Clifford Algebra

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''ℓ(''V'',''Q'') is the "freest" algebra generated by ''V'' subject to the condition¹ : $v^2 = Q(v)\ m{for\ all}\ v\in V.$ If the Characteristic of the ground Field ''K'' is not 2, then one can rewrite this fundamental identity in the form : $uv + vu = \lang u, v ang$ for all $u,v \in V$ where <''u'', ''v''> = ''Q''(''u'' + ''v'') − ''Q''(''u'') − ''Q''(''v'') is the 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). Clifford algebras are closely related to Exterior Algebra s. In fact, if ''Q'' = 0 then the Clifford algebra ''C''ℓ(''V'',''Q'') is just the exterior algebra Λ(''V''). For nonzero ''Q'' there exists a canonical ''linear'' isomorphism between Λ(''V'') and ''C''ℓ(''V'',''Q'') whenever the ground field ''K'' does not have characteristic 2. That is they are Naturally Isomorphic as vector spaces but with different multiplications. Clifford multiplication is strictly richer than the Exterior Product since it makes use of the extra information provided by ''Q''. Quadratic forms and Clifford algebras in Characteristic 2 form an exceptional case. In particular, if char ''K'' = 2 it is not true that a quadratic form 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. UNIVERSAL PROPERTY AND CONSTRUCTION Let ''V'' be a on ''V''. In most cases of interest the field ''K'' is either R or '''C''' (which have characteristic 0) or a finite field. A Clifford algebra ''C''ℓ(''V'',''Q'') is a : Given any associative algebra ''A'' over ''K'' and any linear map ''j'' : ''V'' → ''A'' such that j (where 1 denotes the multiplicative identity of ''A''), there is a unique Algebra Homomorphism ''f'' : ''C''ℓ(''V'',''Q'') → ''A'' such that the following diagram Commutes (i.e. such that ''f'' o ''i'' = ''j''): Working with a symmetric Bilinear Form <·,·> instead of ''Q'' (in characteristic not 2), the requirement on ''j'' is j A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains ''V'', namely the Tensor Algebra ''T''(''V''), and then enforce the fundamental identity by taking a suitable Quotient . In our case we want to take the Two-sided Ideal ''I''_''Q'' in ''T''(''V'') generated by all elements of the form : $v\otimes v - Q(v)1$ for all $v\in V$ and define ''C''ℓ(''V'',''Q'') as the quotient C It is then straightforward to show that ''C''ℓ(''V'',''Q'') contains ''V'' and satisfies the above universal property, so that ''C''ℓ is unique up to isomorphism; thus one speaks of "the" Clifford algebra ''C''ℓ(''V'', ''Q''). It also follows from this construction that ''i'' is Injective . One usually drops the ''i'' and considers ''V'' as a Linear Subspace of ''C''ℓ(''V'',''Q''). The universal characterization of the Clifford algebra shows that the construction of ''C''ℓ(''V'',''Q'') is ''functorial'' in nature. Namely, ''C''ℓ can be considered as a Functor from the 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 of ''V'' is ''n'' and {''e''₁,…,''e''_''n''} is a Basis of ''V'', then the set : $\{e_{i_1}e_{i_2}\cdots e_{i_k} \mid 1\le i_1 < i_2 < \cdots < i_k \le n\mbox{ and } 0\le k\le n\}$ is a basis for ''C''ℓ(''V'',''Q''). The empty product (''k'' = 0) is defined as the multiplicative Identity Element . For each value of ''k'' there are ''n'' Choose ''k'' basis elements, so the total dimension of the Clifford algebra is : $\dim C\ell(V,Q) = \sum_{k=0}^n\begin{pmatrix}n\ k\end{pmatrix} = 2^n.$ Since ''V'' comes equipped with a quadratic form, there is a set of privileged bases for ''V'': the Orthogonal ones. An Orthogonal Basis in one such that : $\langle e_i, e_j angle = 0 \qquad i eq j. \,$ where <·,·> is the symmetric bilinear form associated to ''Q''. The fundamental Clifford identity implies that for an orthogonal basis : $e_ie_j = -e_je_i \qquad i eq j. \,$ 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, 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 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''ℓ(''V'',''Q'') by requiring that distinct elements $e_{i_1}e_{i_2}\cdots e_{i_k}$ are orthogonal to one another whenever the {''e''_''i''}'s are orthogonal. Additionally, one sets : $Q(e_{i_1}e_{i_2}\cdots e_{i_k}) = Q(e_{i_1})Q(e_{i_2})\cdots Q(e_{i_k})$ . The quadratic form on a scalar is just ''Q''(λ) = λ². Thus, orthogonal bases for ''V'' extend to orthogonal bases for ''C''ℓ(''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 The most important Clifford algebras are those over Real and 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: : $Q(v) = v_1^2 + \cdots + v_p^2 - v_{p+1}^2 - \cdots - v_{p+q}^2$ where ''n'' = ''p'' + ''q'' is the dimension of the vector space. The pair of integers (''p'', ''q'') is called the Signature of the quadratic form. The real vector space with this quadratic form is often denoted R^''p'',''q''. The Clifford algebra on R^''p'',''q'' is denoted ''C''ℓ_''p'',''q''(R). The symbol ''C''ℓ_''n''(R) means either ''C''ℓ_''n'',0(R) or ''C''ℓ_0,''n''(R) depending on whether the author prefers positive definite or negative definite spaces. A standard Orthonormal Basis {''e''_''i''} for R^''p'',''q'' consists of ''n'' = ''p'' + ''q'' mutually orthogonal vectors, ''p'' of which have norm +1 and ''q'' of which have norm −1. The algebra ''C''ℓ_''p'',''q''(R) will therefore have ''p'' vectors which square to +1 and ''q'' vectors which square to −1. Note that ''C''ℓ_0,0(R) is naturally isomorphic to R since there are no nonzero vectors. ''C''ℓ_0,1(R) is a two-dimensional algebra generated by a single vector ''e''₁ which squares to −1, and therefore is isomorphic to '''C''', the field of Complex Number s. The algebra ''C''ℓ_0,2(R) is a four-dimensional algebra spanned by {1, ''e''₁, ''e''₂, ''e''₁''e''₂}. The latter three elements square to −1 and all anticommute, and so the algebra is isomorphic to the Quaternion s '''H'''. The next algebra in the sequence is ''C''ℓ_0,3(R) is an 8-dimensional algebra isomorphic to the Direct Sum '''H''' ⊕ '''H''' called Clifford 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 : $Q(z) = z_1^2 + z_2^2 + \cdots + z_n^2$ where ''n'' = dim ''V'', so there is essentially only one Clifford algebra in each dimension. We will denote the Clifford algebra on C^''n'' with the standard quadratic form by ''C''ℓ_''n''(C). One can show that the algebra ''C''ℓ_''n''(C) may be obtained as the Complexification of the algebra ''C''ℓ_''p'',''q''('''R''') where ''n'' = ''p'' + ''q'': : $C\ell_n(\mathbb{C}) \cong C\ell_{p,q}(\mathbb{R})\otimes\mathbb{C} \cong C\ell(\mathbb{C}^{p+q},Q\otimes\mathbb{C})$ . 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 C C where ''M''₂(C) denotes the algebra of 2×2 matrices over C. It turns out that every one of the algebras ''C''ℓ_''p'',''q''(R) and ''C''ℓ_''n''('''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 Λ(''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 Λ(''V'') and ''C''ℓ(''V'',''Q'') considered as vector spaces. This is an algebra isomorphism if and only if ''Q'' = 0. One can thus consider the Clifford algebra ''C''ℓ(''V'',''Q'') as an enrichment 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''_''i''} for ''V'' and extend it to an orthogonal basis for ''C''ℓ(''V'',''Q'') as described above. The map ''C''ℓ(''V'',''Q'') → Λ(''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 {''e''_''i''} is orthogonal. One can show that this map is independent of the choice of orthogonal basis and so gives a natural isomorphism. If the Characteristic of ''K'' is 0, one can also establish the isomorphism by antisymmetrizing. Define functions ''f''_''k'' : ''V'' × … × ''V'' → ''C''ℓ(''V'',''Q'') by : $f_k(v_1, \cdots, v_k) = rac{1}{k!}\sum_{\sigma\in S_k}{ m sgn}(\sigma)\, v_{\sigma(1)}\cdots v_{\sigma(k)}$ where the sum is taken over the Symmetric Group on ''k'' elements. Since ''f''_''k'' is Alternating it induces a unique linear map Λ^''k''(''V'') → ''C''ℓ(''V'',''Q''). The Direct Sum of these maps gives a linear map between Λ(''V'') and ''C''ℓ(''V'',''Q''). This map can be shown to be a linear isomorphism. Yet another way to see the relation is to construct a : $\bigoplus_k F^k/F^{k-1}$ is naturally isomorphic to the exterior algebra Λ(''V''). Grading The linear map on ''V'' defined by $v \mapsto -v$ preserves the quadratic form ''Q'' and so by the universal property of Clifford algebras extends to an algebra Automorphism :α : ''C''ℓ(''V'',''Q'') → ''C''ℓ(''V'',''Q''). Since α is an Involution (i.e. it squares to the Identity ) one can decompose ''C''ℓ(''V'',''Q'') into positive and negative eigenspaces : $C\ell(V,Q) = C\ell^0(V,Q) \oplus C\ell^1(V,Q)$
	{ Border	1 style="margin-left: 2em text-align: center" cellpadding=4

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME

	MORE SHOPPING...

	CATEGORIES ABOUT CLIFFORD ALGEBRA

	clifford algebras

Information About

Clifford Algebra