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In: Acta Applicandae Mathematicae, Kluwer Academic Publishers 23: 65-93, (1991).

The Design of Linear Algebra and Geometry

David Hestenes

Abstract. Conventional formulations of linear algebra do not do justice to

the fundamental concepts of meet, join, and duality in projective geometry.

This defect is corrected by introducing Cliﬀord algebra into the foundations

of linear algebra. There is a natural extension of linear transformations on

a vector space to the associated Cliﬀord algebra with a simple projective

interpretation. This opens up new possibilities for coordinate-free compu-

tations in linear algebra. For example, the Jordan form for a linear trans-

formation is shown to be equivalent to a canonical factorization of the unit

pseudoscalar. This approach also reveals deep relations between the struc-

ture of the linear geometries, from projective to metrical, and the structure

of Cliﬀord algebras. This is apparent in a new relation between additive

and multiplicative forms for intervals in the cross-ratio. Also, various fac-

torizations of Cliﬀord algebras into Cliﬀord algebras of lower dimension are

shown to have projective interpretations.

As an important application with many uses in physics as well as in

mathematics, the various representations of the conformal group in Cliﬀord

algebra are worked out in great detail. A new primitive generator of the

conformal group is identiﬁed.

[Note: Some corrections have been added in brackets.]

1. Introduction

It is sometimes said that projective geometry is just linear algebra. Nevertheless, the two

subjects persist today as separate branches of mathematics. Although they grew up together

in the nineteenth century, projective geometry was left behind as linear algebra ﬂourished

through diverse applications in the twentieth century. This is the second of two articles

aimed at bringing them back together in a mutually beneﬁcial reconciliation. These articles

propose speciﬁc designs for a mathematical system to achieve that end.

Conventional linear algebra is based on the concepts of vector space and linear transfor-

mation. Projective geometry can be formulated within this framework, but only by intro-

ducing coordinate systems without intrinsic geometric signiﬁcance. To achieve a coordinate-

free formulation of projective geometry, a richer algebraic system is needed. In the ﬁrst

article [1] geometric algebra was proposed as the ideal system for that purpose. It was

shown to have exactly the right dimension and algebraic structure to describe all the meet

and join operations essential to projective geometry. This article extends the mathematical

designs of [1] along two lines: (1) The standard concept of a linear transformation is ex-

tended to provide a direct, coordinate-free formulation of projective transformations, and

this extension is shown to be valuable even in standard linear algebra. As an example,

the Jordan canonical form is shown to be equivalent to a particular factoring of the unit

pseudoscalar. (2) The concepts of ‘projective split’ and ‘conformal split’ are introduced

1

and shown to reveal a deep connection between projective structures and the structure of

Cliﬀord algebras. They are used to simplify and coordinate relations between projective,

conformal, aﬃne, and metric geometries. A new relation between additive and multiplica-

tive forms for intervals in the cross-ratio is derived. The conformal groups for metric spaces

of arbitrary dimension and signature are analyzed in great detail, establishing connections

between several alternative approaches in the literature.

Reference [1] is taken as prerequisite to this article, so the notations and nomenclature

introduced there can be taken for granted. An overview of that article and the present one

has been published in [2].

2. Outermorphism

This section reviews the concept of outermorphism developed in [3], with emphasis on its

signiﬁcance for projective geometry. The treatment in [3] is more general and extensive

than the present one. Unfortunately, that may have obscured the essential simplicity and

utility of the outermorphism idea, so the aim here is to clarify the concept. To that end,

some of the results in [3] will be introduced without proof, while others are omitted.

The outermorphism is actually inherent in linear algebra. By making it explicit and

giving it a notation we can exploit it more easily. The conventional abstract formulation of

linear algebra follows the format of matrix algebra. This does not do justice to Grassmann’s

outer product or to duality, two fundamental concepts of projective geometry. The main

point of this section is that the outermorphism concept corrects this defect perfectly. Since

the outermorphism is inherent in linear algebra, it has been repeatedly rediscovered in

various guises. For example, as explained in [3], the so-called ’push-forward’ and ’pull-

back’ mappings of diﬀerential forms are outermorphisms. While preparing this article, it

was discovered that Whitehead [4] had independently deﬁned the outermorphism of a blade

and was probably the ﬁrst to do so. He evidently sensed the importance of the idea, but

he did not develop it far, and no one seems to have followed his lead.

As in [1], Vnis an n-dimensional vector space which generates a geometric algebra

Gn=G(Vn). Every linear transformation fon Vnhas a unique extension to a linear

transformation fon Gndeﬁned by the following four properties:

f(A∧B)=(fA)∧(fB),(2.1)

fa=fa, (2.2)

fα=α, (2.3)

f(αA +βB)=αfA+βfB, (2.4)

where A, B are blades in Gn,ais a vector and α, β are scalars. Thus, fis linear on Gnand

distributive with respect to the outer product. It is called the outermorphism of fbecause,

according to (2.1), it preserves the outer product.

It follows from (2.1) and (2.2) that fis step-preserving, that is,

f(hMik)= hfMi

k(2.5)

2

for any multivector Min Gn, Moreover, for any factoring A=a1∧a2∧···∧a

rof an r-blade

Ainto vectors

fA=(fa1)∧(fa2)∧···∧(far).(2.6)

That is to say, fAdoes not depend on how Amight be factored into vectors. It will be

seen that this property is a great help in ﬁnding and describing the invariants of a linear

transformation.

For the sake of simplicity, we limit our attention here to transformations from Vnto Vn,

so fmaps Gninto Gn. In this case, (2.5) implies that fcan alter the pseudoscalar Ionly

by a scalar multiple. Indeed,

fI= (det f)I(2.7)

deﬁnes the determinant of f, which we write as det for det f. Note that the outermorphism

concept makes it possible to deﬁne the determinant of a linear transformation without

introducing a basis or matrices. As shown in [3], the matrix form of the determinant

is easily introduced when needed, but the determinant can often be evaluated without

introducing matrices.

Let h=gf be the product of linear transformations fand gon Vn. From (2.6) it is

easily proved that

h=gf . (2.8)

In other words, the outermorphism of a product is equal to the product of outermorphisms.

Some important facts about linear transformations follow immediately from (2.8). For

example,

gfI = (det f)gI = (det f)(det g)I.

Therefore,

det (fg) = (det f)(det g).(2.9)

It follows by iteration that, for positive integer k, det (fk) = (det f)k, and if f−1exists,

then (det f−1)(det f)=1.

Every linear transformation fhas an adjoint transformation f, and the adjoint can be

extended to an outermorphism which we denote by falso. The adjoint outermorphism f

can be deﬁned directly by

hMfNi=h(fN)Mi=hNfMi(2.10)

assumed to hold for all M,N in Gn. The multivector derivative deﬁned in [3] can be used

to get

fN=∇MhNfMi.(2.11)

Diﬀerentiation is helpful and much used in [3], especially to generate invariant relations,

but it is not essential and will be avoided in the following. It follows from (2.7) and (2.10)

that

det f=I−1fI =hI−1fI i=I−1f I =detf . (2.12)

Unlike the outer product, the inner product is not generally preserved by an outermor-

phism. However, it is proved in [3] that the inner product obeys the fundamental transfor-

mation law

A·(fB)=f[(f A)·B]or(fB)·A=f[B·(fA)] ,(2.13)

3

for (step A)≤(step B). For clarity, the square brackets are used as parentheses here. Note

that both fand fare needed to describe how the inner product transforms. Also note

that, by symmetry, fand fcan be interchanged in (2.13).

The law (2.13) admits the important special case

A(fI)=f[(fA)I]or(fI)A=f[I(fA)] .(2.14)

For de t f6= 0, this gives us immediately an explicit expression for the inverse outermor-

phism:

f−1A=f(AI)I−1

det f=I−1f(IA)

det f.(2.15)

Applied to ain Vn, of course, this gives us the inverse transformation

f−1(a)=I

−1

f(Ia)

det f.(2.16)

This reveals that the adjoint and double duality are essential relations underlying the

general matrix formula for f−1, which can, in fact, be obtained easily from (2.16).

The inverse of the outermorphism fgiven by (2.15) is equal to the outermorphism of the

inverse f−1. To prove that, we need only establish the distributive property

f−1(A∧B)=(f

−1

A)∧(f

−1

B).(2.17)

For the proof it will be convenient to use the notation A=e

AI−1and put (2.14) in the form

f[(fe

A)I]=Adet f. (2.18)

The proof uses the duality of inner and outer productsalong with (2.13) as follows. Assuming

(step A) + (step B)≤n,

(f−1A)∧(f−1B)=[(fe

A)I]∧[(fe

B)I]

(det f)2=[(fe

A)I]·(fe

B)

(det f)2I

=f[(f[(fe

A)I]) ·e

B]

(det f)2I=f[A·e

B]I

det f=f[(A∧B)I−1]

det f.

(2.19)

Thus (2.17) is established. Now let us turn to the matter of geometrical interpretation.

Outermorphisms can be supplied with a geometrical interpretation by adopting the

projective interpretations for blades introduced in [1]. Recall that vectors representing rays

in Vncan be identiﬁed with points in the projective space Pn−1. Projective transformations

(or projectivities) are of two types, collineations and correlations. The collineations of Pn−1

are simply linear transformations of Vnwith nonvanishing determinants. A correlation is

simply the composite of a collineation with a duality transformation (i.e., multiplication

by a pseudoscalar). Having discussed duality in [1], we can limit our attention here to

collineations without loss of generality.

The most basic fact about collineations is that they map points into points, lines into

lines, planes into planes, etc. This follows from the distributive property of outermorphisms,

4

since points, lines, and planes can be represented by blades of step 1, 2, 3, respectively.

Indeed, a collineation finduces an outermorphism of the equation x∧A= 0 into

x0∧A0=(fx)∧(fA)=0.

This is equivalent to the original equation for point, line, or plane provided fA6= 0 when

A6= 0, and that is assured by the fact that every nonzero blade is a factor of the pseudoscalar

I. For if blade A6= 0, there exists another (not unique) blade Bsuch that A∧B=I. Since

(fA)∧(fB) = (det f)I6=0,

the factor fAcannot vanish.

Thus, we see that the outermorphism fof a collineation fdirectly describes the induced

transformations of lines and planes. We can interpret fAas the transformation of a speciﬁc

line or plane according as blade Ahas step 2 or 3.

To establish the invariance of projective relations under arbitrary projectivities, we need

to prove that the fundamental meet and join relations are preserved under collineations.

Since we can identify the join with the outer product, the invariance of the join relation

is assured by the distributive property (2.1). To ﬁnd the transformation law for the meet,

recall that for (step A) + (step B)≥n, the meet can be expressed in terms of the inner

product by Equation (3.6) of [1]:

A∨B=e

A·B. (2.20)

Proceeding as in the proof of (2.17), we have

(fA)∨(fB)=[(fA)I

−1

]·(fB)= f[(f[( fA)I−1]) ·B

=f[e

A·B](det f).

Thus, the meet obeys the transformation law

(fA)∨(fB) = (det f)f(A∨B).(2.21)

The factor (det f) here is not signiﬁcant in nonmetrical applications of projective geometry,

since the deﬁnition of the meet is arbitrary within a scale factor. However, it can be

removed by a natural change in the deﬁnition of duality, deﬁning duality on the left with

respect to the transformed pseudoscalar I0=f I = (det f)I. Then (2.15) implies that

the transformation A0=fA entails the induced transformation e

A0=fe

Aon the dual

e

A0=A0(I0)−1, and (2.21) can be put in the alternative form

f(e

A·B)= e

A

0·B

0.(2.22)

This transformation law is mathematically equivalent to (2.13), but its geometric meaning

is much more transparent. It tells us that the ‘incidence properties’ in projective geometry

are invariant under collineations or, equivalently, that the ‘subspace intersection property’

is preserved by nonsingular linear transformations. For this reason, it should be counted

as one of the fundamental results of linear algebra, though it does not appear in standard

textbooks on the subject.

5

3. Invariant Blades

The outermorphism is a natural tool for characterizing the invariant subspaces of an arbi-

trary linear transformation. As an illustration, this section shows how to use that tool to

achieve a characterization equivalent to the Jordan canonical form.

The outermorphism gives us a natural generalization of the eigenvector concept. A blade

Aissaidtobeaneigenblade of fwith (scalar) eigenvalue λif

fA=λA . (3.1)

‘Projectively speaking’, this says that Ais a ﬁxed point, line, or plane of a collineation f

according as the step of Ais 1, 2, or 3. Let us call Aa symmetric eigenblade if

fA=f A =λA . (3.2)

This generalizes to arbitrary linear transformations an important feature of the eigenvectors

of a symmetric transformation. Of course, the pseudoscalar Iis a symmetric eigenblade of

every linear transformation, with

fI=f I =µI , (3.3)

where µ=detf. Henceforth, it will be convenient to assume µ6=0.

If Ais an eigenblade of f, then its dual e

A=AI−1is an eigenblade of f. This follows

trivially from (2.14); speciﬁcally, with (3.1) and (3.3), we have

fe

A=µ

λe

A. (3.4)

Since I=A−1e

A=A−1∧e

A,

fI=µI =(fA

−1

)∧(fe

A)= µ

λ(fA

−1

)∧e

A.

We cannot in general remove the wedge from the right side of this expression, but we can

conclude that there exists a blade Bwith the same step as Asuch that

fA=λA +B. (3.5a)

where B∧e

A=(B·A)I

−1= 0. Dually, we have

fe

A=µ

λe

A+e

C. (3.5b)

where A∧e

C=(A·C)I

−1= 0. Clearly, Ais a symmetric eigenblade of fif and only if e

A

is also symmetric; moreover

fA=fe

A=µ

λe

A. (3.6)

A symmetric eigenblade will be called a proper blade of fif it has no factors which

are symmetric eigenblades of lower step. By iterating the argument yielding (3.6), we can

decompose the pseudoscalar Iinto a geometric product of proper blades Ik;thus,

I=I

1

I

2···I

m,(3.7a)

6

where, for k=1,2,...,m ≤n,

fI

k=fI

k=µ

kI

k,(3.7b)

and

det f=µ=µ1µ2···µ

m.(3.7c)

The Ikare unique (within a scale factor) if the µkare distinct. Their supports are precisely

the invariant subspaces of the linear transformation f. If all the Ikhave step 1, then

m=nand (3.7b) is a complete spectral decomposition of f. However, for any Ikwith

(step Ik)>1, we must take the decomposition further to characterize fcompletely. That

we do next.

[Warning: The following analysis does not suﬃce to establish the Jordan form for an

arbitrary linear transformations, though the method of analysis may be of general interest

in linear algbera.]

Suppose that fhas a proper blade Amof step m>1, and let complex scalars be allowed

just to avoid discussing points of peripheral interest. Then the following result is equivalent

to the Jordan decomposition of f: there exists a unique scalar λand a nested sequence of

blades Aksuch that

fAk=λkAk(3.8)

for k=1,2,...,m. The term ‘nested’ means that Ak−1is a factor of Ak. Let e

Ak=AkA−1

m

be the dual with respect to Am; then (3.4) implies that

fe

Ak=λm−ke

Ak.(3.9)

The Akare unique and they completely characterize fon the support of Amk. In fact, the

Akcan be generated from the unique eigenvector a1=A1,by

A

k=a

1∧(fA

k−1

),(3.10)

which iterates to

Ak=a1∧(f a1)∧(f2a1)∧···∧(fk−1a

1).(3.11)

We can prove all this by showing that it is equivalent to the Jordan canonical form.

The blades Akdetermine mvectors akdeﬁned by

ak=A−1

k−1Ak=A−1

k−1·Ak(3.12)

with the convention A0= 1. Conversely, the Akare determined by the akaccording to

Ak=Ak−1ak=a1a2···a

k.(3.13)

Note that this implies the orthogonality relation

ai·aj=0 for i6=j. (3.14a)

To ascertain the action of fon the ak, we apply (3.8) to (3.13) to obtain

λAk=Ak−1∧(fak).

7

This can be solved for fak=fakas follows:

λA−1

k−1Ak=λak=A−1

k−1·(Ak−1∧(fak))

=(A

−1

k−1·A

k−1

)fa

k+(−1)k−1(A−1

k−1·(fak))·A

k−1

=fa

k−a

k−1.

The last step follows from the ‘Laplace expansion’

(a−1

k∧···∧a

−1

2∧a

−1

1)·(fa

k)=(−l)

k−2

a

−1

k−2···a

−1

2a

−1

1(a

−1

k−1·fa

k)

=(−l)

k−2

a

−1

k−2.

Only one term in the expansion survives because

a−1

j·(fak)=a

k

(fa

−1

j)=0 for j<k−1,

as follows from the vanishing of terms in the formula

Aj+1 ·a−1

k=[A

j∧(fa

j

)] ·a−1

k=Aj(a−1

k·f aj)−(Aj·a−1

k)∧A(f aj).

For j=k−1, this formula implies the relation

a−1

k·(f ak−1)=a

k−1·(fa

−1

k)=1

which was also used above.

Thus we have proved that the nested eigenblade formula (3.8) implies that

fak=λak+ak−1(3.14b)

for 1 <k≤m. Of course,

fa1=λa1(3.14c)

since a1=A1. Note that no assumptions about signature were used in the proof, although

it was necessary that A2

m6= 0. Conversely, (3.8) can be derived from (3.14b,c) simply by

inserting them into

fAk=(fa

1

)∧(fa

2

)∧···∧(fa

k).

Equations (3.14a,b,c) will be recognized as the Jordan canonical form for a linear transfor-

mation on a cyclic subspace. We have proved that on the support of Amthe Jordan form

is equivalent to the factoring of Aminto a nested sequence of eigenblades.

If only real scalars are allowed, the above decomposition of a cyclic subspace applies if the

characteristic polynomial has a root λof multiplicity m, in which case there is exactly one

eigenvector. However, when the characteristic polynomial of the subspace does not have

a real root, it has an irreducible factor of some multiplicity m. In this case, the subspace

necessarily has even dimension 2m. There are no eigenvectors in the subspace, but there

is, nevertheless, a scalar γand a nested sequence of blades A2kwith even step 2ksuch that

fA2k=γkA2k(3.15)

for k=1,2,... ,m. This assertion can be proved along lines similar to the previous case,

but we will not go into that here. It should be noted, though, that the determination

of the eigenblades is not suﬃcient, in this case, to characterize the linear transformation

completely.

8

4. Projective Splits

We have seen how the entire geometric algebra Gn+1 is needed to describe the join, meet,

and duality operations on the projective n-space Pnas well as their projective outermor-

phisms. This section shows that the multiplicative structure of geometric algebra reﬂects

deep properties of projective geometry and facilitates connections with aﬃne and metric

geometry.

The reduction of projective geometry to aﬃne geometry can be expressed as a relation

between a vector space Vn+1 and a vector space Vnof one less dimension. Ordinarily Vn

is taken to be a subspace of Vn+1, but geometric algebra admits a more profound way of

relating Vn+1 to Vnwhich will be investigated here. Let Gr

n+1 denote the subspace of all

multivectors of step rin Gn+1. The subspace

G+

n+1 =X

kG2k

n+1 (4.1)

of all multivectors with even step is a subalgebra called the even subalgebra of Gn+1.It

can be identiﬁed with the geometric algebra Gnof an n-dimensional vector space Vn,as

expressed by

Gn=G+

n+1 .(4.2)

This requires the speciﬁcation of a unique relation between the vector spaces Vn+1 =G1

n+1

and Vn=G1

n. A geometrically signiﬁcant way to do this is as follows. Let e0and xbe

vectors in Vn+1, then, for ﬁxed e0with e2

06=0,

xe0=x·e0+x∧e0(4.3)

is a linear mapping of Vn+1 into G0

n+1 +G2

n+1, which deﬁnes a linear correspondence between

Vn+1 and G0

n+G1

nby the identiﬁcation G0

n+1 =G0

nand

Vn=G1

n={x∧e0}.(4.4)

This gives Vna projective interpretation as the pencil of all lines through the point e0.

Moreover, it determines a split of G2

n+1 into a direct sum,

G2

n+1 =G1

n+G2

n.(4.5)

where, in accordance with (4.4) G1

nconsists of all bivectors in G2

n+1 which anticommute

with e0while G2

nconsists of the bivectors which commute with e0. The split of any bivector

Fin G2

n+1 is thus given by

F=Fe

−1

0e

0=(F·e

−1

0)e

0+(F∧e

−1

0)e

0,(4.6)

where

(F·e−1

0)e0=(F·e

−1

0)∧e

0

is in G1

n=Vnand

(F∧e−1

0)e0=(F∧e

−1

0)·e

0

9

is in G2

n.

The linear mapping (4.3) Vn+1 into Gnalong with the identiﬁcation of G+

n+1 with Gn

according to (4.4) and (4.5) will be called a projective split of Vn+1, or if you will, of Gn+1.

In view of its geometrical signiﬁcance, the projective split can be regarded as a canonical

relation between geometric algebras of diﬀerent dimension. Indeed, the underlying geomet-

rical idea played a crucial role in Cliﬀord’s original construction of geometric algebras [5].

However, Cliﬀord’s motivating ideas have been largely ignored in subsequent mathematical

applications of Cliﬀord algebras. The projective split idea was ﬁrst explicitly formulated

and applied to physics in [6]. Accordingly, ﬂat spacetime is represented by a vector space

V4with Minkowski metric, and its associated geometric algebra G4is called the spacetime

algebra. The ‘splitting vector’ e0is taken to be the timelike vector of some inertial frame,

and it determines a projective split of V4and G4into space and time components. This is

called a spacetime split. Each inertial system determines a unique spacetime split. This

relates the invariant properties of objects in spacetime to their ‘observable’ representations

in inertial systems. For example, an electromagnetic ﬁeld is represented by an invariant

bivector-valued function F=F(x)onV

4

. In that case, the split (4.6) describes the frame-

dependent splitting of Finto electric and magnetic components. The spacetime split is thus

a fundamental relation in physics, though its projective character usually goes unrecognized.

The projective split (4.3) can be put in the form

xe0=x0(1 + x),(4.7)

where x0=x·e0and boldface is used to denote a vector

x=x∧e0

x·e0

(4.8)

in Vn. Clearly x∧e0and x·e0or just xrelative to e0amounts to a representation of the

‘point’ xby ‘homogeneous coordinates’.

For ‘points’ a, b in Vn+1, the projective split (4.7) relates products in Gn+1 to products in

Gn;thus,

ab =(ae0)(e0b)

e2

0

=a0b0

e2

0

(1 + a)(1 + b)

=a0b0

e2

0

(1 −a·b+a−b+b∧a).

Separating inner and outer products in Gn+1 we obtain

a·b=a0b0

e2

0

(1 −a·b)(4.9)

and

a∧b=a0b0

e2

0

(a−b+a∧b)

=a0b0

e2

0

(a−b+1

2(a+b)∧(a−b)).(4.10)

10

It is important to distinguish inner and outer products in Gn+1 from inner and outer products

in Gn. The distinction is made here by using boldface for vectors in Gn.

Equation (4.10) relates two diﬀerent forms for the ‘Plucker coordinates’ of a line passing

through points aand b. It tells us that a∧brepresents a line in Vnwith direction a−b

passing through the point 1

2(a+b). We can put (4.10) in the form

a∧b=a0b0

e2

0

(1 + d)(a−b),(4.11)

where

d=(b∧a)·(a−b)

−1(4.12)

can be regarded as the ‘directed distance’ from the origin to the line. For three distinct

points a, b, c on the same line, we have a∧b∧c= 0, and application of (4.11) yields the

invariant ratio a∧c

b∧c=a0

b0µa−c

b−c¶.(4.13)

This is a ‘projective invariant’ in two senses: It is independent of the chosen projective

split, and it is invariant under collineations. Thus, for a collineation fthe linearity of the

outermorphism immediately implies

f(a∧c)

f(b∧c)=a∧c

b∧c.(4.14)

The interval ratio (a−c)/(b−c) is not a projective invariant, but it is an aﬃne invariant

because a0/b0is (see below). The classical cross-ratio for four distinct points a, b, c, d on a

line is given by µa∧c

b∧c¶µb∧d

a∧d¶=µa−c

b−c¶µb−d

a−d¶.(4.15)

This is a projective invariant in both senses mentioned above. Using (4.15), all the well-

known implications of the cross-ratio are easily derived. The considerable advantage of

using geometric algebra here should be obvious. The division in (4.15) is well-deﬁned

without abuse of notation. This is an important example of a projective invariant which is

not so simply expressed in terms of meet and join products alone.

The essential relation between aﬃne and projective geometry is a projective split with

respect to a preferred vector e0. The equation

x·e0=λ(4.16)

determines a 1-parameter family of hyperplanes in Vn+1 with normal e0. The aﬃne group

of Vnis the group of collineations on Vn+1 which leaves the projective split by e0and, hence,

the hyperplane equation (4.16), invariant. Thus, every aﬃne transformation fsatisﬁes the

equation

e0·(fx)=x·(fe

0

)=x·e

0.(4.17)

This is equivalent to the condition that e0is a ﬁxed point of the adjoint transformation f:

fe0=e0.(4.18)

11

(Henceforth, the underbar and overbar notations will be used to denote linear operators as

well as their outermorphisms.)

Of special interest in aﬃne geometry is the translation operator Tadeﬁned for a·e0=0

by

Tax=x+ax ·e0=x+e0·(x∧a).(4.19)

Its adjoint transformation is

Tax=x+e0(a·x)=x+a·(x∧e

0

),(4.20)

which obviously satisﬁes the e0invariance condition (4.18). The translation operator in-

duces an outermorphism of lines in Pn:

Ta(x∧y)=x∧y+a∧(yx ·e0−xy ·e0)

=x∧y+(x∧y∧a)·e

0.

For kpoints, this generalizes to

Ta(x1∧···∧x

k)=x

1∧···∧x

k+(−1)k(x1∧···∧x

k∧a)·e

0.(4.21)

This applies to all vectors in Vn+1 whether or not they ‘lie’ on a single invariant hyperplane.

The linear transformation (4.19) is actually a shear on Vn+1; however, it projects to a

translation on VnTo show that explicitly, write x0=Taxand note that x0·e0=x·e0,so

the projective split (4.8) gives the translation

x0=x+a.(4.22)

By applying the outermorphism (4.20) to x∧e0, we ascertain the important fact that the

projective split is not preserved by the outermorphism.

The advantage of the projective split for aﬃne geometry is precisely the advantage of

homogeneous coordinates, namely, it reduces translations to linear transformations. In the

next section we show how geometric algebra helps with a further simpliﬁcation, reducing

translations to rotations.

The projective split has important implications for metric geometry as well as aﬃne

geometry. For metrical applications we restrict the scalars to real numbers and we must

take signature into account. We write

Vn=V(p, q)(4.23)

when we wish to indicate that the vector space Vnhas the signature (p, q) imposed by its

associated geometric algebra Gn=G(p, q).

When the signature is taken into account in a split by e0, the split relation (4.2) takes

the more speciﬁc form

G(p, q)=G

+

(q+1,p)whene

2

0

>0,(4.24)

or

G(p, q)=G

+

(p, q +1) when e

2

0<0.(4.25)

12

To characterize orthogonal transformations, it is convenient to follow [3] and introduce the

concept of ‘versor.’ A versor Rin G(p, q) is any element which can be factored into a

geometric product

R=v1v2... v

k(4.26)

of unit vectors viin V(p, q). The factorization (4.26) is by no means unique, but there is

always one with k≤n. The multivector Ris even or odd according to whether kis even

or odd. This determines its parity under the main involution:

R∗=(−1)kR. (4.27)

An even versor R=R∗is called a rotor.

The versors in G(p, q) form a multiplicative group Pin(p, q) commonly called a ‘pin group’

[7]. The rotors comprise a subgroupSpin(p, q) called a ‘spin group’.

Now we can simply state what may be regarded as the fundamental theorem of metric

geometry: Every orthogonal transformation Ron V(p, q) can be written in the canonical

form

Rx=R∗xR−1=Rx(R∗)−1,(4.28)

where the underbar distinguishes the linear operator Rfrom its corresponding versor R.

This is called a rotation if R=R∗is a rotor.

A remark on terminology is in order. The word ‘versor’ comes from Cliﬀord [5] who got it

from Hamilton. Cliﬀord explains that it is derived from the word ‘reverse’ in the expression

‘reverse direction’. The term is employed here with the same motivation, but it is deﬁned

diﬀerently. According to (4.26), the most elementary versor is a vector R=v, in which

case (4.28) takes the form

Vx=−vxv−1.(4.29)

This represents a reﬂection Vin a hyperplane with normal v; its net eﬀect is simply to

reverse the direction of vectors collinear with v. Thus, each unit vector vcan be interpreted

as an operator which ‘reverses’ direction; whence ‘versor’. From this it should be clear

that the factorization (4.26) corresponds to the Cartan–Dieudonn´e theorem that every

orthogonal transformation of Vncan be expressed as a product of at most nsymmetries.

The main fact of interest here is that by a projective split with respect to e0Spin(p, q)

is related to Spin(q+1,p)ife

2

0= 1 or to Spin(p, q +1) if e

2

0=−1. Speciﬁcally, every rotor

in Gn+1 can be factored into a product

S=S0R, (4.30a)

where Ris a rotor in Gnand there exists a vector uin Vn+1 such that

S0=ue0.(4.30b)

The vector uis determined by noting that Sdetermines a rotation of e0into a vector

v=Se0S−1(4.31)

and requiring that S0also rotates e0into v.Thenwehave

v=S

0

e

0

S

−1

0=S

2

0

e

−1

0,

13

where the last step follows from assuming that S0has the form (4.30b). Solving for S0we

get

S0=(ve0)1/2=(v+e0)e0

|v+e0|=v(v+e0)

|v+e0|.(4.32)

Thus, u=(v+e

0

)|v+e

0|

−1is a unit vector ‘halfway between’ vand e0.NowRis given

by

R=S−1

0S, (4.33)

and it follows that

Re0R−1=e0.(4.34)

This implies that Ris in Spin(p, q)asclaimed.

Since (4.30b) has the form of the projective split (4.3), we can regard (4.30a) as a

projective split of the spin group. For the spacetime algebra G(1,3), Ref. [5] derives (4.30a)

as a spacetime split of a Lorentz transformation into a boost (or pure Lorentz transforma-

tion) and a spatial rotation. This has many important physical applications. Mathemati-

cians have also used the split (4.30a) to study the transitivity of spin groups on spheres

(e.g. Chap. 21 of [6]) without, however, recognizing its general connection with projective

geometry.

5. Conformal and Metric Geometry

We have seen that the projective split by a vector determines a geometrically signiﬁcant

relation between the geometric algebras Gnand Gn+1. This section shows that a split by a

bivector determines an equally signiﬁcant relation between Gnand Gn+2. Let e0be a ﬁxed

unit 2-blade in G2

n+2 and let xbe a generic vector in Vn+1 =G1

n+2. A linear ‘split’ of Vn+2

into vector spaces V2and Vnis determined by the equation

xe0=x·e0+x∧e0=x0+ρx,(5.1)

where

V2={x0=x·e0=−e0·x}(5.2)

and

Vn={ρx=x∧e0=e0∧x}.(5.3)

The signiﬁcance of the scale factor ρwill be discussed later. The vector space V2generates

a geometric algebra G2with pseudoscalar e0,andV

ngenerates a geometric algebra Gn.We

have the commutative relations

x0x=xx0,(5.4a)

e0x=xe0.(5.4b)

Hence, all elements of G2commute with the elements of Gn, and we can express Gn+2 as

the Kronecker product

Gn+2 =Gn⊗G

2

.(5.5)

14

This is the fundamental multiplicative decomposition theorem for geometric algebras. Equa-

tion (5.5) has been treated previously only as a formal algebraic relation. The present ap-

proach gives it a geometric meaning by developing it as split by a bivector. Note that the

vector space Vndeﬁned by (5.3) has a projective interpretation: The 2-blade e0represents

a ﬁxed line and x=x∧e0represents a plane containing that line; therefore Vnrepresents

a pencil of planes intersecting in a common line. We will not pursue the implications of

this fact for projective geometry. We will concentrate on metrical implications.

The decomposition theorem (5.5) was proved in Cliﬀord’s original article [5] for anti-

Euclidean signature. The signature does not play a crucial role in the proof, as the above

approach shows. However, when the theorem is used to classify geometric algebras by

multiplicative structure, the signature is important, as has been emphasized recently by

several authors ([8], [9], [10]). From the above, we ﬁnd that (5.5) separates into three

diﬀerent cases depending on the signature of G2:

G(p+1,q+1)=G(p, q)⊗G(1,1),(5.6)

G(q+2,p)=G(p, q)⊗G(2,0),(5.7)

G(q, p +2)=G(p, q)⊗G(0,2) .(5.8)

Note that e2

0= 1 in (5.6) and e2

0=−1 in (5.7) and (5.8); this determines the relative

places of pand qon the two sides of the equations. The algebras G(1,1) and G(2,0) are

isomorphic, though not geometrically equivalent, and both are not isomorphic to G(0,2).

There follows a host of theorems about the algebraic equivalence of algebras with diﬀerent

signature [8–10], but that will not concern us here. We will be concerned only with the

geometric signiﬁcance of (5.6), which we abbreviate by

Gn+2 =Gn⊗G(1,1) (5.9)

when the particular signature (p, q) is not a matter of concern. Anticipating the results to

follow, let us refer to this as a conformal split of Gn+2 by the unit 2-blade e0.

From our previous considerations, we know that every orthogonal transformation of

Vn+2 =V(p+1,q+ 1) has the canonical form

x0=Gx(G∗)−1,(5.10)

where Gis an element of the versor group Pin(p+1,q+ 1). The theorem of central interest

in the remainder of this section is that the orthogonal group O(p+1,q + 1) is isomorphic

to the complete conformal group C(p, q)onV

n=V(p, q); therefore Pin(p+1,q+1)isthe

spin representation of C(p, q). We will use the conformal split to prove this theorem and

analyze the structure of the conformal group and its spin representation in detail.

Representations of conformal groups in Cliﬀord algebras have been discussed by many

authors, notably Angles [11], Lounesto and Latvamaa [12], Ahlfors [13], Maks [14], and

Crumeyrolle [15]. The present study aims to show that the conformal split idea simpliﬁes

algebraic manipulations, clariﬁes geometric meanings, and reveals connections among alter-

native approaches. A complete, eﬃcient and systematic treatment of the conformal group

should be helpful in applications to physics. The most thorough discussion of the physical

signiﬁcance of the conformal group for spacetime has been given by Kastrup [16].

Our plan of study for the remainder of this Section is as follows. First, we analyze the

algebra G(1,1) and Pin(1,1) thoroughly. Though the results are well-known, the analysis is

15

necessary to establish notation for subsequent application. Next, we establish homogeneous

coordinates for Vnin Gn+2 and the general relation of the Pin group to the conformal group.

Finally, we determine a canonical factorization of the groups into elementary generators.

5.1. STRUCTURE OF G(1,1) AND Pin(1,1)

An orthonormal basis {e1,e

2}for the vector space V2=V(1,1) is deﬁned by the properties

e2

1=1,e

2

1

=−1,e

1

·e

2

=0,(5.11)

A null basis {e+,e

−}for V(1,1) is deﬁned by

e±=1

2(e1±e2).(5.12)

It has the properties

e2

+=e2

−=0 (5.13)

and

e+e−=1

2(1 + e0).(5.14)

This latter relation separates into

2e+·e−=1=2e

+

·e

1=2e

−

·e

2(5.15)

and

2e+∧e−=e0=e2e1.(5.16)

The bivector e0is the unit pseudoscalar for V2. It has the important properties

e2

0=1,(5.17)

e0e±=±e±=−e±e0.(5.18)

The null basis has the advantage of being unique (up to a scale factor), since it characterizes

intrinsic structure of V(1,1), namely its ‘isotropic cone.’ For this reason, we use the null

basis whenever feasible. However, null vectors are not elements of the versor group, so we

use e1and e2to represent reﬂections.

For comparison with the literature, it should be understood that G(1,1) is isomorphic to

the algebra L2(R)ofreal2×2 matrices, that is,

G(1,1) 'L2(R)(5.19)

To make the isomorphism explicit, we write a generic multivector Min G(1,1) in the

expanded form

M=1

2[A(1 + e0)+B(e

1+e

2

)+C(e

1−e

2

)+D(1 −e0)] ,(5.20)

where A,B,C,Dare scalars. The basis elements have the matrix representation

[e1]=·01

10

¸

,[e

2

]=·01

−10

¸

,[e

0

]=·10

0−1

¸

.(5.21)

16

Consequently (5.20) has the matrix representation

[M]=·AB

CD

¸

.(5.22)

The matrix representation preserves the geometric product, that is

[MN]=[M][N].(5.22)

The conjugate M

ecan be deﬁned by

M

e=(M

∗

)

†,(5.24)

where the dagger denotes reversion in G(1,1). The involute is given by

M∗=e0Me0.(5.25)

It is readily veriﬁed that

det [M]=M

eM. (5.26)

Therefore, we have the isomorphisms

{M|MM

e6=0}'GL2(R),(5.27)

{M|MM

e=1}'SL2(R),(5.28)

to the general linear and special linear groups.

The group (5.28) is a 3-parameter group. Its structure is revealed by the following

canonical decomposition: If MM

e= 1, then there are values of scalar parameters λ, α, β in

the interval [−∞,∞] such that (modulo ±1)

M=KαTβDλ,(5.29)

where

Kα=eαe−=1+αe−,(5.30)

Tβ=eβe+=1+βe+,(5.31)

Dλ=1

2(1 + e0)λ+1

2(1 −e0)λ−1=eφe0(5.32)

for λ=eφ. An alternative to (5.29) is the so-called Iwasawa decomposition

M=VαTβDλ,(5.33)

where, for α= cosh θ,

Vα=α+e1(1 −α2)1/2=eθe1.(5.34)

The Iwasawa decomposition has been much used in the theory of SL2(R) group represen-

tations [17]. However, it will be seen that (5.29) is more signiﬁcant in the present context,

because e+,e−and e0are geometrically unique generators, whereas e1is not.

17

For the matrix representation of (5.29), we get

[M]=[K

α

][Tβ][Dλ]

=·10

α1

¸·1β

01

¸·λ0

0λ

−1¸

=·λβλ

−1

αλ (1 + αβ)λ−1¸.(5.35)

For Vλgiven by (5.34) with λ=αthis gives the matrix representation

[Vλ]=[K

α

T

β

D

λ

]=·λβλ

−1

αλ λ ¸,(5.36)

where λ2=α/β =(1+αβ). This shows the equivalence of (5.29) and (5.33).

An important feature of (5.29) is that it separates distinct and unique 1-parameter sub-

groups Kα,T

β,D

λ, as well as such 2-parameter subgroups as TβDλ. Of these, it will be

seen below that only Dλbelongs to Pin(p+1,q + 1); however, Kαand Tβplay a role in

connecting Pin(1,1) to Pin(p, q).

The spin group of G(1,1) consists of

Spin(1,1) = {e0,D

λ},(5.37)

where Dλis given by (5.32). The rotor Dλrepresents the 1-parameter family of ‘Lorentz

transformations’ of V2. By (5.18),

Dλe±D−1

λ=D2

λe±=λ±2e±.(5.38)

As noted in (5.25), the rotor e0represents the involution

e0e±e0=−e±=e∗

±.(5.39)

The rest of Pin(1,1) is generated by e1, which represents the reﬂection

e1e±e1=e∓.(5.40)

The reﬂection represented by e2=e0e1is a composite of (5.39) and (5.40), and all other

reﬂections of V(1,1) cam be obtained by composites with rotations.

5.2. HOMOGENEOUS COORDINATES AND THE CONFORMAL SPLIT OF

Pin(p+1,q+1)

Like the projective split, the conformal split (5.1) can be regarded as assigning homogeneous

coordinates to the ‘points’ in Vn. There are two extra degrees of freedom in Vn+2, but one

of them can be eliminated by requiring x2= 0 for points xin Vn+2 representing points in

Vn. Then the conformal split (5.1) gives us

x2=(xe0)(e0x)=(x∧e

0+x·e

0

)(x∧e0−x·e0)=0,

18

whence

(x·e0)2=(x∧e

0

)

2=ρ

2

x

2.(5.41)

Unique homogeneous coordinates for xwill be determined by a suitable choice of scale

factor ρ. From (5.16) we get

1

2x·e0=x·(e+∧e−)=(x·e

+

)e

−−(x·e

−

)e

+.(5.42)

Squaring and using (5.15) and (5.41), we obtain

(x·e0)2=−4x·e+x·e−=ρ2x2.

Evidently, the simplest choice of scale factor is

ρ=2x·e

+,(5.43)

for then

x·e−=−x2x·e+.

Inserting (5.42) and (5.43) into (5.1), we get the conformal split in the form

xe0=2x·e

+

(e

−+x

2

e

++2x)≡ρX , (5.44)

where

x=x∧e0

x·e+

(5.45)

expresses xin terms of ‘homogeneous coordinates’ x·e+and x∧e0. Itisofinterestto

note, using (5.20) and (5.21), that (5.44) has the 2 ×2 matrix representation

[X]=·xx

2

ax

¸

.(5.46)

This particular kind of split into homogeneous coordinates should be generally useful in

that part of projective geometry dealing with a null quadratic form (‘Cayley’s absolute’).

We show below that it is very useful in conformal and metric aﬃne geometry.

To make the conformal split of the orthogonal transformation (5.10) explicit, we introduce

the abbreviation b

G=e0(G∗)−1(5.47)

and write (5.10) in the form

GX b

G=σX0.(5.48)

For x2= 0, we can use (5.44) to put this in the explicit form

G(e−+x2e++x)b

G=σ[e−+[g(x)]2e++g(x)] (5.49)

where

x0=g(x)(5.50)

19

is a transformation on Vninvolving a scale change

σ=σg(x)=x

0·e

+

x

0·e

+

.(5.51)

Equation (5.49) is the main theorem relating the group Pin(p+1,q+1) = {G}to the con-

formal group C(p, q)={g}. It has been studied by Angles [11], but without the conformal

split to simplify the analysis.

To prove the main theorem, it is necessary to establish that every function gdetermined

by Gin (5.49) is conformal and that every conformal transformation can be obtained in this

way. Instead of approaching the proof directly, we achieve it as a byproduct of analyzing

the structural relationships between the two groups in detail. The analysis can be carried

out in two complementary ways.

One way is to note, by considering (5.20), that every multivector Gin Gn+2 can be

expressed in the form

G=Ae+e1+Be++Ce−+De−e1,(5.52a)

where the ‘coeﬃcients’A, B, C, D are in Gn. The operations of reversion and involution

yield

G†=D†e+e1+B†e++C†e−+A†e−e1(5.52b)

and

G∗=A∗e+e1−B∗e+−C∗e−+A∗e−e1.(5.52c)

Equation (5.52a) diﬀers from (5.20) only in the replacement of scalars by elements of Gn.

Since the elements of G(1,1) commute with those of Gn, this replacement applies as well to

the matrix representation (5.22). Thus, we have the isomorphism

Gn+2 =Gn⊗G(1,1) 'L2(Gn),(5.53)

where the right side denotes the algebra of 2×2 matrices with elements in Gn. Accordingly,

for the matrix representations of (5.52a,b,c) we have

[G]=·AB

CD

¸

,[G

†

]=·D

†B

†

C

†A

†¸,[G

∗]=·A

∗−B

∗

−C

∗D

∗¸.(5.54)

The matrix representation has the advantage of compactness and easy multiplication, but

the disadvantage of suppressing the relation to vectors in G(1,1) and so complicating geo-

metric interpretation. This kind of matrix representation for the conformal group has been

employed by Ahlfors [13] and Maks [14].

The requirement that Gis an element of Pin(p+1,q+ 1) puts restrictions on the coeﬃ-

cients in (5.52a). First, Gmust satisfy the versor condition

GG†=|G|2,(5.55)

where |G|2is a nonvanishing scalar (possibly negative, unless the signature of Vnis Eu-

clidean). In the matrix representation (5.54), this condition is expressed by

[GG†]=·AD†+BC†AB†+BA†

CD†+DC†CB†+DA†¸=|G|2·10

01

¸

.(5.56)

20

Second, Equation (5.10) or, equivalently, (5.49) must be satisﬁed. The matrix representa-

tion of (5.47) can be written

b

G=·b

Db

B

b

Cb

A¸.(5.57)

with the notation

b

A=(A

†

)

∗

|G|

−2.(5.58)

Then (5.49) gives us the condition

[GXG†]=·(Ax+B)( b

D+xb

C)(Ax+B)( b

B+xb

A)

(Cx+D)( b

D+xb

C)(Cx+D)( b

B+xb

A)¸

=σ·g(x)[g(x)]2

1g(x)¸.(5.59)

where σis a scalar and g(x)∈V

n

. From (5.56) and (5.59) we can easlly read oﬀ a list of

necessary and suﬃcient conditions on A, B, C, D. In particular, from (5.59) we ﬁnd that

σ=(Cx+D)( b

D+xb

C)(5.60)

and gcan be expressed in the homographic form

g(x)=(Ax+B)(Cx+D)−1.(5.61)

Ahlfors [13] calls this a M¨obius transformation. The big advantage of the versor represen-

tation in P(p+1,q + 1) over the elegant M¨obius form for a conformal transformation is

that composition of transformations is reduced to a versor product. This, in turn, simpliﬁes

proofs and applications of a whole host of theorems, such as the canonical decomposition

theorem proved below.

To prove that (5.61) is indeed a conformal transformation, we compute its diﬀerential

g(a)= g(a,x) by taking the directional derivative. Using the fact a·∇x=a, we ﬁrst

compute

a·∇(Cx+D)

−1=a·∇"xb

C+b

D

σ#="ab

C(Cx+D)−a·∇σ

σ#(Cx+D)

−1,

whence

a·∇(Cx+D)

−1=−(Cx+D)

−1

Ca(Cx+D)

−1.(5.62)

Now,

a·∇g=[Aa−(Ax+B)(Cx+D)−1Ca](Cx+D)−1

=(xC

†+D

†

)

−1

[(xC†+D†)A−(xA†+B†)C]a(Cx+D)−1,

where g=g†has been used. Applying the special conditions on the ‘coeﬃcients’, we obtain

g(a)=a·∇g=(xb

C+ˆ

D)

−1

a(Cx+D)

−1.(5.63)

21

This immediately yields the deﬁning condition for a conformal transformation:

[g(a)] ·[g(b)] = a·b

σ2.(5.64)

Thus, the proof that gis conformal is complete.

Now we turn to an alternative way to analyze the group structure, exploiting the fact

(4.26) that every versor element Gcan be generated as a product of vectors in Vn+2. First,

we determine the explicit imbedding of Pin(p, q) in Pin(p+1,q+ 1) By the condition (5.2),

V2is a subspace of Vn+2 and vectors a, b in Vn+2 are in the subspace orthogonal to V2if

a·e0=b·e0= 0. According to (5.3), then,

a=ae0=e0a(5.65)

and

ab =ab ,(5.66)

where a=ae0=a∧e0,b=b∧e0are vectors in Vn. This determines how Pin(p, q) ﬁts

into Pin(p+1,q + 1) as a subgroup. Speciﬁcally, for every versor

R=v1v2...v

k(5.67a)

in Pin(p, q), there are kunit vectors vi=vie0in Vn+2 such that

v1v2...v

k=e

0Rfor kodd, (5.67b)

v1v2...v

k=Rfor keven. (5.67c)

From (5.43) we note that

a∧b=a∧b,(5.68)

which tells us that all bivectors in Gnare also bivectors in Gn+2, even though (5.3) says

that the vectors in Gnare 3-vectors in Gn+2. This implies that the Lie algebra of Spin(p, q)

is contained in the Lie algebra of Spin(p+1,q+ 1), but we will not be using that important

fact.

Now we are prepared to systematically classify the conformal transformations generated

by vectors in Pin(p+1,q + 1). First, we readily check that the imbedding of Pin(p, q)

deﬁned by (5.67a) generates the orthogonal group O(p, q) when employed in the main

theorem (5.49). Indeed, we ﬁnd b

R=(R

∗

)

−1and

g(x)=Rx(R

∗

)

−1(5.69)

as required.

5.3. INVERSIONS, INVOLUTIONS AND DILATATIONS

Next we classify the conformal transformations generated by Pin(1,1), which is also a

subgroup of Pin(p+1,q + 1). For a reﬂection of Vn+2 generated by e1, using (5.40) we

obtain

e1(e−+x2e++x)e1=x2(e−+x−2e++x−1),(5.70a)

22

whence

g(x)=x

−1=x

x

2.(5.70b)

Therefore, e1is a versor representation of inversion in Vn. Of course (5.70b) is not deﬁned

along null lines x2= 0. Similarly,

e0(e−+x2e++x)e0=−(e−+x−2e+−x).(5.71a)

Here σ=−1, and

g(x)=−x.(5.71b)

Therefore, e0is a versor representation of involution in Vn. Since e2=e0e1, it generates

the composite of (5.70b) and (5.71b). Therefore, e2represents an involutory inversion.

For Dλdeﬁned by (5.32), with the help of (5.38) we obtain

Dλ(e−+x2e++x)D−1

λ=λ−2[e++(λ

2

x)

2

e

−+λ

2

x],(5.72a)

whence

g(x)=λ

2

x.(5.72b)

Therefore, Dλis the spin representation of a dilatation by λ2. From (5.32) it is also readily

established that every unit vector in G(1,1) with positive signature can be expressed in the

form

e=Dλe1D−1

λ=e2φe0e1(5.73)

or the form −e=e0ee0. It follows that G(1,1) is a double representation of the group of

inversions, involutions and dilatations on V(p, q).

5.4. TRANSLATIONS, TRANSVECTIONS, AND DIVERSIONS

Our ﬁnal task is to classify the conformal transformation generated by versors with non-

vanishing components in both V2and its orthogonal complement. According to (5.1), any

such vector vcan be written in form

v=e+a, (5.74)

where e=(v·e

0

)e

0and a=ae0is inVn. Also, the versor condition

v2=e2+a26=0 (5.75)

must be satisﬁed, but it can be scaled to any convenient value. We have two distinct cases

to consider. Case I:e2= 0 implies a26= 0, so we can choose e=e+or e−. Case II:e26=0,

so we can choose e=e1or e2, since

e+a=Dλ(e1+a)D−1

λ(5.76)

by (5.73) enables us to get the other possibilitiesby composition with dilatations.

For Case Ilet us write v=e+a−1and consider the rotor

Ta=av =1+ae+.(5.77)

23

Anticipating its interpretation as the spin representation of a translation, let us refer to Ta

as a translator. By (5.18) and (5.65) we have ae+=ae0e+=ae+, so we can write

Ta=1+ae

+=1+e

+

a=1+ae+=Ta(5.78)

showing that T(a)is parameterized by vectors in Vn. Applied in (5.49), this yields

Ta(e−+x2e++x)Ta=e−+(x+a)

2

e

++(x+a),(5.79a)

thus determining the translation

g(x)=x+a,(5.79b)

as anticipated.

The translators form a group with the properties

TaTb=Ta+b(5.80)

T−1

a=T−a=e0Tae0.(5.81)

Note that if a2=−b2, then a+bis a null vector. Hence, (5.80) shows that the translator

(5.78) applies to null vectors, though in that case it cannot be factored into a productof

two vectors as in (5.77). However, (5.80) shows that it can be factored into a product of

four vectors.

Now deﬁne

Ka=e1Tae1=1−ae−=1+ae

−=K

a,(5.82)

where (5.40) and (5.18) have been used. Thus, the interchange of e+with e−in (5.78) is

the equivalent of an inversion followed by a translation and another inversion, so we can

use (5.70a) and (5.79a) to evaluate

Ka(e−+x2e++x)Ka=x2e1[e−+(x

−1+a)

2

e

++(x

−1+a)]e1

=x2(x−1+a)2[e−+(x

−1+a)

−2

e

++(x

−1+a)

−1

].(5.83a)

In this case,

σ(x)=x

2

(x

−1+a)

2=1+2a·x+a

2

x

2,(5.83b)

and

g(x)=(x

−1+a)

−1=x(1 + ax)−1=x+ax2

σ(x),(5.83c)

This kind of conformal translation is called a transversion. This completes the analysis of

Case I.

The analysis of Case Ican be summarized as follows. From (5.77) we conclude that the

vector v=a−1Ta=e++a−1represents a translation-reﬂection. Similarly, v=a−1Ka=

e−+a−1represents a transvection-reﬂection. Turning to Case II, where v=e1+a,we

ﬁrst ask whether vcan be generated from e1by a translation. Computing

Tae1T−1

a=e1+a−a2e+,(5.84)

24

we see that the answer is yes only if a2= 0. So we exclude a2= 0 from this case to assure

that it is completely distinct from the other possibilities.

Now for Case II let us compute the conformal transformation generated by the rotor

Va=ve1=(e

1+a)e

1=1+ae1=1+ae

2.(5.85)

The simplest way to do that is to write e2=e+−e−and, by comparison of (5.52a) with

(5.61), read oﬀ the result

g(x)=(x+a)(1 −ax)−1.(5.86)

This elementary type of conformal transformation and the fact that it is continuously

connected to the identity does not seem to have been noted previously in the literature,

probably because it cannot be generated by the more obvious classical conformal transfor-

mations. Let us call it a diversion.

[On the contrary, P. Lounesto has pointed out that the diversion (5.85) can be factored

into the product Va=λTaDλK−awhere λ2=1+a

2

.]

The eﬀect of interchanging e1and e2in (5.85) is merely to change the sign of the ﬁrst a

in (5.86), but this is a new kind of diversion not reducible to (5.86). According to (5.76), we

can obtain every other diversion by composition with a dilatation and its inverse. Hence,

the most general form for a diversion is

Va=1+ae, (5.87)

and we may call the diversion positive if e2=1ornegative if e2=−1.

An interesting result is obtained from the following product of two diversions:

(1 + ae2)(1 + be1)=(1+ab)e+e1+(a+b)e

++(b−a)e

−+(1−ab)e−e1.(5.88)

Note that all the coeﬃcients on the right are null if a2=−b2. This example was ﬁrst ad-

duced by Maks [14] as a counterexample to the common belief that every special conformal

transformation can be generated by rotations, translations, inversions and dilatations [in

any order] . However, we have shown above that the diversion is a more fundamental coun-

terexample, and its existence does not require an indeﬁnite metric. [Delete the preceding

sentence.] This concludes our analysis of Case II.

5.5. THE CONFORMAL DECOMPOSITION THEOREMS

The main theorem (5.49) establishes the double covering of the conformal group C(p, q)by

the versor group Pin(p+1,q + 1). Similarly, Spin(p+1,q+ 1) is a double covering of the

special conformal group SC(p, q), consisting of all conformal transformations continuously

connected to the identity. The fundamental decomposition theorem for SC(p, q) can be

expressed as follows: The rotor Grepresenting any combination of rotations, translations,

transversions, and dilatations can [usually] be put in the canonical form

G=±KaTbDλR, (5.89)

where the factors are deﬁned by (5.82), (5.78), (5.32) and (5.67c). [The counterexample of

Maks shows that the order of factors may matter in special cases.]

25

This result is easily proved with the help of the following relations to reorder and combine

factors

DλTa=Ta0Dλwhere a0=λ−2a, (5.90)

RTa=Ta0Rwhere a0=R(R∗)−1.(5.91)

From our preceding analysis, it follows that any element of SC(p, q) can be represented as

a product of (5.89) with the positive and negative diversions represented by (5.87).

The most general decomposition theorem for C(p, q) asserts that any element can be

represented in Pin(p+1,q +1)byversorsoftheformVG,e

0

VG,e

1

VG or e0e1VG,

where Vis a diversion and Ghas the form (5.89), but with Rof the general form (5.67a)

representing an orthogonal transformation.

Subgroups of the conformal group can be represented in geometric algebras of lower

dimension. The group Spin(p+1,q+ 1) belongs to the even subalgebra G+

n+2 =Gn+1,so

its action on vectors can be represented in Gn+1 by a projective split. Under a projective

split by e2, Equation (5.44) gives

xe2=xe0e1=1

2(1 −e0)+1

2(1 + e0)x2

1+x1,(5.92a)

where the scale has been ﬁxed by setting x·e+= 1, and the projected point in Vnis now

represented by

x1=(x∧e

0

)·e

1=x∧e

1+(x·e

1

)e

0.(5.92b)

Deﬁning

G−=e1b

Ge1=−e2(G∗)−1e2,(5.93a)

we multiply (5.49) by e1and use (5.92a) to put it in the form

G[1

2(1 −e0)+1

2(1 + e0)x2

1+x1]G−

=σ£1

2(1 −e0)+1

2(1 + e0)[g(x1)]2+g(x1)¤.(5.93b)

This is the representation in Gn+1. Here it is understood that Gis in Spin(p+1,q+1), so g

is in SC(p, q). Equation (5.93b) is the conformal representation obtained by Lounesto and

Latvamaa [12]. Evidently it has no advantage over (5.49). It is mentioned here to show

how alternative representations are related.

The metric aﬃne group A(p, q) is a particularly important subgroup of C(p, q). It is the

group of orthogonal transformations and translations of Vn=V(p, q), so it is represented

by versors in the canonical form

S=TaR. (5.94)

It is represented in the orthogonal group of Vn+2 as the stability subgroup of the null vector

e+. That is to say that Sbelongs to the subgroup of versors satisfying

Se+ˆ

S=e+.(5.95)

Since e+commutes with S, we can simplify (5.49) for this subgroup by multiplying it by

e+and using (5.13) and (5.14) to get

S(1 + e0+xe+)ˆ

S=1+e

0+g(x)e

+.

26

Note that the scale factor σ= 1 is ﬁxed in the subgroup. Since

Se0ˆ

S=e0,

we have the further reduction to the simple canonical form

S(1 + xe+)ˆ

S=1+g(x)e

+,(5.96a)

where ˆ

S=b

RTa=(R

∗

)

−1

T

a(5.96b)

is the spin representation of

g(x)=Rxb

R+a,(5.96c)

the general form for a metric aﬃne transformation.

Note that the entire group representation (5.96a,b,c) can be obtained formally from Gn

by extending the algebra to include an element e+which commutes with all other elements

and satisﬁes e2

+= 0. Cliﬀord himself extended the quaternions in exactly that way to

describe the Euclidean group of rigid rotations and translations in three dimensions [18].

His unfortunate early death prevented him from integrating that idea into his geometric

extension of Grassman algebra; there can be little doubt that he would have done so. Here

we see that the appropriate integration of Cliﬀord’s great ideas entials the identiﬁcation of

e+as a vector in a space of higher dimension. This is more appropriate than alternative

representations of the aﬃne group in geometric algebra, because of its embedding in the

full conformal group.

The aﬃne group representation (5.96a,b,c) has important applications to physics. When

the base algebra Gnis the spacetime algebra G(1,3), it is a representation of the Poincare

group. It is also of interest to note that the factorization

G(1,3) = G(2,0) ⊗G(1,1) (5.97)

implies that the Lorentz group for spacetime is isomorphic to the conformal group for

Euclidean 2-space.

References

1. Hestenes, D. and Ziegler, R.: Projective geometry with Cliﬀord algebra, Acta Appl.

Math. 23 (1991), 25–63.

2. Hestenes, D.: Universal geometric algebra, Simon Stevin 62 (1988), 253–274.

3. Hestenes, D. and Sobczyk, G.: Cliﬀord Algebra to Geometric Calculus, D. Reidel,

Dordrecht, 1985.

4. Whitehead, A. N.: A Treatise on Universal Algebra with Applications, Cambridge

University Press, Cambridge, 1898 (Reprint: Hafner, New York, 1960), p. 249.

5. Cliﬀord, W. K.: Applications of Grassmann’s Extensive Algebra, Amer. J. Math.1

(1878), 350–358.

6. Hestenes, D.: Spacetime Algebra, Gordon and Breach, New York, 1966, 1987.

7. Porteous, 1.: Topological Geometry, 2nd edn., Cambridge University Press, Cam-

bridge, 1981.

27

8. Salingaros, N.: On the classiﬁcation of Cliﬀord algebras and their relations to spinors

in ndimensions, J. Math. Phys.23 (1982), 1.

9. Li, D., Poole, C. P., and Farach, H. A.: A general method of generating and classifying

Cliﬀord algebras, J. Math. Phys.27 (1986), 1173.

10. Dimakis, A.: A new representation for spinors in real Cliﬀord algebras, in J. S. R.

Chisholm and A. K. Common (eds.), Cliﬀord Algebras and their Applications in Math-

ematical Physics, D. Reidel, Dordrecht, 1986.

11. Angles, P.: Construction de revetements du groupe conforme d’un espace vectoriel

muni d’une metrique de type (p, q), Ann. Inst. Henri Poincare 33 (1980), 33.

12. Lounesto, P. and Latvamaa, E.: Conformal transformations and Cliﬀord algebras,

Proc. Amer. Math. Soc. 79 (1980), 533.

13. Ahlfors, L. V.: Cliﬀord numbers and M¨obius transformations in Rn,inJ.S.R.

Chisholm and A. K. Common (eds.), Cliﬀord Algebras and their Applications in Math-

ematical Physics, D. Reidel, Dordrecht, 1986.

14. Maks, J. G.: Modulo (1,1) periodicity of Cliﬀord algebras and generalized (anti-)

M¨obius transformations, Thesis, Delft University of Technology, The Netherlands.

15. Crumeyrolle, A.: Orthogonal and Symplectic Cliﬀord Algebras, Kluwer Academic

Pubs., Dordrecht, 1990, Chap. 12.

16. Kastrup, H. A.: Gauge properties of the Minkowski space, Phys. Rev.150 (1966),

183.

17. Lang, S.: SL2(R), Addison-Wesley, Reading, Mass., 1975.

18. Cliﬀord, W. K.: Preliminary sketch of biquaternions, Proc. Lond. Math. Soc. IV

(1973), 381–395.

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