 | Clifford algebra: Encyclopedia II - Clifford algebra - Properties
Clifford algebra - Properties
Clifford algebra - 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. This is an algebra isomorphism if and only if Q = 0. One can thus consider the Clifford algebra Cℓ(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 {ei} 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
Note that this only works if the basis {ei} 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 fk : V × … × V → Cℓ(V,Q) by
where the sum is taken over the symmetric group on k elements. Since fk is alternating it induces a unique linear map Λk(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.
Yet another way to see the relation is to construct a filtration on Cℓ(V,Q). Recall that the tensor algebra T(V) has a natural filtration: F0 ⊂ F1 ⊂ F2 ⊂ … where Fk 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
is naturally isomorphic to the exterior algebra Λ(V).
Clifford algebra - Grading
The linear map on V defined by preserves the quadratic form Q and so by the universal property of Clifford algebras extends to an algebra automorphism
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
where Cℓi(V,Q) = {x ∈ Cℓ(V,Q) | α(x) = (−1)ix}. Since α is an automorphism it follows that
where the superscripts are read modulo 2. This means that Cℓ(V,Q) is a Z2-graded algebra (also known as a superalgebra). Note that Cℓ0(V,Q) forms a subalgebra of Cℓ(V,Q), called the even subalgebra. The piece Cℓ1(V,Q) is called the odd part of Cℓ(V,Q) (it is not a subalgebra). This Z2-grading plays an important role in the analysis and application of Clifford algebras. The automorphism α is called the main involution or grade involution.
Remark. In characteristic not 2 the algebra Cℓ(V,Q) inherits a Z-grading from the canonical isomorphism with 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. Happily, the gradings are related in the natural way: Z2 = Z/2Z. 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.
If the characteristic of F is not 2 then the even subalgebra Cℓ0(V,Q) of a Clifford algebra is itself a Clifford algebra. 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
for q > 0, and
for p > 0.
In the negative-definite case this gives an inclusion Cℓ0,n−1(R) ⊂ Cℓ0, n(R) which extends the sequence
R ⊂ C ⊂ H ⊂ H⊕H ⊂ …
Likewise, in the complex case, one can show that the even subalgebra of Cℓn(C) is isomorphic to Cℓn−1(C).
Clifford algebra - 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:
.
Since the ideal IQ is invariant under this reversal, this operation descends to an antiautomorphism of Cℓ(V,Q) called the transpose or reversal operation, denoted by xt. The transpose is an antiautomorphism: (xy)t = ytxt. 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
Of the two antiautomorphisms, the transpose is the more fundamental.3
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
where the signs are given by the following table:
Clifford algebra - 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
where <a> denotes the scalar part of a (the grade 0 part in the Z-grading). One can show that
where the vi are elements of V — this identity is not true for arbitrary elements of Cℓ(V,Q).
The associated symmetric bilinear form on Cℓ(V,Q) is given by
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 it is nondegenerate on V.
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,
and
Other related archivesn choose k, Clifford, Dickson invariant, Dirac equation, Dirac matrices, Minkowski space, Paul Dirac, Riemannian geometry, Riemannian manifold, William Clifford, adjoint, algebra homomorphism, algebra homomorphisms, alternating, antiautomorphisms, associative algebra, automorphism, basis, bilinear form, biquaternions, bundle, category, central simple algebra, characteristic, choice of sign, classification of Clifford algebras, commutes, complex, complex numbers, complexification, differential forms, differential geometry, dimension, direct sum, discriminant, electron, exterior algebra, exterior algebras, exterior product, field, filtration, functor, gamma matrices, geometric algebra, geometry, graded algebra, identity, identity element, injective, involution, involutions, linear map, linear subspace, mathematics, matrix algebra, metric, morphisms, multilinear algebra, natural isomorphism, naturally isomorphic, nondegenerate, orthogonal, orthogonal basis, orthogonal transformations, orthonormal basis, paravector, permutation, pseudo, quadratic form, quadratic forms, quantum field theory, quaternions, quotient, real, representations of Clifford algebras, signature, smooth manifold, special orthogonal group, spinor, spinor group, spinors, structure of the corresponding Clifford algebras, subalgebra, superalgebra, symmetric, symmetric group, tangent spaces, tensor algebra, theoretical physics, two-sided ideal, unital, universal property, vector space
 Adapted from the Wikipedia article "Properties", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
