Springer chevalley c the algebraic theory of spinors and clifford algebras (collected works vol 2)(springer 1997)(l)(t)(113s)

In these lectures we shall present a more general algebra called “Clifford algebra” associated to a quadratic form.. If the quadratic form reduces to 0, The applications of the theory of

Claude Chevalley in a kimono gown during a Bourbaki meeting, August 1954

(Photograph by H Cartan Reproduced by kind permission of L-Association

des Collaborateurs de Nicolas Bourbaki)

and Clifford Algebras

Collected Works, Volume 2

Claude Chevalley † Catherine Chevalley

CNRS

Ecole Normale Supérieure 75013 Paris

Includes bibliographical references and index

algsbras,

ISBN 3~-540-57063-2 (v 2 : Bertin :

ISBN 0-367-57063-2 (v 2 : New York :

hardcover ; atk paper) hardcover : atk paper)

Mathematics Subject Classification (1991): 15A66, 15463; 53A50

ISBN 3-540-57063-2 Springer-Verlag Berlin Heidelberg New York

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OF =1:

PENNSYLVANIA LIRHAHIRS

in an edition of collected papers of a mathematician, but quite a number of them are concerned with theoretical politics ( ) they reflect an aspect of myself the omission of which would, I think, give a wrong idea of my lines of thinking” On the other hand, Chevalley thought that the Collected Works

of a mathematician ought to be read not only by other mathematicians, but also by historians of science But the history of mathematics could not be anything pure and detached from the world of general ideas: “I think that history of mathematics should not be what it too often is, namely a collection

of statements of the form ‘in the year X, mathematician A proved theorem B’ , but should study the relationship between such and such a mathematical trend and the general epistemological, philosophical or social trend at the time of a certain publication” For these two reasons, he did not want his technical papers to be published separately from his other work Though he was never fully satisfied with the various ways in which he himself spoke about the connection between his own mathematical achievements and the

“epistemological, philosophical or social trend of ideas” that surrounded him, still he clearly wanted to bear witness to such a connection

Chevalley’s second wish had to do with some out-of-date features, and also typographical defects, of his mathematical papers: “As for the mathematical papers, I know that some of them contain statements which are either false or

at least inaccurate, and I do not see the interest of publishing statements of theorems which might be misleading to the reader Of course, this drawback might be erased by the insertion of appropriate notes; the trouble is that these papers bear upon matters on which I have not thought for a long time, and that it would mean a large amount of work to check every sentence of them,

a work which I do not particularly wish to undertake myself, and a pensum

Pee eee het ee ee

vị FOREWORD

that I would not like to inflict on anybody else” So described, the logic of the

situation would have led directly to the abandonment of the very project of

publishing the Works, if various people had not generously agreed to devote

some of their time to the above-mentioned task of proof-reading

Finally, Chevalley also wished to add to the papers already published

a number of unpublished manuscripts, mathematical and non-mathematical

The mathematical manuscripts were to include two long “rédactions Bourbaki”

that Bourbaki did not accept - a familiar predicament for the members of the

group - namely, “Introduction to Set Theory” and “Elementary Geometry” As

Chevalley wrote:“That choice should in my opinion include two unpublished

papers which were written for ‘Bourbaki’, but were not accepted for publica-

tion, and which are among what I consider as the best of my mathematical

endeavours” Together with these two manuscripts, Chevalley hoped to publish

several other things: “I would like to include some texts concerned with things

I have thought of during these last years, but which are not yet in publishable

form” (letter to K Peters, June 8th, 1983) At the beginning of 1984, he was

working on a list of all the unpublished material that he wanted to bring to

light ‘This list, which he was not able to complete, will be published in Volume

I (Class Field Theory) of the Collected Works, together with the integral text

of the letters which we have been quoting above

Thanks to the help and support of Springer-Verlag and the French

Naitonal Center for Scientific Research (CNRS), it has now become possible

to publish Chevalley’s Works in a way that should fulfill the essence of his

requirements, and we hope that the volumes that will come out will provide

a fitting image of his contributions and personality

Fach volume will be devoted to a special theme and will feature an

introduction by a specialist of the field As it happens chronological ordering

and thematic ordering are almost identical, and the only discrepancy will

be with the non-technical papers and the unpublished manuscripts, often

difficult to date back The first volume, “Class Field Theory” , includes

the two obituaries that were written by J Tits and J Dieudonné after

Chevalley died in 1984 Letters by and to Jacques Herbrand and Emmy

Noether will be published in the volume on epistemology and politics We

hope to establish a complete bibliography, to be included in the last volume,

that will otherwise include large parts of the unpublished material Finally,

most of the mathematical or philosophical correspondence that Chevalley held

with other people is missing This is partly due to the fact that his healthy

disrespect for glory and the absence of a personal need to keep a record of

his own existence had devastating effects on his archives We will therefore

be grateful for copies of any such letters, in case they exist and seem to be

interesting or important

Collected Works of Claude Chevalley

Foreword to this Volume

This volume represents the first step in the ambitious project of publishing Claude Chevalley’s Collected Works The Project is supported by a contract

(code name : GDR 942) with the French Centre National de la Recherche

Scientifique (CNRS), and I am acting as chairman of the editorial committee, Our idea was to collect in this volume the writings of Claude Chevalley about spinors This is a rather minor variation in his scientific work, but well in tune with his long-standing interest in group theory When Chevalley wrote his two books (here reprinted as the main two parts), spinors were a well- established tool in theoretical physics, and E Cartan had already published his account of the theory But Chevalley’s approach to Clifford algebras was quite new in the 1950's, at a time where universal algebra was blossoming and developing fast This explains why we are reprinting his Nagoya lectures about

“Some important algebras” As explained in the review by Jean Dieudonné, originally published in the Bulletin of the American Mathematical Society and appended here, Chevalley’s exposition of the algebraic theory of spinors contains a number of interesting innovations But Chevalley was an algebraist

at heart, and gives no hint of the applications to theoretical physics Since the 1950’s, spinors (and the associated Dirac equation) have developed into

a fundamental tool in differential geometry and especially in the theory of Riemannian manifolds The Postface by Jean-Pierre Bourguignon aims to retrace this new line of mathematical thinking and to provide an up-to-date

account

Some editorial work was required while producing this volume We felt

an obligation to proofread carefully all these texts (see the comment by Dieudonné), and to correct misprints and occasional slips of the pen But the text has remained essentially unaltered

We have to thank a number of people for their cooperation in this project The members of the Chevalley Seminar (and especially Michel Broué, Michel Enguehard and Jacques Tits) gave us their continual moral support and exerted friendly pressure We thank also Jean-Pierre Serre and Armand Borel for their advice and steady insistence S lyanaga, a life-long friend of Chevalley, was instrumental in securing the permissions needed to reprint the Japanese lectures; to him, and to the officers of the Mathematical Society

of Japan, we extend our warmest thanks Henri Cartan lent us his own

copy of Algebraic Theory of Spinors for reproduction purposes and made the

suggestion of appending Dieudonné’s review The staff of the I.H.E.S was very

helpful: we thank especially Marie-Claude Vergne for her dedicated typing and

the directors Marcel Berger and Jean-Pierre Bourguignon for their support of

the project As mentioned above, we have to acknowledge financial support

by the C.N.R.S

Without the faithful friendship of Catherine Chevalley, nothing would have

been possible A special thank-you to her!

Contents

THE CONSTRUCTION AND STUDY

OF CERTAIN IMPORTANT ALGEBRAS

1 Tensor Algebras 2.2.0.0 .cccecescccccsececcces sn

2 Graded Structure of Tensor Algebras

3 Derivations in a Tensor Algebra —

4 Preliminaries About Tensor Product of Modules

5 Tensor Product of Semi-Graded Algebras .,

CHAPTER III CLIFFORD ALGEBRAS

- Clfford Algebras_

- Exterior Algebras 2,

N= - Canonical Anti-Automorphism

- Derivations in the Exterior Algebras; Trace

- Orthogonal Groups and Spinors (a Review)

Oo oa - CHAPTER IV SOME APPLICATIONS OF _ EXTERIOR ALGEBRAS ccesccsscesensesecececececccccce 1 Plũcker Coordinates

Z zxponential Mapping

3 Determinants ọ hii ee ee ee ee i Structure of the Clifford Algebra when M has a Base

No

q

10

13

18

20

20

23

29

32

35

35

37

38

43

45

48

52

52

03

Sĩ

62

xii CONTENTS

THE ALGEBRAIC THEORY OF SPINORS

PRELIMINARIES .cccccevcccccccccccccccecevreececuceessees 69

1 TerminoÌOBY dc c no Ho Ho non HH HH mm ng 69 2 Associative Algebras «se sec s° 70 3 Exterior Algebras .e‹ se sec se c2 70 CHAPTER I QUÁDRATIC FORMS 72

1.1 Bilinear Forms .«««««c n cecccŸ se 72 1.2 Quadratic Forms .-e<.- 75 1.3 Speclal Đas©s co «ch HH mm Ự n6 k1 Tĩ 1.4 The Orthogonal Group ‹ 79

" 1.5 Symmetries 2.0 ccc cece cece cece cece c sete cteecteces 83 1.6 Representation of G on the Multivectors 86

CHAPTETI H THE CLIFFORD ALGEBRA 101-

2.1 Deñnition of the Cliford Algebra 101

2.2 Structure of the CHÑữord Algebra 106

2.3 The Group of CHford - 113

2.4 Spinors (ven Dimension) 119

2.5 Spinors (Odd Dimension) 121

2.6 Imbedded Spaces «eo nen 122 2.7 Extension of the Basic Field 124

2.8 The Theorern of Hurwitz ‹ 125

2.9 Quadratic Forms over the Real Numbers 129

CHAPTER ITI FORMS OF MAXIMAL INDEX HH VY S11 my 134 3.1 Pure SpÏnOorg c‹ccc c n Q S1 v2 135 3.2 Á Bilinear InvarianL c 141

3.3 The Tensor Product of the Spin Representation with lír siiNNGaaiii 148

3.4 The Tensor Product of the Spin Representation with Itself (Characteristic zÝ 2) .se- 153 3.0 Imbedded Spaces 161

3.6 The Kernels of the Half-Spin Representations 165

3.7 The Case m = 6 oo cece ccc cece cece cer enstenvececses 166 3.8 The Case of QOdd Dimension 170

CHAPTER [V THE PRINCIPLE OF TRIALITY 176

4.1 A New Characterization of Pure Spimors 177

4.2 Construction of an Algebra e« 177 4.3 The Principle of Triality cc se cc°› 181 4.4 Geometric Interpretation .- - 185

CONTENTS

SYMBOL INDEX

BOOK REVIEW (J DIEUDONNE)

SPINORS IN 10995 (J.-P BOURGUIGNON)

eee bee es eee eee ee eee eee eee teehee eee eee na ca vn ncCh

TH Tal BAPE on kia arx reo akear Sao acc co

———— — —— — eer re eee tree ee eee eee

ses e+e te ete eee teeta t ear aneeen

Seeeeeteece eee eee treet eee eee ee eae nee tee meee

The Construction and

Study of Certain Important Algebras

TL Le ee oat eed oe ear

In these lectures we shall present a more general algebra called “Clifford algebra” associated to a quadratic form If the quadratic form reduces to 0,

The applications of the theory of exterior algebras are very wide, e.g.: ˆ

theory of determinants, representation of linear variety in a projective space using Plticker coordinates, and the theory of differential forms and their ap- plications to many branches of analysis But I am sorry not to be able to describe them in detail because of the limitation of time

Conventions

Throughout these lectures, we mean by a ring a ring with unit element 1 (or

1’ as the case may be), and also by a homomorphism of such rings a homo-

morphism which maps unit upon unit A will always denote a commutative

ring

By amodule over A, we invariably mean a unitary module Thus a module

over A is a set M such that

1) M has a structure of an additive group,

2) for every a € A and z € M, anelement ar € M called scalar multiple

is defined and we have

i) a(z + y)= az + ay,

ii} (a+ f)r= ox + Bx,

ii) — a(z)= (o/Ø)z,

A map of a module over A into a module over A is called linear if it is

a homomorphism of the underlying additive groups which commutes with

scalar multiplication by every element of A

Analgebra E over A means a module over A with an associative multipli-

cation which makes F a ring satisfying

A homomorphism of algebras will always mean a ring homomorphism which

is linear An ideal of an algebra means always a two-sided ideal A subset S

of an algebra is called a set of generators of E if E is the smallest subalgebra

containing S and the unit 1 of E

In dealing with modules or algebras over A, an element of the basic ring A

is often called a scalar In the case of algebras, any element of the subalgebra

A -1 is called a scalar; a scalar clearly commutes with every element of the

The first basic type of algebras we want to consider is the free algebra

Let & be an algebra over A generated by a given set of generators (*:);e¡

(I: any set of indices) Let ơ = (¡, - sin) be a finite sequence of elements

of I and put yo = 2;, +£;, The number h is called the length of 0 Among the “finite sequences”, we always admit the empty sequence oo, Whose length

is QO, ie., a sequence with no term, and we put Yoo = 1 We define the composition of two finite sequences g = (i), -,i,) and o/ = (71: - 7k}

is the unit for this composition Evidently this composition is associative:

(ga')o” = o(0'o"), and we have yoo’ = YoYo"

Theorem 1.1 Every element of E is a linear combination of the y~’s,

o running over all finite sequences of elements of I

Proof Denote by EF, the module spanned by all the Yo's We shall show

b = £&) First we prove:

Lemma 1.1 £ is closed under multiplication

Proof Let z, 2’ be two elements of F, and put

z= > aa¥o: z'= »¬

ữ ơ

Though these two sums seem apparently infinite, we have in fact a, = 0 and

a, = 0 except for a finite number of o’s Then we have

Ø,ơi

Ta

the sum being finite, we have zz’ € q

Now we return to the proof of Theorem 1.1 The module Fy, is thus a subalgebra of EF, and if ¢ = (i),y, = 2; and also Yoo = 1 Therefore F),

GRADED ALGEBRAS containing the set of generators (z;} and 1, contains E itself, so that we

obtain E = E¡, which proves Theorem 1.1

Definition 1.1 If the y,’s are Jinearly independent over A, then E is

called a free algebra, and the set (#; ier *8 called a free system of generators

of E

Existence and uniqueness of free algebras We first prove the unique-

ness For this, we shall show a more precise condition called “universality”

An algebra F over A with a system of generators (2;),<; is called universal

if, given any algebra E over A generated by a set of elements (€i);¢7 indexed

by the same set J, there is a unique homomorphism y : F — £ such that

yp (ai) = &; for all i

Theorem 1.2 A free algebra F with its free system of generators is

untuersal

Proof By definition, the set {y, = ai, -+24,} forms a base of F as a

module over A Thus there is a linear mapping ~: F — E such that

(1) @(Wø) =ếu ''ếu — for every ở =(in, -, tp)

Ifo = (i1, +,%n), o = (j1, +, je) are two finite sequences of elements of I,

we have

(2) ~ (Yoyo) = (Yoo!) = Gin + bin Ej, ** Ex, =P (Ye) 9(y,")

This proves that is not only linear, but also a homomorphism of F into #,

Especially putting o = (i) resp ¢ = go, we have yp (zi) = & and y(1) = 1,

which proves our assertion

Remark that, in general, any homomorphism y is uniquely determined

when the values y (z;) on a set of generators (xz) are given

Corollary The free algebra generated by (t:),<, is unique up to iso-

morphism More precisely, let F, F’ be two free algebras with free systems of

generators (Xi)cz) (241), er Tespectively, and let I and I’ be equipotent Then

F and F" are isomorphic

Proof We may assume that I = I’ By Theorem 1.2, we have two homo-

morphisms

@: FF such that y(2;) = x!

and

py’: Fi’ oF such that y’(x{) = x;

The composite mapping! y’oy: F — F!’ + F maps each 2; to itself, and by

the uniqueness of homomorphism, yp’ o y must be the identity in F Similarly

` p' op is defined by y' oy(2z) = y'(v(2))

GRADED ALGEBRAS yo’ is the identity in F’ Therefore y is an isomorphism and y = yp}, which proves that F and F’ are isomorphic to each other

Now we shall prove the existence of a free algebra, having any given set

(zi) ;¢y a5 its free system of generators Let 37 be the set of all finite sequences

of elements of I From the theory of linear algebra, we may assume that there

exists a module M over A with a base equipotent to D Let (ys),en be the

base of M; we introduce a structure of algebra into M For this, we have only

to define an associative multiplication for the elements of the base We define

it by

YoYo! = Yao'-

Since the composition in Y is associative, we have the associativity: (YotYor) Yor

= Ye (Yor¥or)- Thus M is a free algebra over A having the free system of gen-

erators (2;),¢7-

2 Graded Algebras

Let F be the free algebra with the free system of generators (1;) ;er: aNd put

Yo = Ti,°** Li, (7 = (i1, +,%,)) We shall classify the elements y, by the

length of oc

Let £, be the module spanned by the y,’s, o being of length A Then F

is the direct sum of Fo, F,, Fa, - as a module:

GRADED ALGEBRAS _

By a homomorphism of a I'-graded algebra E = > Ey tnto another

ver I’-graded algebra E! = » Et, is meant a homomorphism p:E — E of the

algebras such that @ (E,) C đt

Jin a I’-graded algebra F = )~ E, an element belonging to E., is called

homogeneous of degree The zero element 0 of & is homogeneous of any

degree, but each element of E other than 0 is homogeneous of at most one

degree y € I Any element z of F is uniquely decomposed into the sum of

Since #ø - ey € Ez+„, we must have Tg -€g = 2%, and #s - ey = Ú for all + z 6

This implies that eg is a right unit element for all homogeneous elements, and

accordingly for all elements z = 2 /zx„ in & Thus eg = 1, and our assertion is

proved

Corollary Scalars are homogeneous of degree Ð (P : zero element of P )

Among others, the following two special types of I’-gradations are of much

importauce:

case, we say simply “graded” instead of “Z-graded”

ii) I’-gradations where I is the group with two elements 0 and 1 In

this case we write E = + ®E_ in place of E = Ey ® #\; and # is called

semi-graded

GRADED ALGEBRAS

A free algebra F = Fy @ F, @ -@ Fi, ® - can be considered as a graded

algebra with F;, = {0} for all h < 0

Remark A I’-graded algebra is not a special kind of algebra In fact, any algebra may be considered as a I’-graded algebra with degree @ for every element

Homogeneous submodules

Definition 1.3 A submodule M of aI’-graded algebra E = > E, is said

to be homogeneous if the homogeneous components of any element of M still belong to M This is equivalent to the condition that M =) (MN E,)

+ Theorem 1.3 [f a submodule M or an ideal U of a P’-graded algebra EF

is generated by’ homogeneous elements, then it is homogeneous

Proof Let M be a submodule of F spanned by a set § of homogeneous elements and let Af’ be the set of elements of Af whose homogeneous com- ponents belong to M It is evident that S C M’ C M, since S$ consists of

homogeneous elements We shall show that M’ is a submodule If 2 = 3+

and a’ = $a! are in M’, then z-a’ = Dd (2, +25), and 2, kal eM,

so that we have «+ 2’ € M’ Also for a € A, we have similarly az € M’

Thus M’ being a submodule containing the set Š of generators of M, we have M’ > M, and so M = M’, which proves that M is homogeneous

For the case of ideals, we consider the ideal {{ generated by a set S of homogeneous elements Then Lf is spanned, as a module, by all elements of the form zsự, where z € E,s € Ở and g € E Putting z = ồ”#+„, = >)1ø;

Let E = 3) EF, be a I’-graded algebra and Ma homogeneous ideal in EF

We have the direct sum decomposition of Lf into its homogeneous parts:

if every element of M is a linear combination of the elements of S, while in the latter case, an ideal L{ is generated by S if Lf is the smallest ideal containing the

a ain a aa Para "Nợ _

GRADED ALGEBRAS

=3, 1 =UnEy

7 The quotient algebra £/L{ has also the structure of I’- graded algebra, because

k/H= 2 „(Ey/1L,) (direct sum of submodules) and (E,,/31,) (By [iby ) C

Fy4y'/M 47 Therefore £/U is a P-graded algebra and 3+ (By/ÄL,) gives its

homogeneous decomposition The canonical homomorphism w : FE E/i is

a homomorphism not only of algebras, but also of P'-graded algebras

3 Homogeneous Linear Mappings 3

Let E, E’ be two I’-graded algebras over the same ring A, and let A be a

linear mapping of F into £’, i.e., a mapping A: E — E’ such that

A(z+)=À(z) +A(w), — A(œz) = œA(z)

Definition 1.4 Let v be any element of I ; À i3 called homogeneous of

degree v if (Ey) C El, for allyeT

Evidently, if À : # — E’ is homogeneous of degree v and \’: FE’ + FE" is

homogeneous of degree v’, then 4’ o X is homogeneous of degree v + v’

A linear mapping 1: E — E’ cannot always be decomposed into a finite

sum of homogeneous mappings as can be shown by a counter-example But if

the decomposition is possible, it is unique; it is sufficient to prove the following:

Lemma 1.3 Let {AL} , be a family of linear mappings E > E', in

which each X, is homogeneous of degree v If À„(z) =0 (& : any element in

b) except for a finite number of v € I and 3 „Àu = 0, then A, = 0 for all

ver

Proof For an element 2x, of E,, , we have Sov (x) = 0, but since

ư

Av (zy) € El, for each vy € I, we have À„ (zy) = 0 for all e T For an

arbitrary 2 € E, let a= >* x be the homogeneous decomposition of x; then

Av(x) = 37 Av (ty) = 0, which proves that 1, = 0 (v EI’)

3 This notion can be defined not only for graded algebras, but also for “graded

modules” But we shall restrict ourselves to the case of graded algebras, because

we use it in this case only

GRADED ALGEBRAS

4 Associated Gradations and the Main Involution

any I’-graded algebra E = > úy; We associate the following Tgradation of

ver

E For each 7 € I, put

yer (7)

Then obviously FE = »- and E~- E>,C E~,~, In this way E = dL Ee

yer

can be considered as a I’-graded algebra

Definition 1.5 The P’-gradation » E- is called the T'-gradation of E

Lemma 1.4 Every homogeneous element, every homogeneous submod- ule, and every homogeneous ideal in & are also homogeneous in ET

In the special case where I is the group consisting of two elements 0 and

1, and where 7 is onto, we write E* = E} @ E* instead of E7 = Eo @ Fi, and we call it the associated semi-graded algebra of E In that case, the

kernel +—1(0) C I is denoted by P°,, which is a subgroup of index 2, while

T~!{(1) C P is denoted by F'_, which is the coset of P with respect to I, other than 7+ Remark that every subgroup of FP of index 2 can be preassigned

as I’; in some unique associated semi-gradation It may happen that I has

a unique subgroup of index 2 If it is the case, then reference to the map + can be omitted without any ambiguity For example, to every graded (i.e., Z-graded) algebra £& = o_ 4h 3s associated a unique semi-graded algebra

GRADED ALGEBRAS Main involution Fixing a subgroup I, CT of index 2, let FE = > E., be

ver

a Ï-graded algebra, and let E* = £4 @E* be the associated semi-gradation

of & Every element z € # can be decomposed uniquely into the sum of its

&-component 24 and its £*-component z_ :2 = v4 + x_ If we define a

map J: & > E by

\ J(z)=z+—z_ (z=z+y+z_e€E),

maps unit upon unit, and is an involution (i.e, JoJ = identity) Moreover, J

preserves the multiplication In fact let t= 24 +2_, y= Ye +y- (1,4, €

#4:2-, U- € E*) Then (zy), =24y,+2_y_, (zy) = 2_y, +24y_, and

so we have

J(zy) = (r4y4 +2 _y_) — (ty, + #+1—)

= + —#_)(t+ — 9~) = J(z)7(p)

Therefore, J is an involutive automorphism of the IP-graded algebra F,

which we call the main involution of E

For convenience’s sake, we define the symbolical power J “(vy EF) of the

inain involution as follows:

~ | identity if ver, ~

Also we define the power (—1)” (v € I’) of the scalar (—1) of A as follows:

cara {al over

Then we have, just as in the case of usual powers, the following identities:

i) Yo JY = put’

ii) (-1)”(—1)” = (-1)"*”

ii) (J9 = (7)

iv) ((—1)’)” = ((-1)”)”

We shall denote iii) and iv) respectively by J’ and by (—1)”“ for the

sake of simplicity, though no product is defined in general in I’ Any power

of the identity map is understood to be the identity map, and any power of

5 Derivations The definition of derivations in a graded algebra given here is somewhat different from the conventional definition of the derivations in the ordinary algebraic systems In the sequel, when we speak of derivations, we understand that a fixed subgroup I, C I" of index 2 is given

Now, let E, E’ be two I’-graded algebras over A and let y be a homo- morphism of E into £’

Definition 1.6 A ~y-derivation D of E into E’ means a linear mapping D:E -— E’, homogeneous of some given degree v € I", such that for +, in

where J” is the power of the main involution defined above

In the case where F = E’ and ¢ is the identity, D is called simply a

” derivation” Therefore a derivation D of F isa homogeneous linear mapping

of degree v, such that

(2) D(zy) = D(z)y+(J’z)D(y) for 2, ye E

If I = Z, the additive group of integers, (2) can be written as

(2) D(xy) = D(z)w+ (—1)zD(w) for ce En, ye LE

If the elements of F are all of degree @ (@ : zero element of PF), then D

must be of degree 9, and (2) reduces to

(3) D(zy) = D(x)y + zD(y),

which coincides with the ordinary definition of derivation Also, when v be-

longs to Fy formula (2) reduces to (3), while if w belongs to F_ and 2 € E*

then (2) reduces to

A derivation of degree v in Ƒ'_ is sometimes called “anti-derivation”, but we

do not use this terminology in these lectures

GRADED ALGEBRAS The formula (1) can be written in another form Denote by Lz the oper-

ation of left multiplication by 2: Ly = xy Then (1) is equivalent to

In the case where EF = E', and œ is the identity,

Remark that (5) and (6) do not contain the “parameter” y

Lemma 1.5 For every y-derivation D, we have D(1i) = 0

Proof Substituting 2 = y = 1 in (1), we get

]_ p)= Dạ -1)= p()e)+ø(2“) pụ), and since J’1 = 1, (1) = 1, we.obtain D(1) = D(1) + D(1), which proves

D(1) =0

Evidently, if D and D’ are »-derivations of the same degree, D + D’ is

again a y-derivation Also we have

Lemma 1.6 Ifyg: E— E’ andy: E' > E" are homomorphisms and

if D,D' are a @-deriuation of E into E’ and a y’-derivation of E’ into E"

respectively, then yp’ o D and D’ ow are (wp! 0 w)-derivations of E into E”

Proof We have only to check the condition (1) By direct calculation we

Theorem 1.4 Let D be a y-derivation of E into E', F a homogeneous

subalgebra of E, S a set of homogeneous generators of F, and let F’ be a

homogeneous subalgebra of E’ Then if D(S) CF’ and (5S) C E", tue have

D(F) C F' and g(F) C F1,

Proof The latter inclusion is evident, because 9 is a homomorphism

The former is proved as follows Let F, be the set of elements 2 € F such

that D(x) € F’ It is evident that Fy is closed under addition and scalar

multiplication Also if D(x) € F’ and x = 3,+, then the D(z,)’s are the

homogeneous components of D(x) hence D(x) € F’, so we obtain x, #1

Therefore F; is a homogeneous submodule of #, so that z € 1 implies J”z €

#ậ Now for zø,t in F;, we have

GRADED ALGEBRAS

D(xy) = D(x)p(y) + p(J"2)D(y),

and since (+), @(9), @(J”z), D(w) all belong to F’ , we have zy € F\, which proves that F; is a subalgebra containing S$ Since S is a set of generators of

F’, we have F C F\, which proves D(F) C F’ Corollary 1 Let S and if be

homogeneous ideals of E and E! respectively, and S be a set of homogeneous

generators of U If D(S) CL’, (8) C 1Ứ, tue haue D(U) C 1, and (1) C

4E, Proof Again the latter inclusion is evident The former is proved in a similar manner as before, showing that the set

is a homogeneous subalgebra, which proves F C tạ

Corollary 3 Let F, S be as before If two ~p-derivations D,D’ coincide with each other on S, then they coincide on F

Proof From this assumption, D and D’ are of the same degree Then apply Corollary 2 to the derivation D — D’

It follows from this corollary that a derivation D is completely determined

if its values on the elements of a set of generators are given

Theorem 1.5 Let E, E’ be I’-graded algebras, ~ a homomorphism of E

into E’, and D a -derivation of E into E’ Also let U and i be homogeneous ideals in E and E’ respectively such that D(i) C i’, and (1) C 1 Under these assumptions, the induced mapping D : EB/U— E'/i! obtained from D

ts a p-derivation, where @ means the induced homomorphism B/U E!/1

obtained from yp

* Note that this assertion holds without any assumption on ¢

where ý and tý are the canonical mappings

Proof From the theory of mappings of modules, it is easy to see that

D is a linear mapping which makes the diagram commutative The other

conditions (D being homogeneous and satisfying (1)) are proved by direct

calculation from the definitions

D is called the derivation deduced from D by going over to the quotient

algebra E’/L

Hereafter to the end of this paragraph, we assume that = E and ip is

the identity

* In a diagram, let every vertex represent a set, and let each oriented edge rep-

resent a mapping A directed path in a diagram represents a mapping which

is the composition of the\successive mappings assigned to its edges If, for

any two vertices, any two directed paths connecting them give the same map-

ping, then the diagram is said to be commutative For example in the dia-

gram depicted below, for the vertices P and Q and the paths as in it, the

commutativity means f4o fz 0 f2 © ft(Z) = gs 094093092091 0 fi(z) =

is again a derivation 8

Proof It is evident that A is linear and homogeneous of degree v + v’

We have only to check the condition (6) (equivalent to (2)) For D and D’

AL, = [DD'~ (-1)""D'D] Le = Las + E „+0 2A + Le„Ð! + LorgD where

which proves (8) Thus our proof is completed

Corollary 1 Jf v or v’ is in Ly, and in particular when vy = v'! = 8, then

GRADED ALGEBRAS

Corollary 2 If D is a derivation of degree v € I'_, then D? is also a

derivation of degree 2vEI,

Proof If we put D = D’ in the last part in Corollary 1, we conclude that

2D? is a derivation, and the constant coefficient 2 may be omitted, provided

that A is a field of characteristic other than 2

However, we shall prove this assertion directly as follows The character-

istic property that D is a derivation of some degree v in I’_ is

(9) DL¿ = Lp„ + Lạ„D

Then D? is of degree 2v in I’, and we have

D?L„ = DLp„ + DL¿j„D = Lpa¿ + Lạp¿D + LpaxD + Lụy„ D2

But since D is of degree y € _, we have JD = —DJ from (8), and then

D?L„ = Epa„ + L„D3,

which means that D is a derivation of degree 2U € T

6 Existence of Derivations in Free Algebras

Let F be the free algebra with free system of generators (z:);¢7, Over a

commutative ring A Then F is so graded that 2x; is of degree 1 for every

2 € J Let & be a graded algebra over A and y a homomorphism of F into

Theorem 1.7 Assume that for each i € I, a homogeneous element 6C

E of degree v + 1 is preassigned arbitrarily, where v is a fized integer Then

there exists one and only one y-derivation D of F into E, which is of degree

v and satisfies D (x;) = yj

Proof The uniqueness follows from Corollary 3 to Theorem 1.4 So we

Shall prove the existence By Theorem 1.1, the elements Do = 2%, °++2;, form

a base of / where o = (i;, -,i,) runs over the set > consisting of all finite

sequences taken from J We shall define 5(p,) € E by induction on the length

of o First we put

for the empty sequence op If 6 (p,) has already been defined for every o with

length less than h, we set

(2) 6 (wi, +++ 4.) = 6 (zi, +++ ti.) P (BR) +O (TY (Eu #4) ) tu

GRADED ALGEBRAS

In the case where h = 1, we have 6(2;) = y; From the definition, 5(p,) is

homogeneous of degree h-+-v if o has the length h For, if h = 1, 6(2;) = y; is

of degree v+1 by assumption, and if this property has already been proved up

to h—1, the degrees of the terms on the right hand side in (2) are (h—1+v)+1 and (h —1)+(v-+1) respectively, which are both equal to h-+-v Hence 6(p,)

is of degree h + v

Now we define a linear mapping D : F — E such that D(p,) = 6(p,)

for all ơ € 2/ Since (p,) forms a base of F, such D always exists and is determined uniquely Evidently D is linear and homogeneous of degree v Next we shall show the condition

(3) D(uv) = D(u)p(v) + p(J’u)D(v) (u,v € F)

We first remark that

D(po2i) = D(pz)o(2:) + o(J”p.) D(z)

holds by (2), and then forming a linear combination of (p,), we obtain by linearity of D,

(4) D(uz;) = D(u)p(2;) + @(J79)D(s;)

Now we denote by F1 the set of all elements v of F which satisfy the condition (3) for all u in F From (4), we have x; € F; and also 1 € F;, for if v = 1, (3)

reduces to a trivial relation D(u) = D(u) We shall prove that v € F; implies

vz, € F, In fact, substituting uv in (4), we have D(uuz;) = D(uu}¿2(%¡) + @(J”(u9))D(s;)

= D(u)w(u}@(œ¿) + @(J”u)D(u}2(m)

= D(u)@(0z¡) + Ø(2”) [D(u}@(s¿) + @(J79)D(s2))

which proves our assertion Therefore beginning with 2;, € F, and repeating this process, we have p, € Fi for every o = (i1,+ ,%n) Then by the linearity

of D, we have finally that all the elements of F belong to F;, which proves that D is a y-derivation satisfying the conditions of Theorem 1.7

19

CHAPTER II

TENSOR ALGEBRAS

Tensors are usually represented by a quantity with many indices such as

TK: However, we avoid such a representation in these lectures not only on

aesthetic ground, but also due to a more essential reason Tensors have indices

because of the use of bases; on modules without bases, such a representation

is impossible, while tensors can be also defined in such cases

To define a tensor algebra, we shall use the universal algebra, then prove

the existence and uniqueness of the tensor algebra

1 Tensor Algebras

Definition 2.1 Let M be a module over the basic ring A An algebra T

is called a tensor algebra over M if it satisfies the following universality

conditions:

1) T is an algebra containing M as a submodule, and is generated bụ M.1

2) For any linear Yaapping À oƒ M tnto ơn algebra E over A, there is

a homomorphism 6 of T into E which extends » This is represented in the

1 over M It is unique up to isomorphism

Proof Uniqueness: Let T,T’ be two algebras with the above universality

properties over M Then T > M, T’ > M and the injection I’ : A4 —› T"

extends to a homomorphism @ : T -—+ 7” Similarly the injection I: M+ T

1 This means that T is generated by M and 1 in the ordinary sense See the “Conventions”

TENSOR ALGEBRAS extends to a homomorphism 6: T’ + T., The mapping 6’ 0 @ is an endomor- phism of 7’, which coincides with the identity on M But since M generates

T, 9’ 04 is the identity mapping of T Similarly 90 6” is the identity mapping

of T’, which proves that T and TJ” are isomorphic as algebras Therefore the tensor algebra over M is unique up to isomorphism

Existence: First we shall construct an algebra satisfying a somewhat mod-

ified form of condition 2), and then we shall show that this algebra also satisfies 1)

For a while, we forget the structure of module of M and consider M

as a mere set In Chap I, 1, we constructed a free algebra F over A freely generated by the set M To distinguish the addition, substraction and scalar multiplication in this algebra from those of M » we denote the former opera- tions by +,—, and œ- #(œ € A) respectively Therefore we remark that when

z,t € M, we have z-t+y ý M, z¬ụ ý M, and œ-z ý Min general Next, we

denote by S the set of all elements in F of the forms

Lemma 2.1 The algebra T satisfies the following condition:

2’) If X is a linear mapping of M into an algebra E over A, there exists

&@ homomorphism 6:T — E such that

Proof By the universality of free algebras (‘Theorem 1.2), there exists a homomorphism @ : F + E which extends }-

im M_—.———>È,

21

TENSOR ALGEBRAS

Next we prove @() = {0} It is sufficient to prove that @ maps all generators

of ‘% upon QO Since each generator of ¥ has the form (1) or (2), we consider

them separately In fact,

which proves our assertion Hence the kernel of @ containing T, © defines a

homomorphism @ : 7' — FE and if x € M, we have (@oy)(z) = O(z) = A(z):

T

ef No

Now we shall prove that T also satisfies condition 1) in Definition 2.1

From the definition of T and T = F/&, it is clear that the restriction of y to

M is linear Hence it is sufficient to prove that » induces an isomorphism on

M, ie.,

which proves our Lemma

Although (4) may be proved directly, we shall prove it using the above Lemma

2.1 Put B= A® M (direct sun) Since A has a unit element 1, £ is the set

of elements of the form a-1+%, (a € A, x € M) Define a multiplication in

& by

(5) (@-1+2)(b-1+y) =ab-1+ (br + ay) (a,bE A; z,ục M)

Then we have zy = 0 for z,y € M It is easy to verify that E is an associative

algebra over A with unit element, and the injection of M into £ is a linear

mapping Therefore we have a homomorphism Ø : 7' —+ E such that

(6) (@oy)(z)=2 forall zeM,

by Lemma 2.1 If s € MN, we have v(x) = 0 and then (6) asserts that

TENSOR ALGEBRAS This proves that M and the submodule y(M ) of T are isomorphic with each other as modules So we identify them.” Since T is a quotient algebra

of the free algebra generated by M, then M is also a set of generators of 7"

This proves that T satisfies the condition 1) Therefore the algebra T' thus constructed is a tensor algebra over M, which completes our proof of exis-

tence

Example 1 When M has a base consisting of only one element {+}, the

tensor algebra T over M = Ax is the polynomial ring A[z]

Proof Let T be the tensor algebra over M and P be the algebra of polynomials in X with coefficients in A There exists a linear mapping X:

M -+ P which maps x upon X, and we have a homomorphism » :T + P which extends 4 On the other hand, T being an algebra generated by z, an

element y € T has the form )~ agx*, and

y > ay2* ) = So an(y(a))* = > a X*

Thus, py: T — P is surjective Also, » (SS axa") = 0 implies 7 a,X* = 0, and then we must have a, = 0, which means that yp is an isomorphism of J

with P Therefore we may put T = P = Alz]

2 Graded Structure of Tensor Algebras

In the above construction of the tensor algebra T' over M , the ideal T is generated by S whose elements are all of degree 1 in F Hence defining all the

elements of M as of degree 1, the ideal T is homogeneous (cf Theorem 1.3),

and F/< = T is a graded algebra Decomposing F and T into homogeneous

AÌso Tị, is spanned as a module by the products of A elements of MM

We shall give a universality property of T;, similar to that of T

? The identification is made possible by the following property: Given any set X, and a set M, there is a set Y equipotent to X which does not meet M

TENSOR ALGEBRAS Theorem 2.2 Leé h > 1 and B be any h-linear mapping? of M* =

M x: x M into a module N over A Then there exists a linear mapping

w of Ty into N such that

In the left hand side of (2), 2, +-x,, is the product of 21, +,2,, in the tensor

algebra T

Proof Let S be the set of generators of © An element of T is the sum of

a finite number of elements of the form

a=) ax, b= > de (aik © Fe, bie € Fe),

we liave

t= > 2:40 8; Oj

i,k,e

Here a;,05;0bj¢ is homogeneous of degree k+£+-1, because 5; is homogeneous

of degree 1 On the other hand, any homogeneous element of degree k in F

is the sum of products of k elements of M Therefore we have that „

(3) any u in F, NT is the sum of elements of the form:

210 °''O7,OSoy,O+-: ayy, (K+2+ 1 =h;k,£ > 0; T1›°°*;®k;› Ÿ1;'`';ữ/ in M; s in S)

Now the set {z10 - az, | 21, -,2n 4 M} forming a base of F;, fora given

h-linear mapping 8 : M* — N, there exists a linear mapping ¥ : F, - N,

such that

* A h-linear mapping means a function /Ø{(zì, - »2n) of A arguments

2%1,°°*,;%, in M, which is linear with respect to each argument when the other

h— 1 are kept fixed, i.e., we have

8(, +, Xj-1, at; + b2) 81a “.a = afi(x1,- **,2e—1, Vi, L¥41;,°° +, Zn)

+ bB(21,: » Tim 1y Xj, Li4a,°**, Lh),

TENSOR ALGEBRAS

W(z10-+-o2,) = B(2z, -,2z,) for all 21, -,2, in M

Now we shall show that

In fact, by the above remark (3), it is sufficient to show that

(5) (xịn - -+ 1z 1(2-t—(2 + ))ngà d - - - D) = 0,

and

Since W is linear in each of its arguments, we have

(in : nzy(#-†—(# + y))9 n - - ‹ nự¿)

= Ứ(zịn - 12g11 - - - 314//)

+ (zin - - - nzz 1ÿ nÿn d - Dye)

= Ô(\; * **yk; #; 0 + * +; Ue) + Ổ(#1:- © * Ves Ye Wry <= Ye)

— đ(z\, - - -;#k, 4 - , 1: ' * * y 9£)

=0 (because đis hA-linear), and similarly we have (6), and then (4) is proved

Thus, by (1) and (4), W defines a linear mapping w of T, = F,/(F n1)

into N, such that ¥ = wow, (here yp, is the restriction of yp to Fh)

In diagrams this is represented by:

Now we shall define the tensor product of modules using the tensor algebra described above A characteristic property of tensor products will

be given later (cf section 4)

TENSOR ALGEBRAS

Definition 2.2 Let M,N be two modules over A We set

P= M@®N (direct sum), and let T be the tensor algebra over P The

submodule Q of Tz spanned by all products {ry | 2 EM, y E N } és called

the tensor product of M and N,and denoted by M@N The element xy of

From Theorem 2.2, we deduce easily:

Corollary Let there be given a bilinear (= 2-linear) mapping 6 of Mx.N

into a@ third module R Then there exists a linear mapping wv of M@N into

R, such that p(x @ y) = A(x,y) for everyx € M and e N

We leave it to the reader to formulate a similar definition of the tensor

product A4, @ - @ M,, of h modules M; over A

Example 2 If M has a base {2;};., = B, then T is isomorphic to the

free algebra on B Therefore, a tensor is represented in the form aj, 4, once

a base has been chosen

Proof Let U be the free algebra on B and again we use the notations

+,—, and @-2 for the laws of composition in U to distinguish them from

the ones in AZ

Let A: M — U be the linear mapping which is the identity on B:

A(a12;, + -+ OnZi, ) = @1° Zi, + tee +a, ‘Zi

Then there is a homomorphism 6 : T' -+ U which extends by the property 2)

of T’ On the other hand, since B Cc M CT, the universality property of free

algebra U asserts that there exists a homomorphism 6’ : U —» T' which is the

identity on B These relations are represented in the commutative diagram:

Then Ø” o Ø is an endomorphism of T and is the identity on B Since B is

a base of M, 6’ 0 8 is also the identity on M, hence on the algebra T gen-

erated by M Similarly 6 0 9’ is an endomorphism of U and is the identity

on B, hence also on the algebra U generated by B Therefore @ and 6’ are

isomorphisms which are reciprocal with each other Also since maps M into

U, (submodule of elements homogeneous of degree 1 in U ), LF is isomorphic

to U not only as an algebra, but also as a graded algebra, which proves our

assertion If {z:};-, is a base of M, every element in 7T}, is of the form

where a,, ;, € A are the components of the tensor in the familiar way

3 Derivations in a Tensor Algebra

Now, we consider a module M over A and the tensor algebra T = 1,7) over

M We shall prove the following:

Theorem 2.3 If \: M — Ty41 is a linear mapping (v : any integer

> —1), then X may be extended uniquely to a derivation in T (of degree v)

Proof Uniqueness is obvious since M generates T So we prove the ex- istence of an extension Consider the free algebra F on the set M Then

we can write T = F/T, Tyi1 = futi/(Fvii 5), where T is the ideal in

F generated by the elements of the forms

aty—(c+y) (2,y¢M), œ-z—(œz)_ (ae A,reM)

Denote by # : F,41 — Ty41 the canonical map in the factorization T vel =

Fy41/(¥v410%) For each x € M, we select an element A(x) € F++¡ such that A(z) = #(A(z)) This defines a map A: M — F,,,, such that the diagram

TENSOR ALGEBRAS

But now, since x,y, x + are in Ä, we have

D(x) = A(z), D(y)=Aly), D@+y) = A(et+y)

Therefore the right hand side of the equality (2) can be rewritten as

A(z) + A(y) — A(z +9), which is zero, since A is linear This proves that D(x+y—(zx + y)) lies in the

kernel of 7, and therefore in { Likewise we obtain D(a -x—(ax)) € T, and

according to Corollary 1 to Theorem 1.4 this proves (1) Thus D induces a

derivation d of T’ in such a way that the diagram

(x : F + T canonical map) is commutative To see that d is an extension

of A, let x € M ‘Then aw = r(x) and

d(x) = d(a(x)) = 1(D(x)) = a(A(x)) = A(z)

This proves Theorem 2.3

Tensor representation Next, we want to make the following observation

Let M,N be modules over A, T(M), T(.N) their tensor algebras and \: M3

N a linear map Then, as a special case of the universality theorem for tensor

algebras, A extends uniquely to a homomorphism A: T(M) — T(N) In the

special case where M = N, and where À is an automorphism (i.e an invertible

linear mapping) of M, A extends to an endomorphism A : T(M) — TM)

We assert that this endomorphism A is an automorphism To prove this,

let A’ be the inverse of 4 Then 2! extends also to an endomorphism A’ :

TM) — TIỆM), and the composite endomorphism 4o A! : T(M) — TM )

cotucides with the identity on M, so that Ao A’ = identity on T(M) which

is generated by Af The saine is true for A’ o A ‘Thus A, with its inverse A’,

is an automorphism

Now, the restriction of this automorphism A on the h-th part T,,(M) of

(AS) gives au automorphism A, of T,(M) The correspondence \ — A), is a

homomorphism of the group of automorphisms of M into that of the module

In(M) This homomorphism we call the tensor representation of degree h

Remark Suppose M is a submodule of N, for which the injection map

M — W is denoted by 4 Then the homomorphism A : T(M) — T(N)

TENSOR ALGEBRAS induced by 4 is, in general, not an injection However, in some special cases,

A is an injection; for example, in case where N is the direct sum of M and some other module P : N = M @ P, or in case where both M,N are free modules

The following provides an example in which A is not an injection Let

A = Z be the ring of integers, N = {0,1,2,3} the cyclic group of order 4, and let M = {0,2} be the subgroup of N of index 2 Then A maps the non- zero element 2@2 of M@M = M upon the zero element of N @ N = N, for

we have A(2 @2) = 2@2 = 4(1@1) = 0 This shows that A: T(M) - T(N)

is not an injection

4 Preliminaries About Tensor Product of Modules

” vở

“Án -

Before considering the tensor product of semi-graded algebras, we give here ~

_ some preliminaries about tensor product of modules

Characterization Let M,, -,M4, be modules over A Then the tensor

product P = M; @ -@M,, can be characterized in the following manner:

1) P is a module over A into which there is a h-linear map

2) If 8 is a h-linear mapping of M, x - x M), into a module Q, then si

Associativity and commutativity Let M,, -,Mk, Meti, -, Mi, (1<

#ị @+'‹ OL, OLK41 B+ Oxy, upon (21 @ - @L~) @ (E41 © ©-: @ 8g) for | any x; € M, (¢=1, -,A)

Given the characteristic properties 1), 2) for the tensor product, we need only to prove 1) that (21 @-+- @ 2) @ (a41@++- @zp,) € P' (a, EC My, i=

1, -,/) depends linearly on each argument, and P’ is spanned by elements of

the above form, and 2) that, given any multilinear’ map 6: My x -x M,

@, then there is a linear map y: P’ + Q such that

* We say “multilinear” instead of “h-linear” when we don’t want to mention ex-

from My, x -x My, into Q Therefore, there is a linear map, S8Y tzyig, „2

Mĩ: @ - - - @ Mẹ — Q, such that

Presi a(Z1 @ +++ @x~) = ỞỆŒI, ' *' ky #k+1;***y2h)

Now, let ¢ be any element in MM; @ -@ Mg For this fixed ¢, we consider

the mapping

(#k+q, - '- »Zp) —? Doras stn (t)

We assert that this is a multilinear mapping In fact, this is true if t is of the

form t = x; @ +@ ax, because in that case we have

Waray stp (t) — B(x, "** Uk, Ve+iy’** » Zp)

Let now ¢ = 57 a;t, where each É¿ is of the form 71@- - -@zz Since Wags syn

Aq, @ -@ M, — Q is linear, we obtain

Dons th (é) = »› (kg ph Th (t;)

i

Each summand aj z,,1,-,2,, (t:) being multilinear in (x, 41, -,2,), we can

conclude that #z,,,, -,2,(£) is multilinear in (2,41,-+-,2n) Thus for given

e(Te41 Q -@ #n) — 1DZk-Ly»*y#h (é)

Similarly, we can prove that, for any fixed element u in Mp41@-++-@Mn,

the mapping ¢ — -y;(u) is linear Thus, the mapping (t, u) + 7;,(u) is a bilinear

map from (A; @ -@ Mx) x (Mg41 @-+ @ Mp) into Q and so, there is a

linear map

yp: (M; @ -@ My) @ (Mn41 @ -@ Mn) + Q such that

ot @u)=y%(u) (£€ M1 ® -@ Mg, we Mi¿ @ - @ Mụ,)

4hus, for = z¡ @ - ®Zk, u = LE41 @-+- @ Tp, we have

P((t1 @-+- Ory) @ (Te41 @ -@ep)) = B(er,-+*, Te, Legay -yTh)s

which proves 2) Thus our assertion is proved

Let again M,, -,/4;,, be modules over A, and let 7 be any permutation

of {1, -, A} Then there is an isomorphism A, of Mi @ -@M;, onto My) @

cò @ Ä/„(„y such that

In fact, since the mapping

(%1,°°+, Zn) > #x(1ä) © +: © Zz(h)

is h-linear, there exists a linear map À„ : Mĩ @- - -@Mq„, — M,(1)@ -@Ma ny such that

An (Zi @ -@ Ln) = Ly) @ ++ OL ah)

So it remains only to prove that A, is invertible Let A, : My(1)® -@My(n) >

M, @ -@M,, be the linear map obtained similarly from the A-linear mapping

(#x(); * ' * › #z(h)) — Ø1 GÐ - đụ

Then

Nx (Ta(1) B ++ @ Lq(Hy) = L1 @+-+@ Ap,

so that

À„o Nn = identity mapping of Ma) @-.-@ Mừự()›

Alo Aq = identity mapping of M; @ -@ Mp

This proves that A,, with its inverse \/_, is an isomorphism

Remark Identification of (a1 @- @2%)@(xE41® -@rn) with 21@ -@apz

in the case of associativity does not cause any confusion, while identification will not be permitted in the case of commutativity The reader must be careful not to make the following sort of mistakes Consider the case M; = Mz = M,

21,22 in M Can we identify rz @ x2, with 1; @ 22 in M@M 7 No! These

two elements are by no means identical in general

5 Tensor Product of Semi-Graded Algebras

Let FE, E’ be semi-graded algebras over A:

E=E;y®E_., E=E.,@E

31

TENSOR ALGEBRAS Now, we shall give E@E’, the tensor product of the modules E, E’, astructure

of semi-graded algebra To do this, we first define the multiplication in E@E ⁄

in terms of a bilinear map (E @ E’) x (E @ E’) —¬ E @ E!

Since (P@ EL) OE QE) = EOE = (E, @E')O(E_ @P’), it suffices

to define four bilinear maps:

(E@E)x (FE, @E') + EE’, (E@#2)x(E_@E') ¬ E@EẺ,

(Z6E)x(E,@E') ¬ EQ@E!,

(E@E )x(E œ@F') ¬ E@E',

which will be well defined as soon as 4-linear maps:

Ex E', x By x Bf — EE’, ExE)LxE xE' + E@QE', BEXE_xE, xk — EE’, ExE' xE xE' ¬ E@E', are given The first three maps are defined by

xe, y' € E’,and either

f f ‘if

or ze € Eye By

while the last one is defined by

(z,2',y,y') + ~(2y @a'y’) (1 € E,x' € El,ye E_,y' € EB’)

In this way, we obtain a bilinear multiplication (E@E’)-(EQE') C EQE"

Now we assert that this multiplication is agsociative Since every element of

E ®@ E* is a linear combination of elements of the form x @ x’, where both

& and 2’ are nonzero and homogeneous in the semi-gradations, it will be

sufficient to check the associativity of the multiplication for elements of that

form For convenience’s sake, we set, for x #0in E,

e(xy) = e(x) +4(y),

e'(x’) for any nonzero homogeneous element x’ in E’ Then as is easily seen,

we have

(1) (x @2’)-(y @y’) = (~1)* 6*0)(zw @ xy’)

* See p 12 for the definition of (—1)””

32

TENSOR ALGEBRAS

(x € E,y' € E’,z’ homogeneous in E’, y homogeneous in E)

Now we check the identity

for x,y,z Ronzero and homogeneous in E and 2’, y’,z’ nonzero and homoge- neous in E’,

Computing the left hand side of (2), we obtain

((x @ +) ‹ (tự @ y’)) (z &@ z") = (—1)° œ2) (xy @ a'y') - (z @ z')

= (HL) VOOM) yz @ 2'y'2'),

while the right hand side of (2) can be reduced as follows

(x @2')- ((y@y’)-(2@z2')) = (—1)° O(c @ 2) - (yz @y’2')

= (—1)° @eŒ)+e'(2*(2)(yuz @ x'y'z')

= (—L)£ ()£)+e'(x')e(w)+e'(œ')e(2) (zuz ® z''z)

This proves the associativity of the multiplication If 1,1 are the multiplica- —

tive units in E, EB’ respectively, then it is clear that 1 @1'€ E@QE' isthe multiplicative unit in E @ EF’

Thus E @ &’ is an associative algebra, which is semi-graded, namely, if

we put

(E@ BE’), =(£, @ EL) @(E_ QE’),

Observe that, if EB, E’ are -graded algebras and a fixed subgroup I, of

I of index 2 is given, then by the associated semi-gradations

E,= À_ by,E = Ss) Ey,

¬——” TENSOR ALGEBRAS E® E’ is a semi-graded algebra The associative algebra E @ F’ also admits

the following I’-gradation:

E@E = 3 (E@E9a, where

ber

(E@E)a= Ð3_ E,@E,,

x†++y'=â

of which the associated semi-gradation is just the semi-gradation of & @ EF’

given above Direct definition of the multiplication in the I-graded algebra

1) f(ax) = a’ f(x) for alla€é A, 2 € M;

A is bilinear

Then 2 is called the bilinear form associated to f

It is obvious from the definition that @ is symmetric:

B(x, y) = Bly, 2)

and B(z,2) = 2f(z)

Two elements z,y of M such that 6(x,y) = 0 are said to be orthogonal

to each other When M is a free module over A with a base zq, -:-,Z#„ and F(z) = f(D &iai) = + - + G3, then we have

i=l

B(x, y) = BCD > Eas, Sma) = WErm + + + Extn)-

Hence the above definition of orthogonality coincides with the ordinary one

in the n-dimensional Euclidean space

Hereafter we suppose given a quadratic form f on M

Definition 3.2 Let T be the tensor algebra over M, and denote by ®

the multiplication’ in T Let ¢ be the ideal generated in T' by the elements of

the form

* In this chapter, we denote it this way to distinguish it from the various other

multiplications which will be considered later

CLIFFORD ALGEBRAS

called ihe Clifford algebra associated to M and ƒ

Ifa : T — Cis the canonical mapping, (MM) is a submodule of C, which

gcucrates C’ as an algebra Also we have

(x(x))? = f(z)-1 if x+œ<M

We remark that the kernel of ¢ in M is not always 0, and we cannot

identify M and a(M) in general However, if we wish to construct an algebra

satisfying (1), the universality leads to this definition as is shown in the

following:

Theorem 3.1 Assume that we have a linear mapping of M into an

algebra F such that (A(x))? = f(x) -1 for allz in M Then there evists a

A(z) = y(a(x)), for all x in M

This is represented in the diagram:

' a Proof The definition of the tensor algebra asserts the existence of a ho-

momorphism A: 7’ — F which extends \ For z in M, we have

A(w @ x — f(x) -1) = (A(@))?— ƒ(ø) -1 =0

Thus the generators of ¢ being mapped upon 0, we have A(c) = {0}, which

proves that A defines a homomorphism y of C into F satisfying A = pon,

Theorem 3.1 follows since 4 is the restriction of A to M

There exists a quadratic form g on 1(M) with values in the subring

A-1 of C, such that

1ˆ = g(0) -1, for all in 7(M) ; moreover f = gon

Semi-graded structure of Clifford algebras We have shown in the pre-

vious chapter that the tensor algebra T is graded, and a fortiori, T is a

Decomposing T into T; ® T_, (2) belongs to T,, and ¢ is homogeneous in the semi-gradation of 7’, which proves that C = T’/c is a semi-graded algebra

Putting C = Cy @®C_,Cy (resp C_) is generated (as module over A) by the products of an even (resp odd) number of elements of 1( AZ), because

Definition 3.3 When the quadratic form f reduces to 0, the Clifford algebra

One proves easily, for x,y in M, the relations

of a graded algebra

Theorem 3.2 In the case of the exterior algebra E over M, the canonical mapping x of T into E is injective on M, and identifying M with r(M), we may embed M into E

Proof The elements of c are sums of elements of the form

tu @ (x @ +) @0U

37

CLIFFORD ALGEBRAS where = € M, and u,v are homogeneous in 7 If u € Th,U € Tỳ, then

@® (œ @ z) @% belongs to 7},‡r¡¿ and this element has a degree not less

than 2 or else is equal to 0 Therefore the homogeneous components of an

element of c which are not 0 must be of degree > 2 On the other hand, the

eleinents of M being of degree 1, we have cn M = {0}, which proves that m

is an isomorphism of M onto 1(M)

Henceforth we identify M with its image under a in E Then we have

Eo = A-1, BE, = M For A > 1, E;, is spanned by the products of A elements

of M, i.e., by the elements x, -x,, where 2; € M

3 Structure of the Clifford Algebra when M has a

Base

Let M be a module over A and f a quadratic form on M Let C = T/c be

the Clifford algebra associated to M and ƒ

1° First we consider the case M = A-x (i.e., M is freely generated by

a single element 2) As we have already proved in Chap II, 1, the tensor

algebra T over M = A-=z is the polynomial ring A[x], and ¢ is generated by

a* — f (x) -1 If we denote by £ the image of z under 7, C = T/c has the form

A@®A-€ where €? = f(€)-1 Hence A- € being a free module with a base

€, the canonical mapping of M into C is an isomorphism Á -z — A-€ CC

Therefore we may embed M into C in this case

2° Next we consider the case where M = N ® P (direct sum), and N

and P are orthogonal with each other, i.e.,

B(z,y)=0 forall re N,yeP

By the orthogonality property, we have

(1) fl@+y)=f(z)+fly) if <EN and yeP

Theorem 3.3 Under such conditions, let Cu,Cw and Cp be the Clifford

algebras over M,N and P associated to f or the restrictions of f on N and

P respectively Then we have

(2) Cyu=Cn @Cp (tensor product of semi—graded algebras)

Proof Let Ty,Tyw and Tp be the tensor algebras over M,N and P and

7™M,7N,7p the canonical mappings of Ty into Cm, Tn into Cy, Tp into

Oz @a— f(z)-1)=2@x— f(x)-1, for xeEN,

defines a homomorphism of Cn into Cy which will be denoted also by

yy Similarly we have a homomorphism w of C'p into Cy, which extends the

(3) 8(u @%) = Ø(u)J(0) — (ue Cy,v Cp)

By the orthogonality of N and P, we have for x € N,y € P,

where # = TM(()) = @(Tn()) and = mw(W()) = ý(mp(0))

Now Cn = (Cn)+ ® (Cw)_ (semi-graded), where (Cw)+,(CN)_— are

spanned by the products of even or odd numbers of elements of w(j) re- spectively Similarly we put Cp = (Cp), ®(Cp)_ By the anti-commutativity (4), we have

@(u)¿() = /(0)@(u) ifeither ue (Cy), or ve (Cp),

en =-y(v)p(u) if both we (Cy) and œ €(Œp)

Here we shall show:

Lemma 3.1 The linear mapping @ defined above is a homomorphism of

Cn @ Cp into Cy, t.e., 6 satisfies (6) 6((u@v)(u' @v')) =A(u@v)G(u' @v'), for tu CN; 0U € Cp, where the term in the parentheses in the left hand side of (6) is the product ofu@v and u' @v' in Cn @Cp which has been defined in Chap II, 5

CLIFFORD ALGEBRAS

we

Proof It is sufficient to prove that (6) holds when u,v,u’,v’ are all ho-

mogeneous in the semi-graded structure

Putting

?† =

1 if ve (Cp)_, and

{ if wu’ € (Cn)+

ei =

Ll if we (Cy)-,

we have (u@v)(u' @v’) = (-1)"*" uu’ @ vv’ by the definition of the product

in the tensor algebra (Chap IT, 5) Then we have

Ø((u @ 9)(u' @9')) = (—1)?* 8(ww! @ vv’)

= (-1)™ y(uu')p(vv’) by (3)

since y and ~ are homomorphisms

On the other hand (5) is equivalent to

and then

=(-U*“ø(u)e(w)@(w9(0) — by (59,

which proves our assertion (6)

After having constructed a homomorphism @: Cy @ Cp — Cm, we next

construct a homomorphism in the opposite direction ) : Cm — Cn @ Cp

First define a linear mapping Ay : N @ P > Cn @C'p by

(7) Ào(# + ) = “Nn(z) @ 1 +1 @7p(w) (xe N,ye€ P),

where 1 is the unit in Cp or Cy Since Cy @ Cp is an algebra, we have

(Ao(t + y))” = (ay (x) @ 1)? + (1 @ mp(y))? + aN (x) @ xP(®)

+ (1 @ap(y)) - (mw (z) @ 1)

and since ™n(x) € (Cy)_, mp(y) € (Cp)_, the last two terms cancel out

with each other by the definition of the semi-graded tensor product Also

(Ao(z + y))? = f(x)(1@ 1) + f(y)(1@ 1)

and similarly (Ao 6)(1 @xp(y)) =1@ap(y) for y in P But since Cy @Cp '

is generated as an algebra by elements of the forms my (vz) @1 and 1 @ap(y), the homomorphism 40 @ is the identity on Cy @ Cp On the other hand, we

have by (9), (7) and (10)

(0 sÀ)(rM (% + )) = 0(Ào(œ + y)) = O(a (x) @ 1) + 0(1 @ mp(y))

= TM(#) + Tư (U) = Tu( +) (œceN,ueP),

and since the elements Zx¿ (# + ) generate Cạ;, the homomorphism 6 o À is

also the identity on Cy Hence Cy and Cy @ Cp are isomorphic with each other, which proves our Theorem

3° When A is a field K of characteristic 4 2, and M is of dimension 2 over K, it is well known that f is represented in the form

f(ée+ny) =af? +b? = (a, bE K),

by a suitable choice of base z,y If we put N = K-22, P= K-y, x and

y are orthogonal, since f does not contain the term €n Therefore we have

Cm = Cn ® Cp, and since N or P is generated by only one element x or y respectively, the considerations in 1° give now

CLIFEORD ALGEBRAS

which proves that Cyy is spanned as a vector space by four linearly indepen-

dent elements 1 @ 1 = 1, 1@, z @ 1, and z @ The products between these

basic elements are given by the following:

(« @ 1)? =2? @1 = f(x)-1=a-l,

(= @1)(1@y) =2@y = —(1® y)(z ® 1), (since both z @ 1 and 1 @y are of degree 1)

Putting 7@1= X,1@y = Y, we have r @y = XY, and the products are

given by

X?=a, Y?=6b, XY=-YX

This is nothing but a generalized quaternion algebra over K In the case where _

a = 6 = —1 and K is the real number field, this is the ordinary quaternion

algebra of Hamilton

4° Suppose that M has a base consisting of a finite number of elements

%1,°°*,%py which are mutually orthogonal:

It is well known in the theory of quadratic forms that, when A is a fied of

characteristic 4 2, we can always find such a base.”

Theorem 3.4 Under such assumptions, M is identified with the sub-

module m(M) of the Clifford algebra Cy over M Also Cy is spanned by the

linearly independent elements 2;, + + Zi, (i1 <-++ < Gp)

Proof Since this is proved when n = 1 in 1°, we proceed by induction

on n, and assume that this statement has already been proved for n — 1

Put N = Aa, + -+ Azy_1, and P = Az, ; then N and P satisfy the

assumptions of Theorem 3.3, so we have Cyy & Cn @Cp Under this isomor-

phism, 74 ( + y) corresponds to 7n(2z)@1+1@mp(y) (a € N,y € P) By

the inductive assumption, we can identify x with ry(z) and y with mT p(y)

Also 7 @1+1@y being 0 if and only if « = y = 0, the correspondence

M 5 (+) > 2@1+1@y = ry(x +y) is an isomorphism Thus M

may be identified with aas(M/) Next by our inductive assumption, Cy is

spanned by the linearly Independent elements #ịy tt 82,(1 S7 <-' S7 <

n—1) and Cp is generated by 1 and 2, Therefore the tensor product of

the modules Cy and Cp is spanned by the linearly independent elements

7 In the case of characteristic 2, such a base exists only in the trivial case where

the quadratic form f is the square of a linear form

(1 <iy <-+ < i, <2), which proves our assertion

5° In particular when M has a finite base 71, -,%,, and f = 0, the exterior algebra E over M has a base consisting of the 2” elements 2;, - Tụ, (iy < +++ < in) In this case EF is not only semigraded, but also graded, and if

we denote by & = >> E., the decomposition into homogeneous components,

Em is spanned by the products of m elements xj, -xi,, (i1 < - < im) We

have Ey, = {0} if m > n, and £,, is spanned by one element #y - #„ z# 0

This proves that n is uniquely determined by M Therefore if we take another

finite? base y1,°:-,Yp of M, we have p = n, ie., the number of the elements

of the base is invariant

4 Canonical Anti-Automorphism

The notations A, M, f;2,T,¢,C =T/e = Cz ® C_,7z are all as before

Lemma 3.2 For every linear form X: M — A, there exists a derivation

dy in C of odd degree , t.e., dẠ(C+) C C_, and dạ(C_) C +, which satisfies

(1) d\(m(z))=A(z)-1 for zeM,

Prooƒ Since À may be considered as a linear mapping A : T; — To, there

exists a derivation 5, in T of degree —1 which extends 4, as was proved in

the previous chapter (cf Theorem 2.3) We have

ốy( @# — ƒ(#) - 1) =ðA(œ@+}) (since 5,(1) = 0)

=6A(z)®z—z@6a(#) (6, is of degree — 1)

= À(z) : 1® # — À(+) -z @ 1 = À(z)(+ — z) = 0,

3 If M hasa finite base x), -,2n,this property holds if we delete the word “finite”

for the base (y)

43

CLIFFORD ALGEBRAS hence 6(c) = 0 Therefore 5, defines a derivation dy in C,, which satisfies the

condition (1) Also 6% is again a derivation since 5, is of odd degree, and we

have

5x (x2) = 6y(6y(x)) = 5,(A(Z) - 1) = A(z) - 6, (1) = 0, for z in M, which proves (2) since 7(M) generates C as an algebra

Now, if for any element x #4 0 of M, there is a linear form \: M —~ A

such that A(z) # 0, we obtain d,(m(x)) # 0 and then w(x) 4 0 When A is 8

field, every element x # 0 of M satisfies this condition, and we obtain:

Corollary If A is a field, 1: M — x(M) C C is an isomorphism, and

we may identify M with x(M) in C

Canonical anti-automorphism Hereafter we assume that x >: Ms

™(M) c Cis an isomorphism The above corollary asserts that this assump-

tion holds when A is a field

Theorem 3.5 There is an anti-automorphism of C’ of order 2, i.e., a

linear mapping u —» % satisfying UW = vi, and & = u, which leaves the

elements of M fixed

This mapping is called the canonical (or main) anti-automorphism of C -

Proof Let Ở” be the “opposite algebra” of C, i.e., C’ be an algebra with /

the same structure of A-module as C, and a multiplication given byuxv=_

vu (u,v € C) If x € M, we have xxx—f(zx)-1 = xx—f(x)-1 = O and thenthe

injection of M into C’ can be extended to a homomorphism Ở 3 tt — t EC

by the universality of the Clifford algebra This homomorphism is linear and

satisfies

uv = 0% = ZO which proves that u — @ is a endomorphism of C Since

x = @ holds for 2 € M, the map u — @ is the identity of C, and then

& — & is an involution Hence u — @ is an isomorphism of C onto C’ » ie,

an anti-automorphism of C

For 2°3,2%2,-+-+,2, in M, we have

(4) UiTQ°9 Lh = Ipyr+- Rely = op x72

When f = 0 (the case of exterior algebra), we can interchange terms in

the right hand side of (4) by the anti-commutativity xy = —yz, and then we

CLIFFORD ALGEBRAS

Now, since &;, is spanned by the elements x) -2;,, we have

In the case of exterior algebra, (5) can be taken as the definition of the canonical anti-automorphism tứ — @ We can prove directly that u — @ defined by (5) satisfies the conditions of the canonical anti-automorphism, using the property:

5 Derivations in the Exterior Algebras; Trace

In the case of an exterior algebra, we have the decomposition into homoge- neous components T' = S Ths R= » Ep, in the Z-gradation

Lemma 3.3 If a linear mapping py: M — Ep, can be decomposed as row with a linear mapping yb: M —» T;,, and the canonical mapping 1:7), — En, there exists a derivation d of degree h—1 in E, which extends ip It is uniquely determined

The above condition on » is always satisfied when M is a free module,

or when A is a field, or when A = 1 since T; = Fy

Proof The uniqueness follows from the fact that a derivation is uniquely determined by its effect on the generators of an algebra

We shall prove the existence Since M = T), there exists a derivation 6

in T' of degree h — 1 which extends ý, by Theorem 2.3 Let z be in M We have then

CLIFFORD ALGEBRAS since y(x) € w, z € E\, and T(œ) = z for z € M Since the ideal c generated

by z @z (x € M) in T is the kernel of 7, then 6 defines a derivation d of E,

Corollary Any endomorphism of M = E, can be extended to a uniquely

determined derivation of degree 0 in E

Now let §(47) be the set of all endomorphisms of M Then $(M) is again

a module over the basic ring A, and indeed it is also an algebra For every

element y € §(M), we have a derivation d,, of degree 0 in EF by the above

corollary

Lemma 3.4 The derivation d,, depends linearly on y, ie.,

(2) dap+bp' = Ady + bd (a,b € A;y,y’ € 8(M)),

and for the “bracket operation” [y, @'] = œ@' — @'@, the following holds

Proof Since the proof of (2) is similar, we shall prove (3) only The right

hand side of (3) is again a derivation of degree 0 in E, since dy is of degree

0 It is therefore sufficient to prove that both sides of (3) coincide on the

generating set M of E In fact, for x in M, we have

dpe (@) = lv, e'l(x) = (pe" — y'y)(x) = pp" (2) - y'o(x)

= dụ(Ø (z)) — dạ:(@()) = du dự) — đợ dạ(z)

= (dađụ; ™~ yi dp)(2),

which proves our assertion

Now we assume that E,, is a free module of rank 1 for some integer n, and

Ey = {0} if n’ > n For example, this property holds if M is a free module

with a base of n elements Let € be a generator of £,,, that is Eạ = A-€

Since d, maps E,, into B,, we have

dy E = Sf,

where s, is a uniquely determined element of A, which does not depend upon

the special choice of €

ees

dy dy§ = Sys

But since we have assumed that A is commutative, we obtain

and therefore we have

(Tr(py’ 7 'ọ))€ = Dep —p'ot = (dở: — dig thy )E

= (§ø'8¿ — Sạ¿8¿')€ = 0 which proves (4)

Remark By (4) we have, for example,

Trpy'y" = Try"py! = Tryp"

But an equation like Trpy’p” = Try’pyp” is false in general Also y + Try

is not a homomorphism of algebras of §(M) into A

When MM is a free module with a base z1, -,2,, any element y of §(M)

is represented by a square matrix (a,;;) of order n, such that

Tt pai) = À ` quay

j=l

We shall show that the trace defined above coincides with the classical one defined as the sum of diagonal elements of a matrix In our present case, we have E, = Ax, +%, so we may take € = x -a, Then

CLIFFORD ALGEBRAS since d, is a derivation of degree 0 But, since cuz = terru = 0, for ze M,

and u homogeneous in EF, we have

TL Lye Ue (>: cass) Uk41*+ Ly = Gkk 1 “ 1g ' eT = are,

i=]

which proves that

(Tr gg = pe a :

k=l

Le Try = địi + G22 + "') Fann

Our definition of the trace is intrinsic: it is evident that Tr y is determined

6 Orthogonal Groups and Spinors (a Review)

Let I< be a field of characteristic p(> 0), and V a finite dimensional vector

space over Ix Also let f be a quadratic form on V, @ the associated bilinear

form We assume that ( is non-degenerate, ie., G(x, Yo) = 0 for all z € V,

implies yo = 0 We denote by C the Clifford algebra associated to V and ƒ

Definition 3.5 An automorphism s of V is said to be orthogonal with

respect to f if s leaves f invariant, i.e.,

We use the terminology “orthogonal transformation” instead of “orthog-

onal automorphism” The set of all orthogonal transformations is a group

which is called the orthogonal group of f and denoted by O(f)

Definition 3.6 The set I’ of allu in C, such that u has an inverse u-!

ƒ(su(£)) - 1 = (su(œ))? = (uzu~!)? = uz2w~! = u(f(z)- 1)u~ = f(z) -1

Hence the correspondence x : u — s,, is a linear representation of I", which

is called the vector representation of I’ The kernel of this representation is

the set of invertible elements in the center of C

If s is an automorphism of V, it is represented (in a given base of V)

by a matrix whose determinant is taken as the determinant‘ of s If s is

orthogonal, we have det s = +1 The set

{s € O(f) | dets = 1}

is a subgroup of O(f), which is of index 2 unless the characteristic p of K is

2 When p = 2, we have det s = 1 for all s € O(f)

Let C = Cy ® C_ be the homogeneous decomposition of C in the semi-

graded structure and put P+ = INC We define Ot(f) as follows:

if p#2,0+(f) = {se O(f) | dets =1}, (1)

if p= 2,0°(f) = {x(u) [ue rt}

It can be proved that in both cases, {y(u) | u € I+} coincides with Ot(f), and Ot (f) is a subgroup of O(f) of index 2

Let u -—* @ be the canonical anti-automorphism constructed in 4 We

can prove that tu € K-1 for every u € I+ Putting Gu = X(u) - 1,

A is a homomorphism of I’t into K*, where K* is the multiplicative group

of non-zero elements in K The kernel Ij’ of this homomorphism ) is called

the reduced Clifford group Also we denote by §2 the image of Tự under the ẹ

vector representation x, and call it the reduced orthogonal group

When K is R, the real number field, and f(x) = f (37"_, &aj) = €? +

+ + & (positive definite), O+(f) is the ordinary special orthogonal group

It is well known that O+(f) is not simply connected if n > 3 ; the Poincaré

group of Ot (f) is actually of order 2 when n > 3 Also we have 2 = Ot(f)

and x: I — 2 = O+(f) is a covering mapping

We now return to the general case A linear subspace W of V is called totally singular if the restriction of the quadratic form to W is the zero quadratic form on W All maximal totally singular subspaces of V have the same dimension, and the common dimension is called the index of f It is evident that f is of index 0 if and only if there is no x 4 0 with f(x) = 0 We quote without proof the main result about these groups:

If the index of f is not 0, we have ®

(2) O*(f)/2 ~ K*/(K*)?

* See chapter IV, 3 for an intrinsic definition of the determinant

* K* denotes as above the multiplicative group of elements 4 0 in the field K, and

(K*)? the subgroup of squares

CLIFFORD ALGEBRAS Moreover (2 is the commutator subgroup of O(f) except when K has only two

elements, dimV = 4 and f is of index 2 If furthemore n = dimV > 3, 2

ts the commutator subgroup of Ot (f) Also when n = dimV = 2, Ot (f) is

abelian, and its commutator subgroup consists only of {e}

On the other hand, the structure of 92 when the index of f is 0 is quite

unknown

Now we assume that V is of even dimension, namely 2n, and let 2, - a

Y1,°'*;Y¥n be a base of V Suppose that f can be reduced to the following

forin:

When IC is algebraically closed, every quadratic form whose associated bilin-

ear form Ø is non-degenerate can be reduced to this form On the contrary,

if K is not algebraically closed, such a reduction is not always possible, as

shown by the example of the quadratic form €? + 7? over the real number

field Under these assumptions, the Clifford algebra C is isomorphic to a full

matric algebra and has the dimension 2", while Cy is of dimension 2?"—1,

There is a minimal left ideal & in C, of dimension 2" For u € C’, Lis stable

under left multiplication by u and then the transformation Au: € — ué is

a representation of C Moreover u — A, induces a faithful representation

of TC C) This is called the spin representation of the group I, and the

elements of £{ are called spinors

Lhe origin of this name is as follows When E Cartan classified the simple

representations of all simple Lie algebras, he discovered a new representation

of the orthogonal Lie algebra But he did not give a specific name to it, and

much later, he called the elements on which this new representation operates

spinors, gencralizing the terminology adopted by the physicists in a special

case for the rotation group of the three dimensional space

The spin representation of I" is simple except when K has only two ele-

ments, 2 = 1 and f is of index 1 Also the spin representation of P+ is the

sum of two simple representations

Assume now that S{ is homogeneous in the semi-graded structure of C,

i.e.,

This corresponds to the decomposition of the spin representation of + into

two simple ones, and each of them is called the half spin representation Each

half spin representation is of degree 2"—!

* It is then customary to say that the quadratic form is hyperbolic (or split)

R i

CLIFFORD ALGEBRAS When n > 2, the kernel of each half spin representation is of order 1 or

2 On the contrary, if n = 2, ie., if V is of dimension 4, it is not so This corresponds to the fact that the rotation group of dimension 4 is not simple

When n = 2, let A;, Ae be the kernels of the two half spin representations

of If ; we have

Igt = A1-Az (direct product),

and the spin representation of Tử splits into two parts Then A; operates on

Ul, and fixes U_, while Az operates on {{_ and fixes 1, The representation

Au(u € 41) produces all automorphisms of determinant 1 on 44, and then

each of A, and Ag is isomorphic to the multiplicative group of two-by-two matrices of determinant 1

Similar considerations hold for quadratic forms in an odd number of vari- ables For instance, consider a quadratic form in three variables of the type

(5) ƒ(€z + nụ + €z) = €n + ¢?

Then the corresponding reduced Clifford group is isomorphic to the group

of two-by-two matrices of determinant 1, and covers the special orthogonal groups in three variables

When ¥ is R, the real number field, a quadratic form cannot always be written in the form (3) as we have remarked above But if we extend K to the complex number field C, the representation as (3) is possible, and the real quadratic form f is extended to a complex quadratic form This may be

an answer to the question why the spinors in the Euclidean space are usually treated using the complex number field

51

CHAPTER IV

SOME APPLICATIONS OF

EXTERIOR ALGEBRAS

1 Pliicker Coordinates

Let Ix be a field, V a finite n-dimensional vector Space over Jf, and the

exterior algebra over V The decomposition into homogeneous components of

E is denoted by E = 3 Em THÍ #i,'**,#ạ is a base of V, the (7) elements _

m

Mi, +++ 24,, (41 <+++< ¢,,) form a base of Em

Definition 4.1 An element a of E,, is called decomposable if a is the

product of m clements of V

Any element in E,, is the sum of a finite number of decomposable ele-

ments We remark that aa = 0 if a is decomposable

Let W be an m-dimensional linear subspace of V with a base y1,°°-, Ym-

By the canonical mapping of W into V, the exterior algebra F' of W is

naturally isomorphic to the subalgebra of E generated by W, and the ho-

Mogehcous component Ff, of degree m in F is therefore contained in E,,

On the other hand, Fy, is of dimension 1, spanned by 1 '**m Thus to any

linear subspace W in V of dimension m, there corresponds a 1-dimensional

subspace of E,,, namely /, Conversely, if Fy, is a 1-dimensional subspace

of I, spamied by a decomposable element, we have an m-dimensional linear

subspace W, such that the homogeneous component of degree m of the ex-

terior algebra over W is Fi, Also we have 7F,, = 0 if, and only if « € W In

fact, let ay: -+.ym be a base of W Ifa € W and z # 0, we may take + = y;,

and by Fin = I {91 - ym} we have xy, + + yy = 0, and then xFy, = 0 Con-

versely, if « ¢ W, the m+ 1 elements @,Y1,'**5 Ym being linearly independent,

they are part of a base of V, which proves Z1" '* Ưma 7% 0 Also we have:

Theorem 4.1 The elements 21, -, 2m of V are linearly independent if

and only tf x1 -2m #0 in EB

Also the family of all m-dimensional linear subspaces of V, and the fam-

ily of 1-dimensional subspaces of Em which are spanned by decomposable

elements, correspond in a one-to-one manner with each other If we take a

SOME APPLICATIONS OF EXTERIOR ALGEBRAS

fị<** Sim for a base ¥1,-++,%m of W The ratios of various aj, ;,,’8 are invariant if we take another base y}, -,y/, of W, since y; - ym is a base of Fy

Definition 4.2 These ratios of ay, 4,,’s are called the Phicker coordi- nates of W

Since the base of F,, is decomposable, the Pliicker coordinates cannot

be chosen freely, but must satisfy some identities For example, ifn = 4 and

m = 2, the identity reads:

2

For a € E, we may consider the multiplication in E to define x?, 73, -, and

if x is a homogeneous element of degree > 0, we have 2™ = 0 for sufficiently large m But it will cause a difficulty to define exp x by (1), because of the factor S, unless the characteristic of K is 0 So, we shall procced in another

way If x is decomposable, we have x? = 0 and then exp may be defined

sunply by 1+ 2 If we restrict ourselves to elements a, b, - of even degree, we

have the commutativity ab = ba, and we may expect the “addition theorem”

of exponential function:

(2) exp(a -+ b) = (exp a)(exp bd)

Hence exp may be defined through decomposing x into a sum of decompos- able elements However, in order to assert the uniqueness of this definition,

we shall begin with proving some lemmas

di,-:-,d, of degree —1 in E such that d, - d,(z) #0

Since K is a field, we may even assume that d; -d,(x) = 1 by multiply- ing by a suitable scalar

33

meee st + * Hướn vn TT es _ 2 TT

SOME APPLICATIONS OF EXTERIOR ALGEBRAS Prooƒ Lek 1à, - - +, Ua be a base of V Since the elements y;, “Yi, (411 <

++ <%,) form a base of E;,, we can write

fr< -c<iy,

Since œ # 0, there is at least a sequence of indices (ït, -,i„) such that

a(i1, -,%,) 40 Now for each vy = 1, -, A, there exists a linear form Ay on

V such that

(4) Az; )=1 and AL(y:)=0 for all i4i,

By the extension theorem (see Lemma 3.3), there is a derivation d, of degree

1 which extends , We have by the definition of a derivation,

Guys, Yin) = (dy (Yin) ia ++ Yin — Yar (Gu (Yin) is °° Vig Ho

+ (-1)*~1 yi, +++ Yin, (dv (yin))-

But (4) shows that d,(y;) = AL(y;) 4 0 only if i=7,, and then we obtain

Qv(yin-**¥i,) =O if ty ¢ {i, -, tq}

When i, € {t:, -,i,}, namely 7, = i,, we have

dụ (úy *+ Yin) = (1) Mu Bae Mins

where the symbol ~ above y;, means that this factor should be omitted from

the product Then we have

where the summation is taken over the family of indices such that

By successive applications of d,, we have

di: -dn(zx) = +o{(n, eee stn);

by using (3), since d, -d,(y;, -y:,,) vanishes unless (i1, +,4,,) contains all

41,°-+,t,- This proves our assertion since we have assumed that

d(ä, vee »tp) z0, Lemma 4.2 An element x € E has the property that d(x) = 0 for every

derivation d of degree —1 in E, if and only if x € Eo

of degree —1 For the converse, we shall prove the contraposition, ie., the

proposition that if « ¢ Eo, then there exists a derivation of degree —1 such

Since z ¢ Ep, we have an integer h > 1 such that z, 4 0 and z; = 0°

for i > fh By the above Lemma 4.1, we have a derivation d of degree —1,

such that d(x,) # 0 Since d(zg) = 0 and d(x) = d(a,) + + + d(z,) is the homogeneous decomposition of d(x), we have d(x) # 0 from đ(z„) 4 0, which

proves our statement

Lemma 4.3 If a is decomposable of degree > 2, and d is a derivation of degree —1, we have ad(a) = 0

Proof Putting a = xb, where x € V and b is again a decomposable ˆ

element of degree > 1, we have d(a) = d(z)b — xd(b), and then

ad(a) = xbd(x)b — xbard(b) = d(x)xbb + rxbd(b) = 0,

since rz = 0, bb = 0

If the degree of a is even and the characteristic of K is not 2, this lemma & be

Lemma 4.4 Let a1, -,a,% be decomposable elements of strictly positive — even degree, such that ay + -+ a, =0 Then we have

(5) Sai aig ++ ai, = 0,

ti<-<im for every integer m such that 1 <m < k

Proof We first remark that the case m = 2 is easily settled unless the

characteristic of K is 2 In fact, we have a? = 0, and aja; = a;a;, because

the a;’s are decomposable elements of even degree Hence we obtain

0= (a; + - +ag)? = > + “aia; = 25 aay,

it is sufficient by Lemma 4.2 to show that d(u) = 0 for every derivation d

of degree —1 Since the a;’s are all of even degree, they commute with any element in E Thus we have

SOME APPLICATIONS OF EXTERIOR ALGEBRAS

4199 <tin + vee + aj, ‘ee Gi,,, A(a;,, ))

= 3, las, +: G4, 4( ai, ) + Gi, Gig **- Gi, Haig ) 41 << tan

+ - + Qi, °°* Qi,._y d(ai,, )Ì

But since > d(a;) = d(}>a;) = 0 by our assumption, and a;d(a;) = 0 by

Lemina 4.3, we have d(u) = 0 which proves our statement

Now we shall give the definition of the exponential mapping on the space

F of elements with homogeneous components of even degree:

#'= Ea @ lạ @ ‹@ Eạy @-.-

First we define expa = 1 -†ø if a is decomposable For any element u € F,

it is possible in at least one way to represent u in the form u = Qy+-+-+a,z

where each a; is decomposable and of even degree, because each Ez, has a

base consisting of decomposable elements Then we define

(6) exp = (1 -+ ai)(1 + aa) - (1 + ag)

While the decomposition u = a, -+ +-+ a, into decomposable elements is not

tmique, exp u is determined uniquely by u Precisely speaking, if we represent

u in two ways

where a; and b; are decomposable, we have

(7) (1+ ai)(1+a2) - (1+ ag) = (1+ 6,)(1 + 62) -(1 + be)

In fact, putting an41 = —by, -, đị+¿ = —b, we have a,+a9+ - + ape =0,

where @),°:-,@,4¢ are all decomposable Then we have by Lemma 4.4 that

(5) » Gj, °a;,, =O for l<m<k+e

1 vanish because of (5) Thus we obtain, (8) (1-++-ay)++-(1-+a,)(1—b1) + (1—be) = (1+ a1)(1-+a2) +++ (1-+apge) = 1

On the other hand we have (1 + 6;)(1 — 6;) = 1 — 6? = 1, since b; is decom- posable Multiplying by (1+ 6,)(1+ b2) - (1+ 5) both sides of (8), we have

(7), since a;,6; are mutually commutative

Definition 4.3 The mapping u — expu defined above is called the ex- ponential mapping of F into E

It is evident from the definition that exp u satisfies

(2) exp(œ -+ b) = (exp a)(exp d) (a,be F)

In particular when the dimension of V is even, namely 2m, we take a base

¥i,°°-;¥2m- Let I” be a homogeneous element of degree 2 The homogeneous component of degree’ 2m of exp I is a multiple of y; - yam, namely

Definition 4.4 Pp is called the Pfaffian of I € E>

If I is represented as a sum of m decomposable elements! of degree 2, putting I = a; + ++Om, we have

exp I’ = (1+ 4;) -(1 +a), and expanding the right hand side by the polynomial theorem, the term

of degree 2m is merely a,-+-a,, On the other hand, using the polynomial theorem for [™ = (a, + + +am)™, and noticing that a? = 0, we have P™ = mia, -++amy, which proves

(9) mi(exp P)ean = I™

If the characteristic of K is 0 or relatively prime to ml, we obtain

3 Determinants Let V be a finite n-dimensional vector Space over /K Any endomorphism s

of V is extended uniquely to an endomorphism S, of the exterior algebra E,

* This condition is always satisfied according to the theory of skew-symmetric forms, but here we merely assume it

37

SOME APPLICATIONS OF EXTERIOR ALGEBRAS which is homogeneous of degree 0 Since E,, is of dimension 1 and S,(E,) C

E,, there exists a uniquely determined scalar A, such that

Definition 4.5 This A, is called the determinant of the endomorphism

s and denoted by det s

The classical properties of the determinant are easily proved from this

definition For example, we shall show:

Theorem 4.2 1° (det s)(det s’) = det(s 0 8’)

2° dets #0 ¢ and only if s is an automorphism of V

Proof- 1° Let s,s’ be two endomorphisms of V Then S, 0 Sy is an

endomorphism of E which coincides with S,,., in V, and thus we have

Ss 9 Sg: = Ssog Since V generates E Therefore, for z € E,, we obtain

A;ozg2 = SaogZ = (5,9 5,)2Z = S,(Agrz) = Ay (Sz) = Ay A,yz,

which proves our assertion, since K is commutative

2° If 11, +,2n is a base of V, E, is spanned by 271 -2, and we have

(2) A;zi -Zna = S;(Zt -#n) = S3(21) + + Ss(2n) = (21) -8(Zn),

since S, is a homomorphism Therefore by Theorem 4.1, det s 4 0 if and only

if s(x1), -,8(%p) are linearly independent, and in turn this is equivalent to

the fact that s is an automorphism of V

But 2;,-+-2;, = 0 if there exists a pair of indices such that i, = i, (u #4 v),

and when the indices (i, -,‡„) are all distinct, we have 2j, -a4, =

sgn(iz, -,tn)(Z1-+*2n), where sgn(iz,:-+,%,) is +1 or —1 according as

(i1,°-+,%n) is an even or odd permutation of (1, -,n) Thus we obtain

(3) det s = det(aj;) = À_sgn(i, -**y%n)Qi,1°°* inns

where the summation is taken over all the sequences (i1, -,i,) such that 11,°°*,%, are all distinct This shows that det s may be expressed as a poly-

nomial with coefficients +1 in the a;;`s

Now, let U be a vector space of 2n dimensions over K ; we assume that U is given as the direct sum of two n-dimensional linear subspaces V

andW:U=V @W Let 21, -,a2, and y1,°:+,yn be bases of V and W

respectively Taken together they form a base of U We define a bilinear form

(4) 0(œi,#;)= 00,9) =0, Ô0(0:,97) = 0(014Ó) = 6:3 (b3 = 1, -yn)

Then đ is a symmetric non-degenerate bilinear form on U x JU, satisfying

B(V,V) = B(W, W) = {0}

The set of all linear forms on V is again an n-dimensional vector space over K which is called the dual space of V and denoted by V* In our present case, for any y € W, the functional over V defined by

is linear, and belongs to ƒ* Since Ày,(#¡) = ố¿;, the mapping À : — dy is

8 linear isomorphism of W onto V* Therefore we may identify W and V* with each other

If s is an automorphism of V, we can define an automorphism ‘s of V*

by

(sA)(z) = A(sz)

We have easily (s)~! = Xs~1) and this automorphism of V* is đenoted by 8 Since V* is identified with W, 3 is also an automorphism of W Then there exists an automorphism H, of U which coincides with s on V and 3 on W respectively We shall prove the following:

Theorem 4.3 We have det H, = 1

We first prove the following:

SOME APPLICATIONS OF EXTERIOR ALGEBRAS which is an element of degree 2 in the tensor algebra over U Then H, c+-

tends to an automorphism of the tensor algebra over U, and this extended

automorplism leaves © fixed

Proof of Lemma 4.5 What we have to prove is the identity

tt

j=l

which proves that the matrix corresponding to 's is the transposed matrix-of

the matrix corresponding to s Applying $ to (7), we have

Tt

te = > ans(Syi),

{=1 and then

Now we return to the proof of det H, = 1 Since the exterior algebra Ey

over U was defined as a quotient of the tensor algebra over U (see Chap III,

2), we denote the canonical image of @ in Ey by I Then I is represented

2⁄z(#:¡) = 3(z¡)8(¡) are decomposable and sum to I’, we have

exp I’ = (1 +#iu)(1 + #292) 1+ Inn)

= (1 + 2;(11n))(1 + 375 (%2Yo))- ° (1 + Xs(ZnYn))

= £6((1 + t1y1)(1 + coy2)-++ (1+ anyn)) = D5 (exp I’)

Hence £, leaves also invariant the component (exp Ian of the highest di-

mension of exp I" On the other hand, I" being the sum of n decomposable elements, we have

(exp 1 )an = 21112212 - - *#ntfn

as we remarked at the end of section 2, and this is a basic element in (Eu)au - ˆ

(det H;)(t10i - ' + #nn) = Z2(#11 - - ' 1n)

= Z181 ° ' 'TntUn)

which proves det H, = 1

Theorem 4.4 Let U,V,W be as before If o is an automorphism of U,

which leaves V and W invariant, and if we denote by oy,aw the restrictions ofa to V and W respectively, then

det a = (det oy )(det oy)

Proof This theorem follows from Ey & Ey @ Ew, but we shall give a simpler proof Let 21,-:+,2n and y,°-+,Yn be bases of V and W respectively

We denote by © the automorphism of Ey which extends c By definition of ˆ

the determinant, we have Ặ

(mì "Zn) = (det oy)(21 -#n) since Ey is generated by 21, -,2, in Ey and 2(By) Cc By Similarly we

have

Zvi +++ Yn) = (detow)(y1 +++ yn);

and then

(det ơ)(Z: s.« n1 +» Un) = J(X4 9° EnYrer Un)

= 2(t *#n)27(h + - - ta) = (det ơy)(# - - - z„)(detzw)(Ðt - Yn)

= (det oy )(det ow )(x1 °* Pn ' ** Un):

which proves our statement

61

SOME APPLICATIONS OF EXTERIOR ALGEBRAS Corollary The determinant of an automorphism s of V ts equal to the

determinant of its transposed one: det ‘s = det s

Proof The automorphism H, of U which coincides with s on V and >

ou W satisfies the conditions of Theorem 4.4 Then we have, from the two

tlicorems above, that

(det s)(det 3) = det H, = 1

On the other hand (det 3)(det ‘s) = 1, because S = (s)~!, which proves our

assertion

4 An Application to Combinatorial Topology

As au application of the theory of exterior algebras, we shall give a proof of -

a fundamental property of combinatorial topology: that the boundary of a

boundary is 0

Let {Po} be a set of “vertices” We construct a vector space V of which

the P,’s form a base Any element of V is a 0-dimensional chain in the

homology theory Now a simplex a is ordinarily defined as a set of a finite

munber of vertices, namely o = (Py,,-++,P,,) with an orientation which

makes o a skew-symmetric symbol This law of orientation is quite the same

one as in the exterior algebra; it is appropriate to represent the simplex

o = (Pa,,°:+,Pa,) by the element P,, - Po, in the exterior algebra By

over V A p-dimensional simplex is of degree p+1 in Ey Next we define the

boundary operation There exists a linear form 6 on V such that dP = 1 for

all cv ‘hen we have a derivation d of degree —1 in Ey which extends 6 If

we apply ở to a simplex o = (Pg,, -, Pa,), we have

do = (dPa,)Poag*** Pax — Po (dPoz)Pas *** Pox

+ (-1)PTP,, +++ Pons (Pay)

= Pog ++ Poy, — Por Pas *** Poy, +2 0+ + (-1)""' Py, 0+ Pao

apeee ep (—1)*-'P,, TH

This expression coincides with the ordinary definition of the boundary oper-

ation So, we define the boundary operation by d Then d being a derivation

of odd degree, d? is again a derivation and the property

Although there are many other interesting applications of the exterior algebras, we omit them because of limitation of time We only mention an application to physics; the equations of Maxwell in the theory of electro- magnetism may be represented elegantly using the exterior algebra of differ-

.The Algebraic 'Theory

irreducible) representation spaces Cartan did not give a specific name

to this representation; it was only later that, generalizing the terminology adopted in a special case by the physicists, he called the elements on which this new representation operates spinors The simplest case of a spin representation is the one which presents itself for the orthogonal Lie algebra in 3 variables; this Lie algebra is well known to be isomorphic

to the special unitary Lie algebra on 2 variables, which shows that it has a faithful representation of degree 2: this is its spin representation

Similarly, the fact that the orthogonal Lie algebra in 6 variables is isomorphic to the special unitary algebra in 4 variables reflects a

special property of the spin representation of the first one of these algebras

In his book, Legons sur la théorie des spineurs," Cartan recognized the connection between the spinors for a quadratic form Q and the maximal linear varieties of the quadratic cone of equation Q = 0 This connec- tion is similar to the one which exists between subspaces of a vector space V and certain elements (the decomposable ones) of the exterior algebra over V: while every maximal linear variety on the cone Q = 0 -

is represented by a spinor, determined up to a scalar factor, not every spinor is correlated in this manner to 2 linear variety Those which are

we call “pure spinors”; in his book, Cartan indicates that it is possible

to construct quadratic equations in the coefficients of an arbitrary spinor which give necessary and sufficient conditions for the spinor to

be pure,

1 Cartan, Lecons sur la théorie dea spineura (Paris: Hermann et Cie., 1938), 2 volumes,