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Clifford algebra

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Definition

Given a commutative ring RR and RR-modules MM and NN, an RR-quadratic function? on MM with values in NN is a map q:MNq: M \to N such that the following properties hold: * (cube relation) For any x,y,zMx,y,z \in M, we have q(x+y+z)q(x+y)q(x+z)q(y+z)+q(x)+q(y)+q(z)=0q(x+y+z) - q(x+y) - q(x+z) - q(y+z) + q(x) + q(y) + q(z) = 0. * (homogeneous of degree 2) For any xMx \in M and any rRr \in R, we have q(rx)=r 2q(x)q(rx) = r^2q(x).

A quadratic RR-module is an RR-module MM equipped with a quadratic form: an RR-quadratic function on MM with values in RR.

The Clifford algebra Cl(M,q)Cl(M,q) of a quadratic RR-module (M,q)(M,q) can be defined as the quotient of the tensor algebra T R(M)T_R(M) by the ideal generated by the relations xxq(x)x \otimes x - q(x) for all xMx \in M.

Equivalently, it is the initial object in the category whose objects are pairs (A,ϕ)(A,\phi) where AA is an associative unital RR-algebra, and ϕ:MA\phi: M \to A is an RR-linear map satisfying ϕ(x) 2=q(x)1 A\phi(x)^2 = q(x) 1_A for all xMx \in M, and whose morphisms (A,ϕ)(A,ϕ)(A,\phi)\to (A',\phi') are the associative RR-algebra maps χ:AA\chi: A\to A' such that χϕ=ϕ\chi\circ\phi=\phi'.

Examples

Examples in low rank can be calculated easily. If MM is freely generated by a single element ee, with quadratic form q(e)=1q(e) = 1, then Cl(M,q)=R[e]/(e 21)Cl(M,q) = R[e]/(e^2-1). Note that the opposite sign convention is often used in the differential geometry literature, so one may see the assertion that the Clifford algebra of the real line with a positive definite metric is isomorphic to the complex numbers [e]/(e 2+1)\mathbb{R}[e]/(e^2+1). Similarly, the Clifford algebra of a negative definite two dimensional real vector space is isomorphic to the (non-split) quaternions in our convention, but one may see the assertion that it is isomorphic to M 2()M_2(\mathbb{R}). Complexification removes the difference between positive definite and negative definite, and the two complexified algebras are isomorphic.

Let M=LL M = L \oplus L^\vee for LL projective of rank dd over RR, and L =Hom R(L,R)L^\vee = Hom_R(L,R) the dual module. One can define the canonical quadratic form q(f+x)=f(x)q(f+x) = f(x) for fL f \in L^\vee and xLx \in L. In this case, Cl(M,q)M 2 d(R)Cl(M,q) \cong M_{2^d}(R). In general, the Clifford algebra arising from a nondegenerate form is flat-locally (on SpecR\operatorname{Spec} R) isomorphic to a matrix algebra (when rank is even) or a direct sum of two matrix algebras (when rank is odd).

If MM is a projective RR-module of rank dd, then independently of qq, the Clifford algebra Cl(M,q)Cl(M,q) is projective of rank 2 d2^d, and is (noncanonically) isomorphic to M\bigwedge M as an RR-module equipped with a map from MM. The Clifford algebra is isomorphic to the exterior algebra (as algebras equipped with RR-module maps from MM) if and only if q=0q = 0.

If RR is the ring of smooth functions on a pseudo-Riemannian manifold XX, and MM is the RR-module of sections of the tangent bundle, then the metric endows MM with a quadratic structure, and one can form the Clifford algebra of the tangent bundle.

Properties

Classification

Theorem

Let VV be the vector space over the complex numbers of complex dimension dd, equipped with non-degenerate bilinear form, unique up to isomorphism. The Clifford algebra

Cl d()Cl(V) Cl_{d}(\mathbb{C}) \coloneqq Cl(V)

is isomorphic, as a complex associative algebra to a matrix algebra as follows:

Cl d(){Mat 2 d2() fordeven Mat 2 d12()Mat 2 d12() fordodd Cl_d(\mathbb{C}) \simeq \left\{ \array{ Mat_{2^{\tfrac{d}{2}}}(\mathbb{C}) & for \, d \, even \\ Mat_{2^{\tfrac{d-1}{2}}}(\mathbb{C}) \oplus Mat_{2^{\tfrac{d-1}{2}}}(\mathbb{C}) & for \, d \, odd } \right.
Remark

This is one of the incarnations of Bott periodicity.

As a superalgebra

While the tensor algebra of an RR-module MM has a natural integer grading, the quadratic relation collapses this to a natural /2\mathbb{Z}/2\mathbb{Z}-grading on Cl(M,q)Cl(M,q).

Cl(M,q)=Cl(M,q) evCl(M,q) odd Cl(M,q) = Cl(M,q)^{ev} \oplus Cl(M,q)^{odd}

When MM is projective of rank dd, each homogeneous piece is projective of rank 2 d12^{d-1}. When qq is nondegenerate, the even part of the Clifford algebra is also flat-locally isomorphic to a matrix ring or a sum of two matrix rings.

One can view the Clifford algebra multiplication as a quantization of the commutative super algebra RM\bigwedge_R M.

Relation to exterior algebra (quantization)

For VV an inner product space, the symbol map (see there) constitutes an isomorphism of the underlying super vector spaces of the Clifford algebra with the exterior algebra on VV.

One can understand the Clifford algebra as the quantization Grassmann algebra induced from the inner product regarded as an odd symplectic form.

Relation to SpinSpin groups

Let MM be a projective RR-module of finite rank, and let qq be nondegenerate. Write Cl(M,q) ×Cl(M,q)^\times for the group of units of the Clifford algebra Cl(M,q)Cl(M,q).

The Clifford group Γ M,q(R)\Gamma_{M,q}(R) is the subgroup of elements xx for which twisted conjugation stabilizes the submodule MCl(M,q)M \subset Cl(M,q). Here, twisted conjugation is defined by yxyα(x) 1y \mapsto xy\alpha(x)^{-1}, where α\alpha is the automorphism of CL(M,q)CL(M,q) induced by the 1-1 map on MM. Since twisted conjugation by MM-stabilizing elements amounts to reflection yy2(x,y)q(x)xy \mapsto y - 2\frac{(x,y)}{q(x)}x, there is a canonical map Γ M,q(R)O(M,q)\Gamma_{M,q}(R) \to O(M,q), and the Clifford group is in fact a central extension of the orthogonal group by R ×R^\times.

The Clifford group is made of homogeneous elements in the /2\mathbb{Z}/2\mathbb{Z}-grading, and the subgroup of even elements is a normal subgroup of index two. One also has a spinor norm Q:Γ M,q(R)R ×Q: \Gamma_{M,q}(R) \to R^\times on the Clifford group, defined by Q(x)=x txQ(x) = x^tx, where xx tx \mapsto x^t is the anti-involution of the Clifford algebra defined by opposite multiplication in the tensor algebra.

The Pin group Pin M,q(R)Pin_{M,q}(R) is the group elements of the Clifford group with spinor norm 1. The Spin group Spin M,q(R)Spin_{M,q}(R) is the group of elements in the even subgroup of the Clifford group with spinor norm 1.

The restriction of the map Γ M,q(R)O(M,q)\Gamma_{M,q}(R) \to O(M,q) to the Pin group may not be surjective, but it is for positive definite real vector spaces. The kernel is the group μ 2(R)\mu_2(R) of elements of RR that square to 1. Similarly, the Spin group has a map to the special orthogonal group with kernel μ 2(R)\mu_2(R), but it may not be surjective in general.

One can use base change to define the groups given above as functors on commutative RR-algebras.

Spinors

For nondegenerate quadratic forms on real vector spaces, spinors/spin representations are distinguished linear representations of Spin groups? that are not pulled back from the corresponding special orthogonal groups. In other words, the central element 1-1 acts nontrivially. They can be realized as restrictions of representations of the even parts of Clifford algebras. Since even parts of Clifford algebras are (up to complexification) the sum of one or two matrix rings, their representation theory is quite simple.

The specific nature of spinor representations possible depends on the signature of the vector space modulo 8. This is a manifestation of Bott periodicity. One always has a Dirac spinor - the fundamental (spin) representation of the complexified Clifford algebra. In even dimensions, this splits into two Weyl spinors (called half-spin representations). One may also have real representations called Majorana spinors, and these may decompose into Majorana-Weyl spinors.

There are infinite dimensional Clifford algebra constructions that appear in conformal field theory. One may extend the above discussion to topological RR-modules and continuous quadratic forms, and one obtains canonical central extensions of infinite dimensional groups and algebras by a relative determinant construction. Semi-infinite wedge spaces are spinor modules for Clifford algebras of quadratic Tate RR-modules.

Warnings

There is a difference of sign convention between differential geometers (following Atiyah) and everyone else.

Clifford algebras are often defined using bilinear forms instead of quadratic forms (and one often sees incorrect definitions of quadratic forms in terms of bilinear forms). Such definitions will yield wrong (or boring) objects when 2 is not invertible.

Special orthogonal groups are often defined as the kernel of the determinant map on the corresponding orthogonal groups, but in characteristic 2, the determinant is trivial, while the Clifford grading (called the Dickson invariant) is not.

The Clifford group is sometimes defined without the twist in the conjugation, and this means the map to the orthogonal group may not be a surjection, and the action on MM is by negative reflections.

The spinor norm is sometimes defined with the opposite sign.

Special orthogonal groups over the reals are sometimes defined to be the connected component of the identity in the orthogonal group. In indefinite signature, this defines an index two subgroup of the special orthogonal group.

Spin groups in signature (m,n)(m,n) for m,n2m,n \geq 2 have fundamental groups of order two. They are simply connected when mm or nn is at most one.

References

  • William Clifford, Applications of Grassmann’s extensive algebra. American Journal of Mathematics 1 (4): 350–358. (1878)doi:10.2307/2369379.

For some standard material see for instance

  • Eckhard Meinrenken, Clifford algebras and Lie groups (pdf)

  • H. Blaine Lawson, Marie-Louise Michelsohn, (1989), Spin Geometry, Princeton University Press, ISBN 0-691-08542-0.

For a program that promotes the use of Clifford algebra as a good expositional tool in introductory mechanics see Geometric Algebra.

Revised on November 24, 2014 19:08:56 by Urs Schreiber (82.224.164.72)