NationMaster - Encyclopedia: Clifford algebra

In mathematics, Clifford algebras are a type of associative algebra. They can be thought of as one of the possible generalizations of the complex numbers and quaternions. The theory of Clifford algebras is intimately connected with the theory of quadratic forms and orthogonal transformations. Clifford algebras have important applications in a variety of fields including geometry and theoretical physics. They are named for the English geometer William Kingdon Clifford. For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ... For other uses, see Geometry (disambiguation). ... Theoretical physics employs mathematical models and abstractions of physics, as opposed to experimental processes, in an attempt to understand nature. ... William Kingdon Clifford William Kingdon Clifford, FRS (May 4, 1845 - March 3, 1879) was an English mathematician who also wrote a fair bit on philosophy. ...

Some familiarity with the basics of multilinear algebra will be useful in reading this article.

In mathematics, multilinear algebra extends the methods of linear algebra. ...

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 Cℓ(V,Q) is the "freest" algebra generated by V subject to the condition^[1] In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

$v^2 = Q(v) mbox{ for all } vin V.$

If the characteristic of the ground field K is not 2, then one can rewrite this fundamental identity in the form In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...

$uv + vu = 2lang u, vrang mbox{ for all }u,v in V,$

where <u, v> = ½(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). In linear algebra, a symmetric matrix is a matrix that is its own transpose. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V Ã— V â†’ F which is linear in both arguments. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

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. In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...

As quantization of exterior algebra

Clifford algebras are closely related to exterior algebras. In fact, if Q = 0 then the Clifford algebra Cℓ(V,Q) is just the exterior algebra Λ(V). For nonzero Q there exists a canonical linear isomorphism between Λ(V) and Cℓ(V,Q) whenever the ground field K does not have characteristic two. That is, they are naturally isomorphic as vector spaces, but with different multiplications (in the case of characteristic two, they are still isomorphic as vector spaces, just not naturally). Clifford multiplication 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. In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. ... -1... In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ... In physics, quantization is a procedure for constructing a quantum field theory starting from a classical field theory. ... In abstract algebra, a Hopf algebra is a bialgebra H over a field K together with a K-linear map such that the following diagram commutes . (Here Δ is the comultiplication of the bialgebra, ∇ its multiplication, η its unit and ε its counit. ... In abstract algebra, the Weyl algebra is the ring of differential operators with polynomial coefficients (in one variable), More precisely, let F be a field, and let F[X] be the ring of polynomials in one variable, X, with coefficients in F. Then each fi lies in F[X]. âˆ‚X... In mathematics, the symmetric algebra S(V) on a vector space V over a field K is a certain commutative unital associative K-algebra containing V. In fact, it is the most general such algebra, which can be expressed by a universal property. ...

Universal property and construction

Let V be a vector space over a field K, and let Q : V → K be a quadratic form on V. In most cases of interest the field K is either R, C or a finite field. In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ...

A Clifford algebra Cℓ(V,Q) is a unital associative algebra over K together with a linear map i : V → Cℓ(V,Q) satisfying i(v)² = Q(v)1 for all v ∈ V, defined by the following universal property: Given any associative algebra A over K and any linear map j : V → A such that In mathematics, an associative algebra is unital if it contains a multiplicative identity element, i. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ...

j(v)² = Q(v)1 for all v ∈ V

(where 1 denotes the multiplicative identity of A), there is a unique algebra homomorphism f : Cℓ(V,Q) → A such that the following diagram commutes (i.e. such that f o i = j): A homomorphism between two algebras over a field K, A and B, is a map such that for all k in K and x,y in A, F(kx)=kF(x) F(x+y)=F(x)+F(y) F(xy)=F(x)F(y) Categories: Math stubs | Algebra ... In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ...

Working with a symmetric bilinear form <·,·> instead of Q (in characteristic not 2), the requirement on j is Commutative diagram for Clifford algebra. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V Ã— V â†’ F which is linear in both arguments. ...

j(v)j(w) + j(w)j(v) = 2<v, w> for all v, w ∈ V.

A Clifford algebra as described above always exists and can be constructed as follows: start with the most general algebra that contains V, namely the tensor algebra T(V), and then enforce the fundamental identity by taking a suitable quotient. In our case we want to take the two-sided ideal I_Q in T(V) generated by all elements of the form In mathematics, the tensor algebra of a vector space V, denoted T(V) or Tâ€¢(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. ...

$votimes v - Q(v)1$ for all $vin V$

and define Cℓ(V,Q) as the quotient

Cℓ(V,Q) = T(V)/I_Q.

It is then straightforward to show that Cℓ(V,Q) contains V and satisfies the above universal property, so that Cℓ is unique up to a unique isomorphism; thus one speaks of "the" Clifford algebra Cℓ(V, Q). It also follows from this construction that i is injective. One usually drops the i and considers V as a linear subspace of Cℓ(V,Q). One-to-one redirects here. ... The concept of a linear subspace (or vector subspace) is important in linear algebra and related fields of mathematics. ...

The universal characterization of the Clifford algebra shows that the construction of Cℓ(V,Q) is functorial in nature. Namely, Cℓ can be considered as a functor from the category of vector spaces with quadratic forms (whose morphisms are linear maps preserving the quadratic form) to the category of associative algebras. The universal property guarantees that linear maps between vector spaces (preserving the quadratic form) extend uniquely to algebra homomorphisms between the associated Clifford algebras. In category theory, a branch of mathematics, a functor is a special type of mapping between categories. ... In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... A homomorphism between two algebras over a field K, A and B, is a map such that for all k in K and x,y in A, F(kx)=kF(x) F(x+y)=F(x)+F(y) F(xy)=F(x)F(y) Categories: Math stubs | Algebra ...

Basis and dimension

If the dimension of V is n and {e₁,…,e_n} is a basis of V, then the set In mathematics, the dimension of a vector space V is the cardinality (i. ... In linear algebra, a basis is a set of vectors that, in a linear combination, can represent every vector in a given vector space, and such that no element of the set can be represented as a linear combination of the others. ...

${e_{i_1}e_{i_2}cdots e_{i_k} mid 1le i_1 < i_2 < cdots < i_k le nmbox{ and } 0le kle n}$

is a basis for Cℓ(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 For other uses, see identity (disambiguation). ... In mathematics, particularly in combinatorics, a binomial coefficient is a coefficient of any of the terms in the expansion of the binomial (x+1)n. ...

$dim Cell(V,Q) = sum_{k=0}^nbegin{pmatrix}n kend{pmatrix} = 2^n.$

Since V comes equipped with a quadratic form, there is a set of privileged bases for V: the orthogonal ones. An orthogonal basis is one such that In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, an orthonormal basis of an inner product space V(i. ...

$langle e_i, e_j rangle = 0 qquad ineq j. ,$

where <·,·> is the symmetric bilinear form associated to Q. The fundamental Clifford identity implies that for an orthogonal basis

$e_ie_j = -e_je_i qquad ineq j. ,$

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). In mathematics, the permutations of a finite set (i. ... Permutation is the rearrangement of objects or symbols into distinguishable sequences. ...

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 Cℓ(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 Cℓ(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 article: geometric algebra

The most important Clifford algebras are those over real and complex vector spaces equipped with nondegenerate quadratic forms. A geometric algebra is a multilinear algebra with a geometric interpretation. ... In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2. ... In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ...

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 Cℓ_p,q(R). The symbol Cℓ_n(R) means either Cℓ_n,0(R) or Cℓ_0,n(R) depending on whether the author prefers positive definite or negative definite spaces. The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ...

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 Cℓ_p,q(R) will therefore have p vectors which square to +1 and q vectors which square to −1. In mathematics, an orthonormal basis of an inner product space V(i. ...

Note that Cℓ_0,0(R) is naturally isomorphic to R since there are no nonzero vectors. Cℓ_0,1(R) is a two-dimensional algebra generated by a single vector e₁ which squares to −1, and therefore is isomorphic to C, the field of complex numbers. The algebra Cℓ_0,2(R) is a four-dimensional algebra spanned by {1, e₁, e₂, e₁e₂}. The latter three elements square to −1 and all anticommute, and so the algebra is isomorphic to the quaternions H. The next algebra in the sequence is Cℓ_0,3(R) is an 8-dimensional algebra isomorphic to the direct sum H ⊕ H called Clifford biquaternions. In mathematics, a complex number is a number which is often formally defined to consist of an ordered pair of real numbers , often written: In mathematics, the adjective complex means that the underlying number field is complex numbers, for example complex analysis, complex matrix, complex polynomial and complex Lie algebra. ... In mathematics, the quaternions are a non-commutative extension of the complex numbers. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... A Clifford biquaternion is a concept from geometric algebra. ...

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 Cⁿ with the standard quadratic form by Cℓ_n(C). One can show that the algebra Cℓ_n(C) may be obtained as the complexification of the algebra Cℓ_p,q(R) where n = p + q: In mathematics, the complexification of a vector space V over the real number field is the corresponding vector space VC over the complex number field. ...

$Cell_n(mathbb{C}) cong Cell_{p,q}(mathbb{R})otimesmathbb{C} cong Cell(mathbb{C}^{p+q},Qotimesmathbb{C})$ .

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

Cℓ₀(C) = C

Cℓ₁(C) = C ⊕ C

Cℓ₂(C) = M₂(C)

where M₂(C) denotes the algebra of 2×2 matrices over C.

It turns out that every one of the algebras Cℓ_p,q(R) and Cℓ_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. ... In mathematics, in particular the theory of nondegenerate quadratic forms on real and complex vector spaces, finite_dimensional Clifford algebra have been completely classified. ...

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 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 independent of Q). In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...

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 Cℓ(V,Q) as described above. The map Cℓ(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 → Cℓ(V,Q) by In mathematics, the characteristic of a ring R with identity element 1R is defined to be the smallest positive integer n such that n1R = 0, where n1R is defined as 1R + ... + 1R with n summands. ...

$f_k(v_1, cdots, v_k) = frac{1}{k!}sum_{sigmain S_k}{rm sgn}(sigma), v_{sigma(1)}cdots v_{sigma(k)}$

where the sum is taken over the symmetric group on k elements. Since f_k is alternating it induces a unique linear map Λ^k(V) → Cℓ(V,Q). The direct sum of these maps gives a linear map between Λ(V) and Cℓ(V,Q). This map can be shown to be a linear isomorphism, and it is natural. In mathematics, the symmetric group on a set X, denoted by SX or Sym(X), is the group whose underlying set is the set of all bijective functions from X to X, in which the group operation is that of composition of functions, i. ... In linear algebra, a skew-symmetric (or antisymmetric) matrix is a square matrix A whose transpose is also its negative; that is, it satisfies the equation: AT = −A or in component form, if A = (aij): aij = − aji for all i and j. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ...

A more sophisticated way to view the relationship is to construct a filtration on Cℓ(V,Q). Recall that the tensor algebra T(V) has a natural filtration: F⁰ ⊂ F¹ ⊂ F² ⊂ … where F^k contains sums of tensors with rank ≤ k. Projecting this down to the Clifford algebra gives a filtration on Cℓ(V,Q). The associated graded algebra In mathematics, a filtration is an indexed set Si of subobjects of a given algebraic structure S, with an index set I that is a totally ordered set, subject only to the condition that if i ≤ j in I then Si is contained in Sj. ... In mathematics, the tensor algebra of a vector space V, denoted T(V) or Tâ€¢(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ... In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). ...

$Gr_F Cell(V,Q) = bigoplus_k F^k/F^{k-1}$

is naturally isomorphic to the exterior algebra Λ(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^k in F^k+1 for all k), this provides an isomorphism (although not a natural one) in any characteristic, even two.

Grading

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

Clifford algebras are Z₂-graded algebra (also known as superalgebras). Indeed, 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 In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). ... In mathematics and theoretical physics, a superalgebra over a field K is another name for a Z2-graded algebra over K. Specifically, a superalgebra is a super vector space A = A0 âŠ• A1 over K together with a bilinear multiplication which is an even morphism of super vector spaces. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ...

α : Cℓ(V,Q) → Cℓ(V,Q).

Since α is an involution (i.e. it squares to the identity) one can decompose Cℓ(V,Q) into positive and negative eigenspaces In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ... An identity function f is a function which doesnt have any effect: it always returns the same value that was used as its argument. ...

$Cell(V,Q) = Cell^0(V,Q) oplus Cell^1(V,Q)$

where Cℓⁱ(V,Q) = {x ∈ Cℓ(V,Q) | α(x) = (−1)ⁱx}. Since α is an automorphism it follows that

$Cell^{,i}(V,Q)Cell^{,j}(V,Q) = Cell^{,i+j}(V,Q)$

where the superscripts are read modulo 2. This gives Cℓ(V,Q) the structure of a Z₂-graded algebra. The subspace Cℓ⁰(V,Q) forms a subalgebra of Cℓ(V,Q), called the even subalgebra. The subspace Cℓ¹(V,Q) is called the odd part of Cℓ(V,Q) (it is not a subalgebra). The Z₂-grading plays an important role in the analysis and application of Clifford algebras. The automorphism α is called the main involution or grade involution. In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field (or commutative ring) with an extra piece of structure, known as a gradation (or grading). ... In universal algebra, a subalgebra of an algebra A is a subset S of A that also has the structure of an algebra of the same type when the algebraic operations are restricted to A. Since the axioms of algebraic structures in universal algebra are described by equational laws, the... In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...

Remark. In characteristic not 2 the underlying vector space of Cℓ(V,Q) inherits a Z-grading from the canonical isomorphism with the underlying vector space of the exterior algebra Λ(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 Z₂-grading: for instance if $Q(v)neq 0$ , then $vin Cell^1(V,Q)$ , but $v^2in Cell^0(V,Q)$ , not in $Cell^2(V,Q)$ . Happily, the gradings are related in the natural way: Z₂ = Z/2Z. Further, the Clifford algebra is Z-filtered: $Cell^{leq i}(V,Q) cdot Cell^{leq j}(V,Q) subset Cell^{leq i+j}(V,Q)$ . The degree of a Clifford number usually refers to the degree in the Z-grading. Elements which are pure in the Z₂-grading are simply said to be even or odd. In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. ...

The even subalgebra Cℓ⁰(V,Q) of a Clifford algebra is itself a Clifford algebra^[3]. If V is the orthogonal direct sum of a vector a of norm Q(a) and a subspace U, then Cℓ⁰(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

$Cell_{p,q}^0(mathbb{R}) cong Cell_{p,q-1}(mathbb{R})$ for q > 0, and

$Cell_{p,q}^0(mathbb{R}) cong Cell_{q,p-1}(mathbb{R})$ for p > 0.

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

R ⊂ C ⊂ H ⊂ H⊕H ⊂ …

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

Antiautomorphisms

In addition to the automorphism α, there are two antiautomorphisms 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: In mathematics, an antihomomorphism is a type of function defined on sets with multiplication that reverses the order of multiplication. ... In mathematics, the tensor algebra of a vector space V, denoted T(V) or Tâ€¢(V), is the algebra of tensors on V (of any rank) with multiplication being the tensor product. ...

$v_1otimes v_2otimes cdots otimes v_k mapsto v_kotimes cdots otimes v_2otimes v_1$ .

Since the ideal I_Q is invariant under this reversal, this operation descends to an antiautomorphism of Cℓ(V,Q) called the transpose or reversal operation, denoted by x^t. The transpose is an antiautomorphism: $(x y) t = y t x t$ . 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 $bar x$ In informal language, a transposition is a function that swaps two elements of a set. ...

$bar x = alpha(x^t) = alpha(x)^t.$

Of the two antiautomorphisms, the transpose is the more fundamental.^[4]

Note that all of these operations are involutions. One can show that they act as ±1 on elements which are pure in the Z-grading. In fact, all three operations depend only on the degree modulo 4. That is, if x is pure with degree k then In mathematics, an involution is a function that is its own inverse, so that f(f(x)) = x for all x in the domain of f. ...

$alpha(x) = pm x qquad x^t = pm x qquad bar x = pm x$

where the signs are given by the following table:

k mod 4	0	1	2	3
$alpha(x),$	+	−	+	−	(−1)^k
$x^t,$	+	+	−	−	(−1)^k(k−1)/2
$bar x$	+	−	−	+	(−1)^k(k+1)/2

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 Cℓ(V,Q) as explained earlier (which we also denoted by Q). A basis independent definition is

$2Q(x) = lang x^t xrang$

where <a> denotes the scalar part of a (the grade 0 part in the Z-grading). One can show that

$Q(v_1v_2cdots v_k) = Q(v_1)Q(v_2)cdots Q(v_k)$

where the v_i 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

$lang x, yrang = lang x^t yrang.$

One can check that this reduces to the original bilinear form when restricted to V. The bilinear form on all of Cℓ(V,Q) is nondegenerate if and only if it is nondegenerate on V. In mathematics, a degenerate bilinear form f(x,y) on a vector space V is one such that for some non-zero x in V for all y ∈ V. A nondegenerate form is one that is not degenerate. ...

It is not hard to verify that the transpose is the adjoint of left/right Clifford multiplication with respect to this inner product. That is, In mathematics, the conjugate transpose or adjoint of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A by taking the transpose and then taking the complex conjugate of each entry. ...

$lang ax, yrang = lang x, a^t yrang,$ and

$lang xa, yrang = lang x, y a^trang.$

Structure 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. In ring theory and related areas of mathematics a central simple algebra (CSA) over K, also called a Brauer algebra after Richard Brauer, is a finite-dimensional (associative) algebra A, which is a simple ring, and for which the center is exactly K. For example, the complex numbers C form...

If V has even dimension then Cℓ(V,Q) is a central simple algebra over K.
If V has even dimension then Cℓ⁰(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 Cℓ⁰(V,Q) is a central simple algebra over K.

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

$Cl_{p+2,q}(mathbb{R}) = M_2(mathbb{R})otimes Cl_{q,p}(mathbb{R})$

$Cl_{p+1,q+1}(mathbb{R}) = M_2(mathbb{R})otimes Cl_{p,q}(mathbb{R})$

$Cl_{p,q+2}(mathbb{R}) = mathbb{H}otimes Cl_{q,p}(mathbb{R})$

These formulas can be used to find the structure of all real Clifford algebras; see the classification of Clifford algebras. In mathematics, in particular the theory of nondegenerate quadratic forms on real and complex vector spaces, finite_dimensional Clifford algebra have been completely classified. ...

Notably, the Morita equivalence class of a Clifford algebra (its representation theory: the equivalence class of the category of modules over it) depends only on the signature $p - q$ mod 8. This is an algebraic form of Bott periodicity. In abstract algebra, Morita equivalence is a relationship defined between rings that preserves many ring-theoretic properties. ... In mathematics, the Bott periodicity theorem is a result from homotopy theory which was discovered by Raoul Bott during the latter part of the 1950s, and proved to be of foundational significance for much further research, in particular in K-theory. ...

The Clifford group Γ

In this section we assume that V is finite dimensional and the bilinear form of Q is non-singular.

The Clifford group Γ is defined to be the set of invertible elements x of the Clifford algebra such that

$x v alpha(x)^{-1}in V$

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 − <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 xvα(x)⁻¹ by xvx⁻¹. This produces the same Clifford group, but the action of the Clifford group on V is changed slightly: the action of the odd elements Γ¹ of the Clifford group is multiplied by an extra factor of −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^*. Using the action α(x)vx⁻¹ instead of xvα(x)⁻¹ makes no difference: it produces the same Clifford group with the same action on V.

The Clifford group Γ is the disjoint union of two subsets Γ⁰ and Γ¹, where Γⁱ is the subset of elements of degree i. The subset Γ⁰ is a subgroup of index 2 in Γ.

If V is finite dimensional with nondegenerate bilinear form then the Clifford group maps onto the orthogonal group of V (by the Cartan-Dieudonné theorem) and the kernel consists of the nonzero elements of the field K. This leads to exact sequences In mathematics, the Cartan-DieudonnÃ© theorem, named after Ã‰lie Cartan and Jean DieudonnÃ©, is a theorem on the structure of the automorphism group of symmetric bilinear spaces. ...

$1 rightarrow K^* rightarrow Gamma rightarrow O_V(K) rightarrow 1,,$

$1 rightarrow K^* rightarrow Gamma^0 rightarrow SO_V(K) rightarrow 1.,$

In arbitrary characteristic, the spinor norm Q is defined on the Clifford group by In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

$Q(x) = x^tx,$

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.

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 of a vector r has image Q(r) in K^*/K^*2, and this property uniquely defines it on the orthogonal group. This gives exact sequences:

$1 rightarrow {pm 1} rightarrow Pin_V(K) rightarrow O_V(K) rightarrow K^*/K^{*2},,$

$1 rightarrow {pm 1} rightarrow Spin_V(K) rightarrow SO_V(K) rightarrow K^*/K^{*2}.,$

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

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 Pin group Pin_V(K) is the subgroup of the Clifford group Γ of elements of spinor norm 1, and similarly the Spin group Spin_V(K) is the subgroup of elements of Dickson invariant 0 in Pin_V(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. ... In mathematics the spin group Spin(n) is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie groups For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n). ... In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

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 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. In mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group of n-by-n orthogonal matrices with entries from F, with the group operation that of matrix multiplication. ...

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. Please note, however, that this is not true in general: if V is R^p,q for p and q both at least 2 then the spin group is not simply connected. In this case the algebraic group Spin_p,q is simply connected as an algebraic group, even though its group of real valued points Spin_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.

See spinor group, spinor. In mathematics the spinor group or spin group Spin(n) is a particular double cover of the special orthogonal group SO(n, R). ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Spinors

Clifford algebras Cℓ_p,q(C), with p+q=2n even, are matrix algebras which have a complex representation of dimension 2ⁿ. By restricting to the group Pin_p,q(R) we get a complex representation of the Pin group of the same dimension, called the spinor representation. If we restrict this to the spin group Spin_p,q(R) then it splits as the sum of two half spin representations (or Weyl representations) of dimension 2^n-1. In mathematics, a spinor representation is a particular kind of projective representation of a special orthogonal group, or orthogonal group. ...

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 Pin_p,q(R). On restriction to the spin group Spin_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 the 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 spinors. In mathematics, in particular the theory of nondegenerate quadratic forms on real and complex vector spaces, finite_dimensional Clifford algebra have been completely classified. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Real spinors

For more details on this topic, see spinor.

To describe the real spin representations, one must know how the spin group sits inside its Clifford algebra. The Pin group, Pin_p,q is the set of invertible elements in Cl_p,q which can be written as a product of unit vectors: To meet Wikipedias quality standards, this article or section may require cleanup. ... ...

${mathit{Pin}}_{p,q}={v_1v_2dots v_r |,, forall i, |v_i|=pm 1}$

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 Spin group consists of those elements of Pin_p,q 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). In mathematics the spin group Spin(n) is the double cover of the special orthogonal group SO(n), such that there exists a short exact sequence of Lie groups For n > 2, Spin(n) is simply connected and so coincides with the universal cover of SO(n). ... In mathematics, the Cartan-DieudonnÃ© theorem, named after Ã‰lie Cartan and Jean DieudonnÃ©, is a theorem on the structure of the automorphism group of symmetric bilinear spaces. ...

Let α : Cℓ → Cℓ be the automorphism which is given by -Id acting on pure vectors. Then in particular, Spin_p,q is the subgroup of Pin_p,q whose elements are fixed by α. Let

$Cl_{p,q}^0 = { xin Cl_{p,q} |, alpha(x)=x}$ .

(These are precisely the elements of even degree in Cℓ_p,q.) Then the spin group lies within Cℓ⁰_p,q.

The irreducible representations of Cℓ_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 Cℓ⁰_p,q

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) In mathematics, in particular the theory of nondegenerate quadratic forms on real and complex vector spaces, finite_dimensional Clifford algebra have been completely classified. ...

Cℓ⁰_p,q ≈ Cℓ_p,q-1, for q > 0

Cℓ⁰_p,q ≈ Cℓ_q,p-1, 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 bundle of differential forms on a smooth manifold. In the case of a (pseudo-)Riemannian manifold, the tangent spaces come equipped with a natural quadratic form induced by the metric. Thus, one can define a Clifford bundle in analogy with the exterior bundle. This has a number of important applications in Riemannian geometry. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In mathematics, in particular in topology, a fiber bundle (or fibre bundle) is a space which locally looks like a product of two spaces but may possess a different global structure. ... A differential form is a mathematical concept in the fields of multivariate calculus, differential topology and tensors. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In mathematics, a Clifford bundle is a algebra bundle whose fibers have the structure of a Clifford algebra and whose local trivializations respect the algebra structure. ... In mathematics, the exterior bundle of a manifold M is the subbundle of the tensor bundle consisting of all antisymmetric covariant tensors. ... In differential geometry, Riemannian geometry is the study of smooth manifolds with Riemannian metrics, i. ...

Physics

Clifford algebras have numerous important applications in physics. Physicists usually consider a Clifford algebra to be an algebra spanned by matrices γ₁,…,γ_n called Dirac matrices which have the property that The Dirac equation is a relativistic quantum mechanical wave equation invented by Paul Dirac in 1928. ...

$gamma_igamma_j + gamma_jgamma_i = 2eta_{ij},$

where η is the matrix of a quadratic form of signature (p,q) — typically (1,3) when working in Minkowski space. These are exactly the defining relations for the Clifford algebra Cl_1,3(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. In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einsteins theory of special relativity is most conveniently formulated. ... In mathematics, in particular the theory of nondegenerate quadratic forms on real and complex vector spaces, finite_dimensional Clifford algebra have been completely classified. ...

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 bilinears. Paul Adrien Maurice Dirac, OM, FRS (IPA: [dÉªrÃ¦k]) (August 8, 1902 â€“ October 20, 1984) was a British theoretical physicist and a founder of the field of quantum physics. ... For other uses, see Electron (disambiguation). ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ... Quantum field theory (QFT) is the quantum theory of fields. ... In physics, the Dirac equation is a relativistic quantum mechanical wave equation formulated by British physicist Paul Dirac in 1928 and provides a description of elementary spin-Â½ particles, such as electrons, consistent with both the principles of quantum mechanics and the theory of special relativity. ...

Footnotes

^ Mathematicians who work with real Clifford algebras and prefer positive definite quadratic forms (especially those working in index theory) sometimes use a different choice of sign in the fundamental Clifford identity. That is, they take v² = −Q(v). One must replace Q with −Q in going from one convention to the other.
^ Thus the group algebra K[Z/2] is semisimple and the Clifford algebra splits into eigenspaces of the main involution.
^ We are still assuming that the characteristic is not 2.
^ The opposite is true when uses the alternate (−) 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 $v - 1 = v t / Q (v)$ while in the (−) convention it is given by $v^{-1} = bar{v}/Q(v)$ .

In physics, a sign convention is a choice of the signs (plus or minus) of a set of quantities, in a case where the choice of sign is arbitrary. ... In the theory of group representations, the group algebra is any of various constructions to assign to a group (either a locally compact topological group, or a group without a topology, i. ... In mathematics, the term semisimple is used in a number of related ways, within different subjects. ...

References

Carnahan, S. Borcherds Seminar Notes, Uncut. Week 5, "Spinors and Clifford Algebras".
Lawson and Michelsohn, Spin Geometry, Princeton University Press. 1989. ISBN 0-691-08542-0. An advanced textbook on Clifford algebras and their applications to differential geometry.
Lounesto, P., Clifford Algebras and Spinors, Cambridge University Press. 2001. ISBN 0-521-00551-5.
Porteous, I., Clifford Algebras and the Classical Groups, Cambridge University Press. 1995. ISBN 0-521-55177-3.

External links

Planetmath entry on Clifford algebras
A history of Clifford algebras (unverified)
John Baez on Clifford algebras

Categories: Clifford algebras | Ring theory | Quadratic forms

Results from FactBites:

Encyclopedia4U - Clifford algebra - Encyclopedia Article (502 words)
	Clifford algebras are associative algebras of importance in mathematics, in particular in the theories of quadratic forms and of orthogonal groups, and in physics.
	The associated graded algebra is canonically isomorphic to the exterior algebra Λ V of the vectorspace.
	In case the field k is the field of real numbers the Clifford algebra of a quadratic form of signature p,q is usually denoted C(p,q).

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Contents

Introduction and basic properties

As quantization of exterior algebra

Universal property and construction

Basis and dimension

Examples: Real and complex Clifford algebras

Properties

Relation to the exterior algebra

Grading

Antiautomorphisms

The Clifford scalar product

Structure of Clifford algebras

The Clifford group Γ

Spin and Pin groups

Spinors

Real spinors

Applications

Differential geometry

Physics

See also

Footnotes

References

External links

Contents

Introduction and basic properties GA_googleFillSlot("encyclopedia_square");

As quantization of exterior algebra

Universal property and construction

Basis and dimension

Examples: Real and complex Clifford algebras

Properties

Relation to the exterior algebra

Grading

Antiautomorphisms

The Clifford scalar product

Structure of Clifford algebras

The Clifford group Γ

Spin and Pin groups

Spinors

Real spinors

Applications

Differential geometry

Physics

See also

Footnotes

References

External links

Introduction and basic properties