v^t/Q(v)</math> while in the (&minus;) convention it is given by <math>v^{-1} = \bar{v}/Q(v)</math>.</ref> Note that all of these operations are [[involution]]s. One can show that they act as &plusmn;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 :<math>\alpha(x) = \pm x \qquad x^t = \pm x \qquad \bar x = \pm x</math> where the signs are given by the following table: {| border=1 style="margin-left: 2em; text-align: center;" cellpadding=4 ! ''k'' mod 4 || 0 || 1 || 2 || 3 || |- | <math>\alpha(x)\,</math> || + || &minus; || + || &minus; | (&minus;1)<sup>''k''</sup> |- | <math>x^t\,</math> || + || + || &minus; || &minus; | (&minus;1)<sup>''k''(''k''&minus;1)/2</sup> |- | <math>\bar x</math> || + || &minus; || &minus; || + | (&minus;1)<sup>''k''(''k''+1)/2</sup> |} ===The 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 ''C''&#x2113;(''V'',''Q'') as explained earlier (which we also denoted by ''Q''). A basis independent definition is :<math>2Q(x) = \lang x^t x\rang</math> where <''a''> denotes the scalar part of ''a'' (the grade 0 part in the '''Z'''-grading). One can show that :<math>Q(v_1v_2\cdots v_k) = Q(v_1)Q(v_2)\cdots Q(v_k)</math> where the ''v''<sub>''i''</sub> are elements of ''V'' &mdash; this identity is ''not'' true for arbitrary elements of ''C''&#x2113;(''V'',''Q''). The associated symmetric bilinear form on ''C''&#x2113;(''V'',''Q'') is given by :<math>\lang x, y\rang = \lang x^t y\rang.</math> One can check that this reduces to the original bilinear form when restricted to ''V''. The bilinear form on all of ''C''&#x2113;(''V'',''Q'') is [[nondegenerate]] if and only if it is nondegenerate on ''V''. It is not hard to verify that the transpose is the [[adjoint of an operator|adjoint]] of left/right Clifford multiplication with respect to this inner product. That is, :<math>\lang ax, y\rang = \lang x, a^t y\rang,</math> and :<math>\lang xa, y\rang = \lang x, y a^t\rang.</math> ==Structure of Clifford algebras== {{details|classification of Clifford algebras}} In this section we assume that the vector space ''V'' is finite dimensional and that the 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 ''C''&#x2113;(''V'',''Q'') is a central simple algebra over ''K''. *If ''V'' has even dimension then ''C''&#x2113;<sup>0</sup>(''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 ''C''&#x2113;(''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 ''C''&#x2113;<sup>0</sup>(''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 (&minus;1)<sup>dim(''U'')/2</sup>''dV'', which is the space ''V'' with its quadratic form multiplied by (&minus;1)<sup>dim(''U'')/2</sup>''d''. Over the reals, this implies in particular that :<math> Cl_{p+2,q}(\mathbb{R}) = M_2(\mathbb{R})\otimes Cl_{q,p}(\mathbb{R}) </math> :<math> Cl_{p+1,q+1}(\mathbb{R}) = M_2(\mathbb{R})\otimes Cl_{p,q}(\mathbb{R}) </math> :<math> Cl_{p,q+2}(\mathbb{R}) = \mathbb{H}\otimes Cl_{q,p}(\mathbb{R}) </math> These formulas can be used to find the structure of all real 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 <math>p-q</math> mod 8. This is an algebraic form of [[Bott periodicity]]. ==The Clifford group &Gamma;== In this section we assume that ''V'' is finite dimensional and the quadratic form ''Q'' is [[nondegenerate]]. The invertible elements of the Clifford algebra act on it by twisted conjugation: conjugation by ''x'' maps <math>y \mapsto x y \alpha(x)^{-1}</math>. The Clifford group &Gamma; is defined to be the set of invertible elements ''x'' that ''stabilize vectors'', meaning that :<math>x v \alpha(x)^{-1}\in V</math> for all ''v'' in ''V''. This formula also defines an action of the Clifford group on the vector space ''V'' that preserves the norm ''Q'', and so gives a homomorphism from the Clifford group to the orthogonal group. The Clifford group contains all elements ''r'' of ''V'' of nonzero norm, and these act on ''V'' by the corresponding reflections that take ''v'' to ''v'' &minus; <''v'',''r''>''r''/''Q''(''r'') (In characteristic 2 these are called orthogonal transvections rather than reflections.) Many authors define the Clifford group slightly differently, by replacing the action ''xv''&alpha;(''x'')<sup>&minus;1</sup> by ''xvx''<sup>&minus;1</sup>. This produces the same Clifford group, but the action of the Clifford group on ''V'' is changed slightly: the action of the odd elements &Gamma;<sup>1</sup> of the Clifford group is multiplied by an extra factor of &minus;1. This action used here has several minor advantages: it is consistent with the usual superalgebra sign conventions, elements of ''V'' correspond to reflections, and in odd dimensions the map from the Clifford group to the orthogonal group is onto, and the kernel is no larger than ''K''<sup>*</sup>. Using the action &alpha;(''x'')''vx''<sup>&minus;1</sup> instead of ''xv''&alpha;(''x'')<sup>&minus;1</sup> makes no difference: it produces the same Clifford group with the same action on ''V''. The Clifford group &Gamma; is the disjoint union of two subsets &Gamma;<sup>0</sup> and &Gamma;<sup>1</sup>, where &Gamma;<sup>''i''</sup> is the subset of elements of degree ''i''. The subset &Gamma;<sup>0</sup> is a subgroup of index 2 in &Gamma;. If ''V'' is a finite dimensional real vector space with positive definite (or negative definite) quadratic form then the Clifford group maps onto the orthogonal group of ''V'' with respect to the form (by the [[Cartan-Dieudonné theorem]]) and the kernel consists of the nonzero elements of the field ''K''. This leads to exact sequences :<math> 1 \rightarrow K^* \rightarrow \Gamma \rightarrow O_V(K) \rightarrow 1,\,</math> :<math> 1 \rightarrow K^* \rightarrow \Gamma^0 \rightarrow SO_V(K) \rightarrow 1.\,</math> 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=== {{details|Spinor_norm#Galois_cohomology_and_orthogonal_groups}} In arbitrary characteristic, the [[Orthogonal group#The spinor norm|spinor norm]] ''Q'' is defined on the Clifford group by :<math>Q(x) = x^tx\,</math> <!-- Note that (-1)^D(x) is the same as \alpha(x), so this expression is the same as \alpha(x)^t x (-1)^D(x). --> It is a homomorphism from the Clifford group to the group ''K''<sup>*</sup> 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 &minus;1, 2, or &minus;2 on &Gamma;<sup>1</sup>. The difference is not very important in characteristic other than 2. The nonzero elements of ''K'' have spinor norm in the group ''K''<sup>*2</sup> 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''<sup>*</sup>/''K''<sup>*2</sup>, also called the spinor norm. The spinor norm of the reflection of a vector ''r'' has image ''Q''(''r'') in ''K''<sup>*</sup>/''K''<sup>*2</sup>, and this property uniquely defines it on the orthogonal group. This gives exact sequences: :<math> 1 \to \{\pm 1\} \to \mbox{Pin}_V(K) \to O_V(K) \to K^*/K^{*2},\,</math> :<math> 1 \to \{\pm 1\} \to \mbox{Spin}_V(K) \to SO_V(K) \to K^*/K^{*2}.\,</math> Note that in characteristic 2 the group {±1} has just one element. From the point of view of [[Galois cohomology]] of [[algebraic group]]s, the spinor norm is a [[connecting homomorphism]] on cohomology. Writing &mu;<sub>2</sub> for the [[Group scheme of roots of unity|algebraic group of square roots of 1]] (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 :<math> 1 \to \mu_2 \rightarrow \mbox{Pin}_V \rightarrow O_V \rightarrow 1\,</math> yields a long exact sequence on cohomology, which begins :<math> 1 \to H^0(\mu_2;K) \to H^0(\mbox{Pin}_V;K) \to H^0(O_V;K) \to H^1(\mu_2;K)\,</math> The 0th Galois cohomology group of an algebraic group with coefficients in ''K'' is just the group of ''K''-valued points: <math>H^0(G;K) = G(K)</math>, and <math>H^1(\mu_2;K) \cong K^*/K^{*2}</math>, which recovers the previous sequence :<math> 1 \to \{\pm 1\} \to \mbox{Pin}_V(K) \to O_V(K) \to K^*/K^{*2},\,</math> where the spinor norm is the connecting homomorphism <math>H^0(O_V;K) \to H^1(\mu_2;K)</math>. ==Spin and Pin groups== {{details3|[[Spin group]], [[Pin group]] and [[spinor]]}} 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 [[Pin group]] ''Pin''<sub>V</sub>(''K'') is the subgroup of the Clifford group &Gamma; of elements of spinor norm 1, and similarly the [[Spin group]] ''Spin''<sub>V</sub>(''K'') is the subgroup of elements of [[Dickson invariant]] 0 in ''Pin''<sub>V</sub>(''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 [[special orthogonal group]] to be the image of &Gamma;<sup>0</sup>. 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 &isin; ''K''<sup>*</sup>/''K''<sup>*2</sup>. The kernel consists of the elements +1 and &minus;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. Please note, however, that this is not true in general: if ''V'' is ''R''<sup>''p'',''q''</sup> for ''p'' and ''q'' both at least 2 then the spin group is not simply connected. In this case the algebraic group ''Spin''<sub>''p'',''q''</sub> is simply connected as an algebraic group, even though its group of real valued points ''Spin''<sub>''p'',''q''</sub>(''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==<!-- This section is linked from [[Spinor]] --> Clifford algebras ''C&#x2113;''<sub>''p'',''q''</sub>('''C'''), with ''p+q=2n'' even, are matrix algebras which have a complex representation of dimension 2<sup>n</sup>. By restricting to the group ''Pin''<sub>''p'',''q''</sub>('''R''') we get a complex representation of the Pin group of the same dimension, called the [[Spinor representation|spinor representation]]. If we restrict this to the spin group ''Spin''<sub>''p'',''q''</sub>('''R''') then it splits as the sum of two ''half spin representations'' (or ''Weyl representations'') of dimension 2<sup>''n''-1</sup>. If ''p+q=2n+1'' is odd then the Clifford algebra ''C&#x2113;''<sub>''p'',''q''</sub>('''C''') is a sum of two matrix algebras, each of which has a representation of dimension 2<sup>n</sup>, and these are also both representations of the Pin group ''Pin''<sub>''p'',''q''</sub>('''R'''). On restriction to the spin group ''Spin''<sub>''p'',''q''</sub>('''R''') these become isomorphic, so the spin group has a complex spinor representation of dimension 2<sup>''n''</sup>. More generally, spinor groups and pin groups over any field have similar representations whose exact structure depends on the [[classification of Clifford algebras|structure 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 on [[spinor]]s. ===Real spinors=== {{details|spinor}} To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The [[Pin group]], ''Pin''<sub>p,q</sub> is the set of invertible elements in Cl<sub>p,q</sub> which can be written as a product of unit vectors: :<math>{\mathit{Pin}}_{p,q}=\{v_1v_2\dots v_r |\,\, \forall i, \|v_i\|=\pm 1\}</math> 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 [[Spin group]] consists of those elements of ''Pin''<sub>''p'',''q''</sub> which are products of an even number of unit vectors. Thus by the [[Cartan-Dieudonné theorem]] ''Spin'' is a cover of the group of proper rotations ''SO''(''p'',''q''). Let ''α'' : ''C''&#x2113; → ''C''&#x2113; be the automorphism which is given by -''Id'' acting on pure vectors. Then in particular, ''Spin''<sub>''p'',''q''</sub> is the subgroup of ''Pin''<sub>''p'',''q''</sub> whose elements are fixed by ''α''. Let :<math>Cl_{p,q}^0 = \{ x\in Cl_{p,q} |\, \alpha(x)=x\}</math>. (These are precisely the elements of even degree in ''C''&#x2113;<sub>''p'',''q''</sub>.) Then the spin group lies within ''C''&#x2113;<sup>0</sup><sub>''p'',''q''</sub>. The irreducible representations of C&#x2113;<sub>''p'',''q''</sub> 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 C&#x2113;<sup>0</sup><sub>p,q</sub> To classify the pin representations, one need only appeal to the [[classification 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) :C&#x2113;<sup>0</sup><sub>p,q</sub> ≈ C&#x2113;<sub>p,q-1</sub>, for ''q'' > 0 :C&#x2113;<sup>0</sup><sub>p,q</sub> ≈ C&#x2113;<sub>q,p-1</sub>, 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). ==Applications== ===Differential geometry=== One of the principal applications of the exterior algebra is in [[differential geometry]] where it is used to define the [[fiber bundle|bundle]] of [[differential form]]s on a [[smooth manifold]]. In the case of a ([[pseudo-Riemannian manifold|pseudo]]-)[[Riemannian manifold]], the [[tangent space]]s come equipped with a natural quadratic form induced by the [[metric tensor|metric]]. Thus, one can define a [[Clifford bundle]] in analogy with the [[exterior bundle]]. This has a number of important applications in [[Riemannian geometry]]. ===Physics=== Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra spanned by matrices &gamma;<sub>1</sub>,&hellip;,&gamma;<sub>''n''</sub> called [[Dirac matrices]] which have the property that :<math>\gamma_i\gamma_j + \gamma_j\gamma_i = 2\eta_{ij}\,</math> where &eta; is the matrix of a quadratic form of signature (''p'',''q'') &mdash; typically (1,3) when working in [[Minkowski space]]. These are exactly the defining relations for the Clifford algebra ''Cl''<sub>1,3</sub>(''C'') (up to an unimportant factor of 2), which by the [[classification of Clifford algebras]] is isomorphic to the algebra of 4 by 4 complex matrices. 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]]. The entire Clifford algebra shows up in [[quantum field theory]] in the form of [[Dirac field bilinear]]s. ==See also== <div style="-moz-column-count:2; column-count:2;"> *[[Algebra of physical space]], APS *[[Classification of Clifford algebras]] *[[Representations of Clifford algebras]] *[[Gamma matrices]] *[[Exterior algebra]] *[[Geometric algebra]] *[[Spin group]] *[[Spinor]] *[[Paravector]] </div> ==Footnotes== <references/> ==References== * {{Citation | last1=Bourbaki | first1=Nicolas | author1-link= Nicolas Bourbaki | title=Algebra | publisher=[[Springer-Verlag|Springer-Verlag]] | location=Berlin, New York | isbn=978-3-540-19373-9 | year=1988}}, section XI.9. *Carnahan, S. ''Borcherds Seminar Notes, Uncut.'' Week 5, "Spinors and Clifford Algebras". * {{Citation | last1=Lawson | first1=H. Blaine | last2=Michelsohn | first2=Marie-Louise | title=Spin Geometry | publisher=[[Princeton University Press|Princeton University Press]] | location=Princeton, NJ | isbn=978-0-691-08542-5 | year=1989}}. An advanced textbook on Clifford algebras and their applications to differential geometry. * {{Citation | last1=Lounesto | first1=Pertti | title=Clifford algebras and spinors | publisher=[[Cambridge University Press|Cambridge University Press]] | location=Cambridge | isbn=978-0-521-00551-7 | year=2001}} * {{Citation | last1=Porteous | first1=Ian R. | title=Clifford algebras and the classical groups | publisher=[[Cambridge University Press|Cambridge University Press]] | location=Cambridge | isbn=978-0-521-55177-9 | year=1995}} ==External links== *[http://planetmath.org/encyclopedia/CliffordAlgebra2.html Planetmath entry on Clifford algebras] *[http://members.fortunecity.com/jonhays/clifhistory.htm A history of Clifford algebras] (unverified) *[http://www.math.ucr.edu/home/baez/octonions/node6.html John Baez on Clifford algebras] [[Category:Clifford algebras|*]][[Category:Ring theory]][[Category:Quadratic forms]] [[az:Klifford cəbri]] [[de:Clifford-Algebra]] [[es:Álgebra de Clifford]] [[fr:Algèbre de Clifford]] [[it:Algebra di Clifford]] [[ru:Алгебра Клиффорда]] [[sv:Cliffordalgebra]]