Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

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associated with the so-called ‘‘elliptic r-matrix.’’

Set

¼ diagð1;";...;"

n1

Þ;

01 ... 0

10... 0

;"¼ e

2i=n

½12

Put Z

= Z=n Z  Z=n Z; for a = (a

, a

) 2 Z

set I

= I

; mat rices I

define an irreducible

projective representation of Z

(they form the so-

called ‘‘finite Heisenberg group)’’. Let us denote

the elliptic curve of modulus  by E = C=Z þ Z

and let P !E be the n-dimensional holomorphic

vector bundle with flat connection and with

monodromies given by

z 7!z þ1 : h

¼ Ad I

; z 7!z þ : h

¼ Ad I

Let G

 Lg be the subspace of Laurent expansions

at zero of the global meromorphic sections of P

with a unique pole at 0 2E.Then(Lg, Lg

, G

)is

again a Manin triple. The associated classical

r-matrix is the kernel of a singular integral operator

which associates a meromorphic section of P to its

principal part at 0. Explicitly, it is given by

rðz  z

Þ¼

n1

a;b¼0



z  z

 a  b



ðAd I

a;b

 IÞt

½13

where  is the Weierstrass zeta function.

5. Let g be an arbitrary semisimple Lie algebra

again. Let us equip the loop algebra Lg with the

inner product

hhX; Yii

¼ Res

z¼0

hXðzÞ; YðzÞiz

1

Set N

= n

þ g  zC[[z]], N



= n



þ g 

1

C[z

1

]. We have Lg = N

þh

þN



, where

we identify h, n



 g with the corresponding

subalgebras of constant loops in Lg. Let P



, P

be the projection operators onto N



, h in this

decomposition and r



=  P



 (1=2)P

. The

classical r-matrices r



define on Lg the structure

of a factorizable Lie bialgebra. The associated

tensor kernels are called the trigonometric classi-

cal r-matrices.

Classical r-matrices described above are associated

with factorization problems in the infinite-dimensional

loop groups: matrix Riemann problems or matrix

Cousin problems (in the elliptic case). Belavin and

Drinfeld have given a complete classification of

factorizable Lie bialgebra structures for semisimple

Lie algebras; in the loop algebra case, the problem they

solved consists of classification of all meromorphic

solutions of the classical Yang–Baxter equation. In

other words, we assume that the distribution kernel

associated with the classical r-matrix is represented by

a meromorphic function (of two complex variables).

Up to an equivalence, any such solution depends

only on one variable and belongs to the rational,

trigonometric, or elliptic type (in the latter case, the

underlying Lie algebra is necessarily sl( n)). Classifi-

cation of solutions in the elliptic case is completely

rigid; in the trigonometric case, the moduli space is

finite dimensional and admits an explicit descrip-

tion. In the rational case, the classification is

somewhat less explicit (it has been completed by A

Stolin under some nondegeneracy condition). Con-

trary to to the popular belief, there are many other

structures of a factorizable Lie bialgebra on loop

algebras, for which the associated r-matrices are

given by more singular distribution kernels.

Poisson Lie Groups

If the tangent Lie bialgebra of a Poisson Lie group is

of coboundary type, the cocycle  is also trivial,

(g ) = r  Ad g  Ad g  r. Hence, the Poisson

bracket on G is given by

f’; g¼hr; r

’ ^r

ihr; r’ ^r i; r 2 g ^ g

where r’, r

’ 2 g



are left and right differentials of

’ 2 C

(G). This is the so-called ‘‘Sklyanin bracket’’.

Let us assume that G is a matrix group; its affine

ring generated by evaluation functions 

which

assign to L 2 G its matrix coefficients, 

(L) = L

The Poisson bracket on G is completely determined

by its values on 

. Explicitly, we get



;



ðLÞ¼½r; L  L

ikjm

½14

the commutator in the RHS is in Mat(n

). By a

variation of languag e, evaluating functions and their

values on a generic element L 2 G are denoted by

the same letter; using tensor notation to suppress

matrix indices, we get

; L

¼ r; L

½; L

¼ L I ; L

¼ I L ½15

In the case of loop algebras, these Poisson bracket

relations take the form

ðÞ; L

ðÞfg¼½rð; Þ; L

ðÞL

ðÞ

Let us assume that G is factorizable and the

associated factorization problem is globally solvable.

The Poisson bracket on the dual group G



’

Classical r-Matrices, Lie Bialgebras, and Poisson Lie Groups 515



)  G  G may be characterized in terms of the

matrix coefficients of (h

, h



) = i

(h), or of their

quotient h = h

1



. Explicitly, we get



; h







¼ r; h





; h







¼ r

; h





½16

; h



¼ rh

þ h

r  h

 h



;

r ¼

þ r



ðÞ

½17

The key question in the geometry of Poisson

groups consists in description of symplectic leaves in

G, G



. This question is already nontrivial when G



abelian (and hence may be identified with the dual of

the Lie algebra g = Lie(G)). The Poisson bracket on



is linear; this is the well-known Lie–Poisson (alias,

Beresin–Kirillov–Kostant) bracket. Its symplectic

leaves coincide with the orbits of the coadjoint

representation of G in g



. The natural way to prove

this fundamental result (which goes back to Lie) is to

consider first the natural action of G on the

cotangent bundle T



G ’ G  g



;thisactionis

Hamiltonian, and the coadjoint orbits arise as a

result of Hamiltonian reduction associated with this

action. The generalization of the theory of coadjoint

orbits to the case of arbitrary Poisson groups starts

with the notion of symplectic double, which is the

nonlinear analog of the cotangent bundle.

Let D be the double of (G, G



); assume for

simplicity that D = G  G



globally and hence the

associated factorization problem is always solvable.

Let r

= (1=2)(P

 P



). Set

f’; g



¼hr

r’; r ihr

’; r

i½18

The bracket { , }



is the usual Sklyanin bracket which

defines the structure of a Poisson group on D,while

{,}

is nondegenerate and defines a symplectic

structure on D. Let us denote the copies of D equipped

with the bracket { , }



by D



. The bracket on D

is not

multiplicative, but it is covariant with respect to the

action of D



by left and right translations; in other

words, the natural mappings D



 D

and

 D



, associated with multiplication in D,

preserve Poisson brackets. Since G,G



 D



are

Poisson subgroups, natural actions G  D

and G



 D

by left and right translations are

Poisson mappings. Consider the natural projections





’ D=GGnD ’ G



G ’ D=G



nD ’ G

onto the space of left and right coset classes. It is easy

to see that functions on D

which are constant on each

projection fiber are closed with respect to the Poisson

bracket. This means that the quotient spaces inherit

the Poisson structure. Moreover, the maps , 

and

p, p

form the so-called ‘‘dual pairs’’, that is, the

algebras of functions which are constant on the fibers

of  and 

(or of p and p

) are mutual centralizers of

one another in the big Poisson algebra F(D

Since D = G  G



= G



 G, we have G



=D ’ G,

G=D ’ G



; it is easy to check that the quotient

Poisson structure induced on G, G



coincides with

the original one. Applying the fundamental theorem

on dual pairs of Poisson mappings (going back to S.

Lie), we conclude that symplectic leaves in G and G



respectively, coincide with the orbits of G



(respec-

tively, G) in these quotient spaces. The actions G 



, G



 G !G are called ‘ ‘dressing transfor-

mations’’. Unit elements in G and G



are fixed points

of dressing transformations; their linearizations at the

tangent spaces T



’ g



, T

G ’ g coincide with the

coadjoint actions of G and G



,respectively.

When D 6¼ G  G



(i.e., the factorization problem in

D is not always solvable), dressing actions are still well

defined as global transformations of the quotient

spaces; in this case G, G



may be identified with open

cells in D=G



, D=G, respectively, which means that

dressing action on G, G



is,ingeneral,incomplete.

If the group G is factorizable, symplectic leaves in the

dual group G



admit a nice uniform description: since

in this case D = G  G and G  D is the diagonal

subgroup, the quotient D=G may be modeled on G

itself. The quotient Poisson bracket in this realization

coincides with [17], while the dressing action coin-

cides with conjugation in G (and is independent of

r). Hence, symplectic leaves in D/G coincide with

conjugacy classes in G; the equivalence of this model

with G



(equipped with the bracket [16])isprovided

by the factorization map. The description of sym-

plectic leaves in G is more subtle (and already

crucially depends on the choice of r!); for semisimple

Lie groups with the standard Poisson structure, it is

related to the geometry of double Bruhat cells.

For loop groups with rational, trigonometric, or

elliptic r-matrices, dressing action is associated with

auxiliary factorization problems in the loop group.

Roughly speaking, symplectic leaves correspond to

rational loops with prescribed singularities. Many

important examples have been described in connection

with integrable lattice systems, although a complete

classification theorem is still not available. For

g = sl(2), the elliptic Manin triple described earlier

leads to the Poisson structure on the group of ‘‘elliptic

loops’’ with values in SL(2); its simplest symplectic

leaves (corresponding to loops with simple poles) are

associated with a remarkable Poisson algebra, the

Sklyanin algebra (with four generators and two

Casimir functions), which admits an interesting

explicit quantization.

516 Classical r-Matrices, Lie Bialgebras, and Poisson Lie Groups

Dressing action is a nontrivial example of a

Poisson group action. In general, such actions are

not Hamiltonian in the usual sense; the appropriate

generalization is provided by the notion of the

nonabelian moment map. Let G M!M be an

action of a Poisson group G on a Poisson manifold

M, g !Vect M, the associated homomorphism of

Lie algebras. A mapping  : M!G



is called the

nonabelian moment map associated with this action,

if for any X 2 g and ’ 2 F(M), we have

X  ’ ¼h

1

f; ’g

; Xi

In this case, G M!M is a fortiori a Poisson

map. Both dressing actions G



 G !G and G 



admit nonabelian moment maps, which are

just the identity maps  = id

and 



= id



. For

compact Poisson groups, the nonabelian moment

map has good convexity properties, which general-

ize the convexity properties of the ordinary moment

map for Hamiltonian group actions.

The general theory of homogeneous Poisson spaces

has some peculiarities. Typically, the G-covariant

Poisson structure on a given homogeneous space is

not unique (when it exists); this is true already for

principal homogeneous spaces (a simple example is

provided by the symplectic double D

). Let G be a

Poisson Lie group, (g, g



) its tangent Lie bialgebra, d

its double, U its Lie subgroup, u = Lie U.Asubalgebra

l  d is called Lagrangian if it is isotropic with respect

to the canonical inner product in d. The general

classification result, according to Drinfeld, asserts that

there is a bijection between G-covariant Poisson

structures on G=U and the set of all Lagrangian

subalgebras l  d such that l \ g = u. Various non-

trivial examples arise, notably in the study of integr-

able systems. For instance, the geometric proof of the

factorization theorem for lattice zero-curvature equa-

tion, which is stated in the following section, uses a

different Poisson structure on the double (the so-called

‘ ‘twisted symplectic double).’’

Applications to Integrable Systems

The definition of Poisson–Lie groups was motivated

by key examples which arise in the theory of

integrable systems. In applications, one often deals

with nonlinear differential equations which may be

written in the form of the so-called ‘‘lattice zero

curvature equations’’

¼ L

 M

mþ1

; m 2 Z ½19

where L

, M

are matrices, possibly depending on

an additional parameter (or, more generally, abstract

linear operators). Equations [19] give the compat-

ibility conditions for the auxiliary linear system

mþ1

¼ L

;

¼M

; m 2 Z ½20

The use of finite-difference operators associated with

a one-dimensional lattice, as in [20], is particularly

well suit ed for the study of ‘‘multiparticle’’ lattice

models. Let we assume that the ‘‘potential’’ L

in [20]

is periodic, L

mþN

= L

; the period N may be

interpreted as the number of copies of an ‘‘elemen-

tary’’ system. It is natural to presume that ‘‘Lax

matrices’’ L

in [19] are elements of a matrix Lie

group G (or of a loop group, if they depend on an

extra parameter). The auxiliary linear problem [20]

leads to a family of dynamical systems on G

which

remain integrable for any N. Let T : G

!G be the

‘‘monodromy map’’ which assigns to the set

, ..., L

of local Lax matrices their ordered

product T

= L

N1

L

. Let us assume that G is

equipped with the Sklyanin bracket associated with a

factorizable r-matrix r. Then T is a Poisson map. Let

I(G ) be the algebra of central functions on G; for ’ 2

I(G ), set H

’

= ’  T. All functions H

’

, ’ 2 I( G) are

in involution with respect to the pro duct Poisson

bracket on G

and give rise to lattice zero-curvature

equations of the same form as [19]; for a given ’,we

may choose the M-matrix in either of the two forms:



¼ r



r’ðT

1



;

1km

Let L

(t), m = 1, ..., N, be the integral curve of

this equation which starts at L

. The construction of

this curve reduces to the factorization problem asso-

ciated with the chosen r-matrix. Explicitly, we get

ðtÞ¼g

mþ1

ðtÞ

1

ðtÞ

¼ g

mþ1

ðtÞ

1



ðtÞ



where (g

(t)

, g

(t)



) is the curve in G



which

solves the factorization proble m

ðtÞ

1



expðtr’ðTðL

ÞÞÞ

1

;

ðL

This result exhibits the double role of the r-matrix.

On the one hand, it serves to define the Poisson

structure on G

which is adapted to the study of

lattice zero-curvature equations; in particular, the

dynamical flow associated with these equations is

automatically confined to symplectic leaves in G

(In applications, G is usually a loop group equipped

with a factorizable r-matrix; despite the fact that

dim G = 1, it admits plenty finite-dimensional sym-

plectic leaves.) In its second incarnation, the r-matrix

serves to define the factorization problem which

solves these zero-curvature equations. In the loop

Classical r-Matrices, Lie Bialgebras, and Poisson Lie Groups 517

group case, this is a matrix Riemann problem; its

explicit solution is based on the study of the spectral

curve associated with the ‘‘monodromy matrix’’ T

and uses the technique of algebraic geometry.

The monodromy map T : G

!G may be regarded

as a nonabelian moment map associated with an

action of the dual Lie algebra g



on the phase space.

This action actually extends to an action of the (local)

Lie group G



which transforms solutions into solu-

tions again. This is the prototype ‘‘dressing’’ action

(originally defined by Zakharov and Shabat in their

study of zero-curvature equations related to Riemann–

Hilbert problems). Dressing provides an effective tool

to produce new solutions of zero-curvature equations

from the ‘‘trivial’’ ones; it was also the first nontrivial

example of a Poisson group action.

See also: Affine Quantum Groups; Bicrossproduct

Hopf Algebras and Noncommutative Spacetime;

Bi-Hamiltonian Methods in Soliton Theory; Deformations

of the Poisson Bracket on a Symplectic Manifold;

Functional Equations and Integrable Systems;

Hamiltonian Fluid Dynamics; Hopf Algebras and

q-Deformation Quantum Groups; Integrable Systems

and Recursion Operators on Symplectic and Jacobi

Manifolds; Integrable Systems: Overview; Lie, Symplectic

and Poisson Groupoids, and their Lie Algebroids; Multi-

Hamiltonian Systems; Poisson Reduction; Recursion

Operators in Classical Mechanics; Toda Lattices;

Yang–Baxter Equations.

Further Reading

Babelon O, Bernard D, and Talon M (2003) Introduction to Classical

Integrable Systems. Cambridge: Cambridge University Press.

Belavin AA and Drinfel’d VG (1984) Triangle equations and simple

Lie algebras. In: Mathematical physics reviews,vol.4,Soviet

Scientific Reviews Section C Mathematical Physics Reviews,

pp. 93–165. Chur: Harwood Academic Publishers, Reprinted in

1998, Classic Reviews in Mathematics and Mathematical

Physics, vol. 1. Amsterdam: Harwood Academic Publishers.

Chari V and Pressley A (1995) A Guide to Quantum Groups.

Cambridge: Cambridge University Press.

Drinfeld VG (1987) Quantum groups. In: Proceedings of the

International Congress of Mathematicians, (Berkeley, Calif.,

1986) vol. 1, pp. 798–820. Providence, RI: American

Mathematical Society.

Etingof P and Schiffman O (1998) Lectures on Quantum Groups.

Boston: International Press.

Frenkel E, Reshetikhin N, and Semenov-Tian-Shansky MA (1998)

Drinfeld–Sokolov reduction for difference operators and

deformations of W-algebras. I. The case of Virasoro algebra.

Communications in Mathematical Physics 192(3): 605–629.

Lu J-H (1991) Momentum mappings and reduction of Poisson

actions. Symplectic Geometry, Groupoids, and Integrable Sys-

tems (Berkeley, CA, 1989), Mathematical de Sciences Research

Institute Publications vol. 20: 209–226. New York: Springer.

Lu J-H and Weinstein A (1990) Poisson–Lie groups, dressing

transformations, and Bruhat decompositions. Journal of

Differential Geometry 31(2): 501–526.

Reshetikhin N (2000) Characteristic systems on Poisson–Lie

groups and their quantization. In: Integrable Systems:

From Classical to Quantum (Montre´al, QC, 1999), CRM

Proceedings Lecture Notes, vol. 26, pp. 165–188. Providence,

RI: American Mathematical Society.

Reshetikhin NY and Semenov-Tian-Shansky MA (1990) Central

extensions of quantum current groups. Letters in Mathema-

tical Physics 19(2): 133–142.

Reyman AG and Semenov-Tian-Shansky MA (1994) Group-

theoretical methods in the theory of finite-dimensional integrable

systems. In: Encyclopaedia of Mathematical Sciences, Dynamical

Systems VII, ch. 2, vol. 16, pp. 116–225. Berlin: Springer.

Semenov-Tian-Shansky MA (1994) Lectures on R-matrices,

Poisson–Lie groups and integrable systems. In: Babelon O,

Cartier P, and Kosmann-Schwarzbach Y (eds.) Lectures on

Integrable Systems (Sophia-Antipolis, 1991), pp. 269–317.

River Edge: World Scientific.

Terng C-L and Uhlenbeck K (1998) Poisson actions and scattering

theory for integrable systems. In: Surveys in Differential Geome-

try: Integrable Systems, pp. 315–402. Lectures on geometry and

topology, sponsored by Lehigh University’s Journal of Differential

Geometry. A supplement to the Journal of Differential Geometry.

Edited by Chuu Lian Terng and Karen Uhlenbeck. Surveys in

Differential Geometry IV, Boston: International Press.

Clifford Algebras and Their Representations

A Trautman, Warsaw University, Warsaw, Poland

Introduction

Introductory and Historical Remarks

Clifford (1878) introduced his ‘‘geometric algebras’’

as a generalization of Grassmann algebras, complex

numbers, and quaternions. Lipschitz (1886) was the

first to define groups constructed from ‘‘Clifford

numbers’’ and use them to represent rotations in a

Euclidean space. Cartan discovered representations of

the Lie algebras so

(C)andso

(R), n > 2, that do

not lift to representations of the orthogonal groups.

In physics, Clifford algebras and spinors appear for

the first time in Pauli’s nonrelativistic theory of the

‘‘magnetic electron.’’ Dirac (1928), in his work on the

relativistic wave equation of the electron, introduced

matrices that provide a representation of the Clifford

algebra of Minkowski space. Brauer and Weyl (1935)

connected the Clifford and Dirac ideas with Cartan’s

spinorial representations of Lie algebras; they found,

in any number of dimensions, the spinorial, projective

representations of the orthogonal groups.

518 Clifford Algebras and Their Representations

Clifford algebras and spinors are implicit in

Euclid’s solution of the Pythagorean equation x



þ z

= 0, which is equivalent to

y  xz

zyþ x

= 2

pqðÞ ½1

and gives x = q

 p

, y = p

þ q

, z = 2pq. If the

numbers appearing in [1] are real, then this equation

can be interpreted as providing a representation of a

vector (x, y, z) 2 R

, null with respect to a quadratic

form of signature (1, 2), as the ‘‘square’’ of a spinor

(p, q) 2 R

. The pure spinors of Cartan (1938)

provide a generalization of this observation to

higher dimensions.

Multiplying the square matrix in [1] on the left by

a real, 2  2 unimodular matrix, on the right by its

transpose, and taking the determinant, one arrives at

the exact sequence of group homomorphisms:

1 ! Z

! SL

ðRÞ= Spin

1;2

! SO

1;2

! 1

Multiplying the same matrix by

" =

0 1

½2

on the left and computing the square of the product,

one obtains

zxþ y

x  y z

= ðx

 y

þ z

This equation is an illustration of the idea of

representing a quadratic form as the square of a

linear form in a Clifford algebra. Replacing y by iy,

one arrives at complex spinors, the Pauli matrices,



;

= i"; 

0 1

Spin

= SU

, etc.

This article reviews Clifford algebras, the asso-

ciated groups, and their representations, for quad-

ratic spaces over complex or real numbers. These

notions have been generalized by Chevalley (1954)

to quadratic spaces over arbitrary number fields.

Notation

If S is a vector space over K = R or C, then S



denotes its dual, that is, the vector space over K

of all K-linear maps from S to K. The value of ! 2



on s 2 S is sometimes written as hs, !i.

The transpose of a linear map f :S

! S

is the

map f



! S



defined by hs, f



(!)i= hf (s), !i for

every s 2 S

and ! 2 S



.IfS

and S

are complex

vector spaces, then a map f :S

! S

is said to be

semilinear if it is R-linear and f (is) = if (s). The

complex conjugate of a finite-dimensional complex

vector space S is the complex vector space



S of all

semilinear maps from S



to C. There is a natural

semilinear isomorphism (complex conjugation) S !



s 7!



s such that h!,



si =

hs, !i for every ! 2 S



The space



S can be identified with S and then



s = s.

The spaces (



and S



are identified. If f : S

! S

is a complex-linear map, then there is the complex-

conjugate map



f :



given by



f (



s) = f (s)and

the Hermitian conjugate map f

def



! S



AlinearmapA : S !



such that A

= A is said to

be Hermitian. K(N) denotes, for K = R, C or H,the

set of all N by N matrices with elements in K.

Real, Complex, and Quaternionic Structures

A real structure on a complex vector space S is a

complex-linear map C:S !



S such that



CC = id

A vector s 2 S is said to be real if



s = C(s). The set of

all real vectors is a real vector space; its real

dimension is the same as the complex dimension of S.

A complex-linear map C:S !



S such that



CC =  id

defines on S a quaternionic structure; a

necessary condition for such a structure to exist is

that the complex dimension m of S be even, m = 2n,

n 2 N. The space S with a quaternionic structure

can be made into a right vector space over the field

H of quaternions. In the context of quaternions, it is

convenient to represent the imaginary unit of C as

ﬃﬃﬃﬃﬃﬃﬃ

1

. Multiplication on the right by the quaternion

unit i is realized as the multiplication (on the left) by

ﬃﬃﬃﬃﬃﬃﬃ

1

. If j and k = ij are the other two quaternion

units and s 2 S, then one puts sj =



s) and sk = sij.

A real vector space S can be complexified by

forming the tensor product C 

S = S  iS.

The realification of a complex vector space S is the

real vector space having S as its set of vectors so that

dim

S = 2dim

S. The complexification of a realifica-

tion of S is the ‘‘double’’ S  S of the original space.

Inner-Product Spaces and Their Groups

Definitions: quadratic and symplectic spaces A

bilinear map B :S  S ! K on a vector space S over

K is said to make S into an inner-product space. To

save on notation, one also writes B: S ! S



so that

hs, B(t)i= B(s, t) for all s, t 2 S. The group of

automorphisms of an inner-product space,

AutðS; BÞ= fR 2 GLðSÞjR



 B  R = Bg

is a Lie subgroup of the general linear group GL(S).

An inner-product space (S, B) is said here to be

Clifford Algebras and Their Representations 519

quadratic (resp., symplectic) if B is symmetric (resp.,

antisymmetric and nonsingular). A quadratic space is

characterized by its quadratic form s 7! B(s, s). For

K = C, a Hermitian map A: S !



defines a

Hermitian scalar product A(s, t) = h



s, A(t)i.

An orthogonal space is defined here as a quadratic

space (S, B) such that B:S ! S



is an isomorphism.

The group of automorphisms of an orthogonal space

is the orthogonal group O(S, B). The group of

automorphisms of a symplectic space is the sym-

plectic group Sp(S, B). The dimension of a symplec-

tic space is even. If S = K

is a symplectic space

over K = R or C, then its symplectic group is

denoted by Sp

(K). Two quaternionic symplectic

groups appear in the list of spin groups of low-

dimensional spaces:

ðHÞ= fa 2 Hð2Þja

a = Ig

and

1;1

ðHÞ= fa 2 Hð2Þja



a = 

Here a

denotes the matrix obtained from a by

transposition and quaternionic conjugation.

Contractions, frames, and orthogonality From now

on, unless otherwise specified, (V, g) is a quadratic

space of dimension m.Let^V = 

p = 0

V be its

exterior (Grassmann) algebra. For every v 2 V and

w 2^V there is the contraction gðvÞcw characterized

as follows. The map V ^V !^V, ðv, wÞ7!

gðvÞcw, is bilinear; if x 2^

V,thengðvÞcðx ^ wÞ=

ðgðvÞcxÞ^w þð1Þ

x ^ðgðvÞcwÞ and gðvÞcv=gðv,vÞ.

A frame (e



) in a quadratic space (V, g) is said to

be a quadratic frame if  6¼  implies g(e



, e



) = 0.

For every subset W of V there is the orthogonal

subspace W

containing all vectors that are ortho-

gonal to every element of W.

If (V, g) is a real orthogonal space, then there is an

orthonormal frame (e



),  = 1, ..., m,inV such that

k frame vectors have squares equal to 1, l frame

vectors have squares equal to 1 and k þ l = m. The

pair (k, l) is the signature of g. The quadratic form g

is said to be neutral if the orthogonal space (V, g)

admits two maximal totally null subspaces W and

such that V = W  W

. Such a space V is 2n-

dimensional, either complex or real with g of

signature (n, n). A Lorentzian space has maximal

totally null subspaces of dimension 1 and a

Euclidean space, characterized by a definite quad-

ratic form, has no null subspaces. The Minkowski

space is a Lorentzian space of dimension 4.

If (V, g) is a complex orthogonal space, then an

orthonormal frame (e



),  = 1, ..., m, can be

chosen in V so that, defining g



= g(e



, e



), one

has g



= (1)

þ1

and, if  6¼ , then g



= 0.

If A:S !



is a Hermitian isomorphism, then

there is a (pseudo)unitary frame (e



)inS such that

the matrix A





= A(e



, e



) is diagonal, has p 1’s

and q 1’s on the diagonal, p þ q = dim S.Ifp = q,

then A is said to be neutral. A is definite if either p

or q = 0.

Algebras

Definitions An algebra over K is a vector space A

over K with a bilinear map AA!A,(a, b) 7!ab,

which is distributive with respect to addition.

The algebra is associative if (ab)c = a(bc) holds for

all a, b, c 2A. It is commutative if ab = ba for all

a, b 2A.Anelement1

is the unit of A if

a = a1

= a holds for every a 2A.

From now on, unless otherwise specified, the bare

word algebra denotes a finite-dimensional, associa-

tive algebra over K = R or C, with a unit element.

If S is an N-dimensional vector space over K, then the

set End S of all endomorphisms of S is an N

dimensional algebra over K, the product being

defined by composition; if f , g 2 End S,thenone

writes fg instead of f g; the unit of End S is

the identity map I. By definition, homomorphisms

of algebras map units into units. The map K !A,

a 7!a1

is injective and one identifies K with its

image in A by this map so that the unit can be

represented by 1 2 K A. A set BAis said to

generate A if every element of A can be represented

as a linear combination of products of elements of B.

For example, if V is a vector space over K,thenits

tensor algebra

TðVÞ= 

p = 0



is an (infinite-dimensional) algebra over K generated

by K  V. The algebra of all N  N matrices

with entries in an algebra A is denoted by A(N).

Its unit element is the unit matrix I. In particular,

R(N), C(N), and H(N) are algebras over R. The

algebra R(2) is generated by the set f

, 

g.Asa

vector space, the algebra R(2) is spanned by the set

fI, 

, ", 

The direct sum AB of the algebras A and B

over K is an algebra over K such that its underlying

vector space is ABand the product is defined by

(a, b)  (a

, b

) = (aa

, bb

) for every a, a

2A and

b, b

2B. Similarly, the product in the tensor

product algebra A

B is defined by

ða  bÞða

 b

Þ= aa

 bb

½3

520 Clifford Algebras and Their Representations

For example, if A is an algebra over R, then the

tensor product algebra R(N) 

A is isomorphic to

A(N) and

KðNÞ

KðN

Þ= KðNN

Þ½4

for K = R or C and N, N

2 N. There are isomorph-

isms of algebras over R:

C 

C = C C

C 

H = Cð2Þ

H 

H = Rð4Þ

½5

An algebra over R can be complexified by complex-

ifying its underlying vector space; it follows from [5]

that C(2) is the complex algebra obtained by

complexification of the real algebra H.

The center of an algebra A is the set

ZðAÞ= fa 2Ajab = ba 8 b 2Ag

The center is a commutative subalgebra containing

K. An algebra over K is said to be central if its center

coincides with K. The algebras R(N) and H(N) are

central over R. The algebra C(N) is central over C,

but not over R.

Simplicity and representations Let B

and B

be subsets of the algebra A. Define B

= fb

, b

g. A vector subspace B of A is said

to be a left (resp., right) ideal of A if AB  B (resp.,

BA  B). A two-sided ideal – or simply an ideal – is

a left and right ideal. An algebra A 6¼f0g is said to

be simple if its only two-sided ideals are f0g and A.

For example, the algebras R(N) and H(N) are

simple over R; the algebra C(N) is simple when

considered as an algebra over both R and C; every

associative, finite-dimensional simple algebra over R

or C is isomorphic to one of them.

A representation of an algebra A over K in a vector

space S over K is a homomorphism of algebras :A!

End S.If is injective, then the representation is said to

be faithful. For example, the regular representation :

A!End A of an algebra A, defined by (a)b = ab

for all a, b 2A, is faithful. A vector subspace T of

the vector space S carrying a representation  of A

is said to be invariant for

 if (a)T  T for every

a 2A; it is proper if distinct from both f0g and S.

For example, a left ideal of A is invariant for the

regular representation. Given an invariant subspace

T of  one can reduce  to T by forming the

representation 

:A!End T, where 

(a)s = (a)s

for every a 2A and s 2 T. A representation is

irreducible if it has no proper invariant subspaces.

AlinearmapF : S

! S

is said to intertwine the

representations 

:A!End S

and 

:A!End S

F

(a) = 

(a)F holds for every a 2A.IfF is an

isomorphism, then the representations 

and 

are

said to be equivalent, 

 

. The following two

propositions are classical:

Proposition (A)

(i) An algebra over K is simple if and only if it

admits a faithful irreducible representation in a

vector space over K. Such a representation is

unique, up to equivalence.

(ii) The complexification of a central simple algebra

over R is a central simple algebra over C.

For real algebras, one often considers complex

representations, that is, representations in complex

vector spaces. Two such representations 

:A!

End S

and 

:A!End S

are said to be complex

equivalent if there is a complex isomorphism F: S

intertwining the representations; they are real

equivalent if there is an isomorphism among the

realifications of S

and S

, intertwining the

representations. For example, C, considered as an

algebra over R, has two complex-inequivalent

representations in C : the identity representation

and its complex conjugate. The realifications of

these representations, given by i 7!" and i 7!",

respectively, are real equivalent: they are intertwined

by 

. The real algebra H, being central simple, has

only one, up to complex equivalence, representation

in C

: every such representation is equivalent to the

one given by

i 7!

ﬃﬃﬃﬃﬃﬃﬃ

1

; j 7!

ﬃﬃﬃﬃﬃﬃﬃ

1

; k 7!

ﬃﬃﬃﬃﬃﬃﬃ

1

This representation extends to an injective homo-

morphism of algebras i : H(N) !C(2N)whichisused

to define the quaternionic determinant of a matrix a 2

H(N)asdet

ðaÞ=det iðaÞ,sothatdet

(a)5 0and

det

(ab)=det

(a)det

(b) for every a,b 2H(N). In

particular, if q 2H and , 2R,thendet

(q)=



qq and

det

 q



q 

= ð þ

qqÞ

½6

There are quaternionic unimodular groups

ðHÞ = fa 2 HðNÞjdet

ðaÞ= 1g. For example,

the group SL

(H) is isomorphic to SU

and SL

(H)

is a noncompact, 15-dimensional Lie group, one of

the spin groups in six dimensions.

Antiautomorphisms and inner products An auto-

morphism of an algebra A is a linear isomorphism :

A!A such that (ab) = (a)(b). An invertible

element c 2Adefines an inner automorphism Ad(c) 2

GL(A), Ad(c)a = cac

1

. Complex conjugation in C,

considered as an algebra over R, is an automorphism

that is not inner. An antiautomorphism of an

Clifford Algebras and Their Representations 521

algebra A is a linear isomorphism  :A!Asuch that

(ab) = (b)(a) for all a, b 2A. An (anti)auto-

morphism  is involutive if 

= id. For example,

conjugation of quaternions defines an involutive

antiautomorphism of H.

Let :A!End S be a representation of an algebra

with an involutive antiautomorphism . There is then

the contragredient representation



: A!End S



given



(a) = (((a)))



. If, moreover, A is central simple

and  is faithful irreducible, then there is an isomorph-

ism B: S ! S



intertwining  and



 which is either

symmetric, B



= B,orantisymmetric,B



= B.It

defines on S the structure of an inner-product space.

This structure extends to End S: there is a symme-

tric isomorphism B  B

1

:End S ! (End S)



= End S



given, for every f 2 End S,by(B  B

1

)(f ) = Bf B

1

Let K



= Knf0g be the multiplicative group of the

field K. Given a simple algebra A with an involutive

antiautomorphism , one defines N(a) = (a)a and

the group

GðÞ= fa 2AjNðaÞ2K



Let  :A!End S be the faithful irreducible represen-

tation as above, then, for a 2Aand s, t 2 S, one has

BððaÞs;ðaÞtÞ= NðaÞBðs; tÞ

If a 2G() and  2 K



, then a 2G() and the norm

N satisfies N(a) = 

N(a). The inner product B is

invariant with respect to the action of the group

ðÞ= fa 2GðÞjNðaÞ= 1g

Proposition (B) Let A be a central simple algebra

over K with an involutive antiautomorphism  and a

faithful irreducible representation  so that



ðaÞ= BðaÞB

1

The map h : AA!K defined by

hða; bÞ= tr ððaÞbÞ

is bilinear, symmetric, and nondegenerate. The map

 is an isometry of the quadratic space (A, h) on its

image in the quadratic space (End S, B  B

1

Graded Algebras

Definitions An algebra A is said to be Z-graded

(resp., Z

-graded) if there is a decomposition of the

underlying vector space A = 

p2Z

(resp.,

A= A

A

)suchthatA

A

pþq

.InaZ

-graded

algebra, it is understood that p þ q is reduced mod 2. If

a 2A

,thena is said to be homogeneous of degree p.

The exterior algebra ^V of a vector space V is

Z-graded. Every Z-graded algebra becomes Z

-graded

when one reduces the degree of every element

mod 2. A graded isomorphism of graded algebras

is an isomorphism that preserves the grading.

A Z

-grading of A is characterized by the

involutive automorphism  such that, if a 2A

then (a) = (1)

a. From now on, grading means

-grading unless otherwise specified. The elements

of A

(resp., A

) are said to be even (resp., odd). It

is often convenient to denote the graded algebra as

!A ½7

Given such an algebra over K and N 2 N, one

constructs the graded algebra A

(N) ! A(N). Two

graded algebras over K, A

!Aand A

are

said to be of the same type if there are integers N

and N

such that the algebras A

(N) !A(N) and

) !A

) are graded isomorphic. The prop-

erty of being of the same type is an equivalence

relation in the set of all graded algebras over K.

Given an algebra A, one constructs two ‘‘canoni-

cal’’ graded algebras as follows:

1. the double algebra

A!AA

graded by the ‘‘swap’’ automorphism, (a

, a

) =

, a

) for a

, a

2A;

2. the algebra

AA!Að2Þ

is defined by declaring the diagonal (resp., anti-

diagonal) elements of A(2) to be even (resp., odd).

The real algebra R(2) has also another grading,

given by the involutive automorphism  such that

(a) = "a"

1

, where a 2 R(2) and " is as in [2].In

this case, [7] reads

C ! Rð2Þ

There are also graded algebras over R:

R ! C; C ! H; and H ! Cð2Þ

The grading of the last algebra can be defined by

declaring the Pauli matrices and iI to be odd.

Super Lie algebras A super Lie algebra is a graded

algebra A such that the product (a, bÞ7!½a, b is

super anticommutative, ½a, b=  (1)

½b, a, and

satisfies the super Jacobi identity,

½a; ½b; c= ½½a; b; cþð1Þ

½b; ½a; c

for every a 2A

, b 2A

and c 2A. To every graded

associative algebra A there corresponds a super Lie

algebra GLA: its underlying vector space and

grading are as in A and the product, for a 2A

522 Clifford Algebras and Their Representations

and b 2A

, is given as the supercommutator ½a, b=

ab  (1)

ba.

Supercentrality and graded simplicity A graded

algebra A over K is supercentral if Z(A) \A

= K.

The algebra R ! C is supercentral, but the real

ungraded algebra C is not central.

A subalgebra B of a graded algebra A is said to be

a graded subalgebra if B= B\A

B\A

graded ideal of A is an ideal that is a graded

subalgebra. A graded algebra A 6¼f0g is said to be

graded simple if it has no graded ideals other than

f0g and A. The double algebra of a simple algebra is

graded simple, but not simple.

The graded tensor product Let A and B be graded

algebras; the tensor product of their underlying

vector spaces admits a natural grading, (AB)



B

pq

. The product defined in [3] makes

ABinto a graded algebra. There is another ‘‘super’’

product in the same graded vector space given by

ða  bÞða

 b

Þ= ð1Þ

 bb

for a

and b 2B

. The resulting graded algebra

is referred to as the graded tensor product and

denoted by A

B. For example, if V and W are

vector spaces, then the Grassmann algebra ^(V 

W) is isomorphic to ^V

^W.

Clifford Algebras

Definitions: The Universal Property and Grading

The Clifford algebra associated with a quadratic

space (V, g) is the quotient algebra

C‘ðV; gÞ= TðVÞ=JðV; gÞ½8

where J(V, g) is the ideal in the tensor algebra T (V)

generated by all elements of the form v  v 

g(v, v)1

T (V)

, v 2 V.

The Clifford algebra is associative with a unit

element denoted by 1. One denotes by  the

canonical map of T (V) onto C‘(V, g) and by ab

the product of two elements a, b 2C‘(V, g)sothat

(P  Q) = (P)(Q)forP, Q 2T(V). The map  is

injective on K  V, and one identifies this subspace of

T (V) with its image under . With this identification,

for all u, v 2 V, one has

uv þ vu = 2gðu; vÞ

Clifford algebras are characterized by their universal

property described in the following proposition.

Proposition (C) Let A be an algebra with a unit 1

and let f : V !Abe a Clifford map, that is, a linear

map such that f (v)

= g(v, v)1

for every v 2 V. There

then exists a homomorphism

f: C‘(V, g) !A of

algebras with units, an extension of f, so that f (v) =

f(v)

for every v 2 V.

As a corollary, one obtains

Proposition (D) If f is an isometry of (V, g) into

(W, h), then there is a homomorphism of algebras

C‘(f ):C‘(V, g) !C‘(W, h) extending f so that there

is the commutative diagram

C‘ðV; gÞ!

C‘ðf Þ

C‘ðW; hÞ

V !

For example, the isometry v 7!v extends to the

involutive main automorphism  of C‘(V, g), defin-

ing its Z

-grading:

C‘ðV; gÞ= C‘

ðV; gÞC‘

ðV; gÞ

The algebra C‘(V, g) admits also an involutive cano-

nical antiautomorphism  characterized by (1) = 1

and (v) = v for every v 2 V.

The Vector Space Structure of Clifford Algebras

Referring to proposition (D),letA= End( ^V)and,for

every v 2 V and w 2^V,putf (v)w = v ^ w þ g(v)cw,

then f :V ! End( ^V) is a Clifford map and the map

i : C‘ðV; gÞ!^V ½9

given by i(a) =

f(a)1

is an isomorphism of vector

spaces. This proves

Proposition (E) As a vector space, the algebra

C‘(V, g) is isomorphic to the exterior algebra ^V.

If V is m-dimensional, then C‘(V, g)is

-dimensional. The linear isomorphism [9] defines a

Z-grading of the vector space underlying the Clifford

algebra: if i(a

) 2^

V,thena

is said to be of

Grassmann degree k. Every element a 2C‘(V, g)

decomposes into its Grassmann components,

a =

k2Z

. The Clifford product of two elements of

Grassmann degrees k and l decomposes as follows:

p2Z

)

,and(a

)

= 0ifp < jk  lj or

p  k  l þ1mod2orp > m jm  k  lj.

One often uses [9] to identify the vector spaces ^V

and C‘(V, g); this having been done, one can write,

for every v 2 V and a 2C‘(V, g),

va = v ^ a þ gðvÞca ½10

so that [v, a] = 2g(v)ca, where [ , ] is the supercommu-

tator. It defines a super Lie algebra structure in the

vector space K  V. The quadratic form defined by g

need not be nondegenerate; for example, if it is the

Clifford Algebras and Their Representations 523

0-form, then [10] shows that the Clifford and exterior

multiplications coincide and C‘(V, 0) is isomorphic, as

an algebra, to the Grassmann algebra.

Complexification of Real Clifford Algebras

Proposition (F) If (V, g) is a real quadratic space,

then the algebras C C‘(V, g) and C‘(C  V, C  g)

are isomorphic, as graded algebras over C.

From now on, through the end of the article, one

assumes that (V, g) is an orthogonal space over

K = R or C.

The Clifford algebra associated with the orthogo-

nal space C

is denoted by C‘

. The Clifford

algebra associated with the orthogonal space

kþl

, g), where g is of signature (k, l), is denoted

by C‘

k, l

, so that C C‘

k, l

= C‘

kþl

Relations between Clifford Algebras in Spaces of

Adjacent Dimensions

Consider an orthogonal space (V, g)overK and the

one-dimensional orthogonal space (K, h

), having a

unit vector w 2 K, h

(w, w) = ",where" = 1or1.

The map V 3 v 7!vw 2C‘

(V  K, g  h

)satisfies

(vw)

= "g(v, v) and extends to the isomorphism

of algebras C‘(V, "g) !C‘

(V  K, g  h

). This

proves

Proposition (G) There are isomorphisms of algebras:

C‘

!C‘

mþ1

and C‘

k, l

!C‘

kþ1, l

Consider the orthogonal space (K

, h) with a

neutral h such that, for ,  2 K, one has

h(, ), h(, )i= . The map

! Kð2Þ; ð; Þ7!

0 

 0

has the Clifford property and establishes the

isomorphisms represented by the horizontal arrows

in the diagram

C‘ðK

; hÞ!Kð2Þ

C‘

ðK

; hÞ!K  K

½11

Proposition (H) If (K

, h) is neutral and (V, g) is

over K, then the algebra C‘(V  K

, g  h) is

isomorphic to the algebra C‘(V, g)  K(2)_ Specifically,

there are isomorphisms

C‘

kþ1;lþ1

= C‘

k;l

 Rð2Þ

C‘

mþ2

= C‘

 Cð2Þ

½12

The Chevalley Theorem and the Brauer–Wall

Group

If (V, g) and (W, h) are quadratic spaces over K, then

their sum is the quadratic space (V  W, g  h)

characterized by g  h : V  W ! V



 W



so that

(g  h)(v, w) = (g(v), h(w)). By noting that the map

V W 3(v,w)7!v 1 þ1 w 2C‘(V,g)

C‘(W,h)

has the Clifford property, Chevalley proved

Proposition (I) The algebra C‘(V  W, g  h) is

isomorphic to the algebra C‘(V, g)

C‘(W, h).

The type of the (graded) algebra C‘(V  W, g  h)

depends only on the types of C‘(V,

g)andC‘(W, h).

The Chevalley theorem (I) shows that the set of types

of Clifford algebras over K forms an abelian group for

a multiplication induced by the graded tensor product.

The unit of this Brauer–Wall group of K isthetypeof

the algebra C‘(K

, h) described in [11];forafull

account with proofs, see Wall (1963).

The Volume Element and the Centers

Let e = (e



) be an orthonormal frame in (V, g). The

volume element associated with e is

 = e

e

If 

is the volume element associated with another

orthonormal frame e

in the same orthogonal space,

then either 

=  (e and e

are of the same

orientation) or 

=  (e and e

are of opposite

orientation). For K = C, one has 

= 1; for K = R

and g of signature (k, l) one has



= ð1Þ

ð1=2ÞðklÞðklþ1Þ

½13

It is convenient to define  2f1, ig so that 

= 

.For

every v 2 V one has v = (1)

mþ1

v. The structure of

the centers of Clifford algebras is as follows:

Proposition (J) If m is even, then Z(C‘(V, g)) = K

and Z(C‘

(V, g)) = K  K. If m is odd, then

Z(C‘(V, g)) = K  K and Z(C‘

(V, g)) = K.

The graded algebra C‘(V, g) is supercentral for

every m.

The Structure of Clifford Algebras

The complex case Using [4] one obtains from [11]

and [12] the isomorphisms of algebras

C‘

2nþ1

= C‘

= Cð2

Þ½14

C‘

2nþ1

= C‘

2nþ2

= Cð2

ÞCð2

Þ½15

for n = 0, 1, 2 , ... : Therefore, there are only two types

of complex Clifford algebras, represented by

C !C C and C  C ! C(2): the Brauer–Wall

group of C is Z

524 Clifford Algebras and Their Representations