Principal Structures and Methods of Representation Theory 9780821837313

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Principal Structure s and Method s o f Representation Theor y

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10.1090/mmono/228

Translations o f

MATHEMATICAL MONOGRAPHS Volume 2 2 8

Principal Structure s and Method s o f Representation Theor y D. Zhelobenk o Translated b y Alex Martsinkovsk y

American Mathematica l Societ y !? Providence , Rhod e Islan d °^VDED^*

EDITORIAL COMMITTE E AMS Subcommitte e R o b e r t D . MacPherso n Grigori i A . Marguli s J a m e s D . Stashef f (Chair ) A S L S u b c o m m i t t e e Steffe n L e m p p (Chair ) I M S S u b c o m m i t t e e Mar k I . Freidli n (Chair ) H . I I . >Kejio6eHK O OCHOBHblE CTPYKTYPb l H METOHb l TEOPMM riPEHCTABJIEHM M MIIHMO, MOCKBA , 200 4 This wor k wa s originall y publishe d i n Russia n b y MIIHM O unde r th e titl e "OcHOBHbi e CTpyKTypu TeopH H npe,a;cTaBJieHHM " ©2004 . Th e presen t translatio n wa s create d unde r license fo r th e America n Mathematica l Societ y an d i s publishe d b y permission . Translated fro m th e Russia n b y Ale x Martsinkovsk y 2000 Mathematics Subject Classification. Primar y 20-01 , 20Cxx ; Secondary 1 7B1 0 , 20G05 , 20G42 .

For additiona l informatio n an d update s o n thi s book , visi t www.ams.org/bookpages/mmono-228

Library o f Congres s C a t a l o g i n g - i n - P u b l i c a t i o n D a t a Zhelobenko, D . P. (Dmitri i Petrovich ) [Osnovnye struktur y i metody teori i predstavlenii . English ] Principal structure s an d method s o f representatio n theor y / D . Zhelobenk o ; translate d b y Alex Martsinkovsky . p. cm . — (Translation s o f mathematical monograph s ; v. 228) "Originally publishe d i n Russia n b y MTSNM O unde r th e title 'Osnovny e struktur y i metod y teorii predstavlenii ' c2004"—T.p . verso . Includes bibliographica l reference s an d index . ISBN 0-821 8-3731 - 1 (alk . paper) 1. Representation s o f groups. 2 . Representations o f algebras. I . Title . II . Series . QA176.Z5413 200 4 512 / .22—dc22 200505235 2

C o p y i n g a n d reprinting . Individua l reader s o f thi s publication , an d nonprofi t librarie s acting fo r them, ar e permitted t o mak e fai r us e of the material, suc h a s to cop y a chapte r fo r use in teachin g o r research . Permissio n i s grante d t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgmen t o f the source i s given . Republication, systemati c copying , o r multiple reproductio n o f any materia l i n this publicatio n is permitte d onl y unde r licens e fro m th e America n Mathematica l Society . Request s fo r suc h permission shoul d b e addressed t o the Acquisitions Department , America n Mathematica l Society , 201 Charle s Street , Providence , Rhod e Islan d 02904-2294 , USA . Requests ca n als o b e mad e b y e-mail t o reprint-permission@ams . org . © 200 6 b y the American Mathematica l Society . Al l rights reserved . The America n Mathematica l Societ y retain s al l right s except thos e grante d t o the United State s Government . Printed i n the United State s o f America . @ Th e paper use d i n this boo k i s acid-free an d falls withi n th e guideline s established t o ensur e permanenc e an d durability . Visit th e AMS home pag e a t http://www.ams.org / 10 9 8 7 6 5 4 3 21 1

1 10 09 08 07 0 6

Contents Preface i

x

Part 1 . Introductio n Chapter 1 . Basi c Notion s 3 1. Algebrai c structure s 3 2. Vecto r space s 8 1 3. Element s o f linea r algebr a 4. Functiona l calculu s 2 5. Unitar y space s 2 6. Tenso r product s 3 7. 5-module s 4 Comments t o Chapte r 1 4 Part 2 . Genera l Theor y 4

4 0 6 5 0 6 9

Chapter 2 . Associativ e Algebra s 5 1 8. Algebra s an d module s 5 1 9. Semisimpl e module s 5 8 10. Grou p algebra s 6 4 11. System s o f generator s 7 0 12. Tenso r algebra s 7 5 13. Forma l serie s 8 0 14. Wey l algebra s 8 6 15. Element s o f rin g theor y 9 3 Comments t o Chapte r 2 9 8 Chapter 3 . Li e Algebra s 9 16. Genera l question s 9 17. Solvabl e Li e algebra s 0 18. Bilinea r form s 0 19. Th e algebra. /7(g ) 1 20. Semisimpl e Li e algebra s 2 21. Fre e Li e algebra s 2 1 22. Example s o f Li e algebra s 3 Comments t o Chapte r 3 3

9 9 5 9 5 0 5 0 7

1 Chapter 4 . Topologica l Group s 3 23. Topologica l group s 3 1 24. Topologica l vecto r space s 4

9 9 5

CONTENTS

25. Topologica l module s 26. Invarian t measure s 27. Grou p algebra s 28. Compac t group s 29. Solvabl e group s 30. Algebrai c group s Comments t o Chapte r 4

152 157 164 170 175 181 185

lapter 5 . Li e Group s 31. Manifold s 32. Li e group s 33. Forma l group s 34. Loca l Li e group s 35. Connecte d Li e group s 36. Representation s o f Li e group s 37. Example s an d exercise s Comments t o Chapte r 5

187 187 192 198 203 209 214 219 224

Part 3 . Specia l Topic s

225

Chapter 6 . Semisimpl e Li e Algebra s 38. Carta n subalgebra s 39. Classificatio n 40. Verm a module s 41. Finite-dimensiona l g-module s 42. Th e algebr a Z(g) 43. Th e algebr a F ext (g) Comments t o Chapte r 6

227 227 233 238 244 250 256 262

Chapter 7 . Semisimpl e Li e Group s 44. Reductiv e Li e group s 45. Compac t Li e group s 46. Maxima l tor i 47. Semisimpl e Li e group s 48. Th e algebr a A(G) 49. Th e classica l group s 50. Reductio n problem s Comments t o Chapte r 7

263 263 268 272 277 283 289 294 300

Chapter 8 . Banac h Algebra s 51. Banac h algebra s 52. Th e commutativ e cas e 53. Spectra l theor y 54. C*-algebra s 55. Representation s o f C*-algebra s 56. Vo n Neuman n algebra s 57. Th e algebr a C*(G) 58. Abelia n group s Comments t o Chapte r 8

301 301 307 312 317 323 329 335 340 346

CONTENTS

vii

Chapter 9 . Quantu m Group s 59. Hop f algebra s 60. Wey l algebra s 61. Th e algebr a U q($) 62. Th e categor y 0\ nt 63. Th e algebr a A q($) 64. Gaussia n algebra s 65. Projectiv e limit s Comments t o Chapte r 9

347 347 353 359 365 370 377 383 388

Appendix A . Roo t System s Comments t o Appendi x A

391 402

Appendix B . Banac h Space s

403

Appendix C . Conve x Set s

407

Appendix D . Th e Algebr a B(H)

413

Bibliography

421

Index

425

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Preface The titl e o f thi s boo k admit s tw o interpretations , wit h emphasi s o n eithe r th e "principal structures " o r the "representatio n theory" . Th e latter i s more preferable , as i t i s difficul t t o identif y wha t th e basi c structure s o f moder n mathematic s are . Nevertheless, i n a sense , th e tw o interpretation s agree . Indeed, representatio n theor y deal s wit h fundamenta l aspect s o f mathemat ics, beginnin g wit h algebrai c structure s lik e semigroups , groups , rings , associativ e algebras, Li e algebras , etc . Eventuall y topolog y enter s th e pla y b y wa y o f algebro topological an d algebro-analytica l structure s lik e topological groups , manifolds , Li e groups, etc . Formall y speaking , th e subjec t o f representation theor y i s the stud y o f homomorphisms (representations ) o f abstract structure s int o linea r structure s con sisting, a s a rule , o f linea r operator s o n vecto r spaces . Bu t i n fac t representatio n theory i s tie d u p wit h structur e theory . Ver y earl y th e student s o f mathematic s learn tha t "rin g theor y i s inseparabl y linke d wit h modul e theory" . A n importan t feature o f this settin g i s that th e abov e structure s ar e eithe r linea r o f hav e suitabl e linearizations (linea r hull s o f semigroups , tangen t Li e algebra s o f Li e groups , etc.) . Here we come to th e questio n o f the rol e representation theor y play s i n moder n mathematics. Originall y (i n the beginnin g o f the 20t h century ) tha t rol e was rathe r modest an d wa s confine d t o th e representatio n theor y o f finite group s and , eventu ally, finite-dimensional (associative ) algebras . W e shoul d mentio n th e connection s of that theor y wit h problems of symmetry i n algebra and geometry, includin g Galoi s theory (th e symmetrie s o f algebrai c equations) , an d wit h problem s o f crystallog raphy. Eventuall y th e subjec t o f representatio n theor y significantl y expande d i n response t o genera l question s fro m analysis , geometry , an d physics . Fundamenta l discoveries i n theoretica l physics , suc h a s the theor y o f relativity an d quantu m me chanics, playe d a significan t rol e i n tha t process . Fo r example , i t turne d ou t tha t logical foundation s o f quantu m mechanic s ca n b e adequatel y expresse d i n term s o f automorphisms o f certai n algebra s (th e algebra s o f observables) . Th e proces s o f describing observable s reduce s t o representatio n theor y o f certai n Li e group s an d algebras. Amon g classica l result s o f tha t perio d w e specificall y mentio n th e work s of E . Car t an an d H . Wey l o n th e genera l aspect s o f th e theor y o f Li e group s an d on harmoni c analysi s o n compac t groups . The underlyin g ide a o f harmoni c analysi s o n group s i s base d o n th e connec tion betwee n a grou p G an d th e "dua l object " G consisting , roughl y speaking , o f irreducible representation s o f G. Usuall y G ca n b e recovered , u p t o isomorphism , from it s dual object G. A remarkabl e featur e o f harmonic analysi s is that numerica l functions o n G ca n b e recovere d fro m thei r (operator ) "Fourie r images" , wher e th e role of elementary harmonic s i s played b y irreducible representation s o f G. A mean ingful definitio n o f Fourie r image s o n locall y compac t group s i s possible becaus e o f the fundamenta l result s o f A. Haar , J . vo n Neumann , an d A . Weil on th e existenc e ix

x PREFAC

E

(and uniqueness ) o f invarian t measure s o n suc h groups . I n tha t sense , th e classi cal Fourie r analysi s (Fourie r serie s an d integrals ) i s subsume d int o a n impressiv e development progra m o f harmoni c analysi s o n topologica l groups . Logical foundation s o f Fourie r analysi s ca n b e significantl y clarifie d withi n th e framework o f "abstrac t harmoni c analysis" , wher e th e grou p G i s replace d b y a C*-algebra. Fundamenta l result s i n tha t directio n ar e du e t o I . M . Gelfan d an d M. A. Naimark (i n the 1 940s) . Beginnin g wit h th e 1 950s , the theor y o f C*-algebra s develops ver y rapidl y and , t o a larg e extent , characterize s th e functiona l analysi s of th e 20t h century . I t i s importan t t o observ e tha t tha t theor y ha s fundamenta l applications t o operato r algebras , Hop f algebras , dynamica l systems , statistica l mechanics, quantu m field theory , etc . Modern representatio n theor y deal s with a wide variety o f associative algebras , including structur e algebra s o f manifolds an d Li e groups, universa l envelopin g alge bras o f Lie algebras, grou p (convolution ) algebras , Hop f algebras , quantu m groups , etc. Notic e tha t th e theor y o f Lie groups, bor n withi n th e contex t o f differential ge ometry, i s now included i n the framework o f functional analysi s by way of bialgebras and forma l group s associate d wit h Li e groups . One ma y als o expan d th e definitio n o f representatio n theor y t o include , i f de sired, suc h neighborin g discipline s a s abstrac t theor y o f differentia l equations , the ory o f sheaves o n homogenou s spaces , microanalysis , quantu m field theory , etc . There i s a know n thesi s accordin g t o whic h "mathematic s i s representatio n theory". Th e correspondin g antithesi s ca n b e state d a s "mathematic s doe s no t reduce to representation theory" . I t i s worthwhile to note the nature of the question . Whatever i s true , i t appear s tha t th e scop e o f representatio n theor y i s alread y comparable wit h tha t o f th e entir e mathematics . It ma y b e tha t th e desir e t o systematiz e mathematic s i n th e spiri t o f represen tation theor y mad e N . Bourbak i writ e th e multi-volum e se t "Element s o f mathe matics" . Despit e certai n shortcoming s o f tha t titani c wor k (excessiv e formalism , unfinished parts ) on e finds origina l treatment o f several fundamenta l issues , includ ing general aspect s o f algebra, topology , th e theor y o f integration, th e theor y o f Lie groups an d Li e algebras , etc . At present , ther e i s a large numbe r o f monographs dealin g wit h variou s aspect s of representation theory , includin g Li e groups an d Li e algebras ([4] , [1 0] , [1 4] , [31], [35], [61 ]) . Banac h algebra s ([6] , [8] , [1 3] , [22] , [49] , [58]) , algebrai c group s ([3] , [29], [64] , [73]) , infinite-dimensiona l group s ([53]) , genera l representatio n theor y ([40]). Th e author' s monograp h [75 ] ca n b e use d a s a n easil y accessibl e sourc e of informatio n o n representation s o f Li e groups , especiall y suitabl e fo r physicists . However, ther e i s still n o monograp h whic h woul d pu t togethe r al l of those aspect s of representatio n theory . To fill th e gap , thi s boo k wa s conceive d a s a compilatio n o f canonica l text s on representatio n theory . I t provide s a systemati c descriptio n o f a wid e spectru m of algebro-topologica l structures . O n on e hand , th e concep t o f suc h a boo k i s appealing becaus e i t allow s u s t o compar e idea s an d method s fro m differen t part s of representatio n theory . O n th e othe r hand , i t i s als o risk y jus t becaus e o f th e sheer volum e o f th e materia l t o b e covered . Nevertheless , th e autho r think s tha t a partia l resolutio n o f this dilemm a i s possible becaus e th e offere d text s hav e bee n carefully worke d upo n an d refined .

PREFACE x

i

The content s o f th e boo k spli t int o thre e parts . Par t I (Introduction ) contain s general fact s fo r beginners , includin g linea r algebr a an d functiona l analysis . Th e survey-type section s o n topology , theor y o f integration , etc . (se e [23] , [24] , [26] , [31]) a s wel l a s Appendice s A , B , C , an d D ar e writte n i n th e sam e spirit . I n the mai n Par t I I (Genera l theory ) w e conside r associativ e algebras , Li e algebras , topological groups , an d Li e groups . W e als o mentio n som e aspect s o f rin g theor y and th e theor y o f algebrai c groups . W e provid e a detaile d accoun t o f classica l results i n thos e branche s o f mathematics , includin g invarian t integratio n an d Lie' s theory o f connection s betwee n Li e group s an d Li e algebras . I n Par t II I (Specia l topics) w e conside r semisimpl e Li e algebr a an d Li e groups , Banac h algebras , an d quantum groups . The boo k bring s th e reade r clos e t o th e moder n aspect s o f "noncommutativ e analysis", includin g harmoni c analysi s o n locall y compac t groups . Th e autho r regards th e content s o f thi s boo k a s a prerequisit e fo r thos e wh o wan t t o seriousl y study representatio n theory . The styl e o f th e boo k allow s th e autho r t o choos e th e dept h o f th e expositio n to hi s taste. Fo r example , w e prove th e theore m o n th e conjugac y o f Car t an subal gebras (i n comple x Li e algebras) bu t omi t a simila r resul t fo r Bore l subalgebra s (i n semisimple Li e algebras) . Ye t th e autho r hope s tha t th e reade r wil l se e a detaile d enough panorami c descriptio n o f representatio n theory . The divers e natur e o f the compile d materia l unavoidabl y lead s to discrepancie s in traditions , whic h sometime s caus e certai n redundanc y i n th e definition s an d notation. Fo r example , th e notatio n E n d X i n th e categor y o f vecto r space s i s sometimes replace d b y L(X) wher e dim X < oo . The exercises included i n the book, a s a rule, ar e designed a s tests for beginners . Sometimes (i n moderation ) th e result s o f th e exercise s ar e use d t o shorte n certai n proofs. Onl y th e exercise s marke d wit h a n asteris k ca n b e viewe d a s mor e o r les s serious problems . While workin g o n th e boo k th e autho r fel t himsel f a chronicler . Indeed , th e book cover s a centur y i n th e developmen t o f mathematics , a perio d whic h i s prob ably no t ye t full y appreciated . The content s o f th e boo k are , t o a larg e extent , base d o n tw o electiv e course s the author gav e at th e Independent Universit y o f Moscow in 1 996-1 998 . Th e lectur e notes o f one o f those course s wer e publishe d i n 200 1 ([78]) . Th e wor k o n thi s boo k was partiall y supporte d b y th e RFF I Gran t 01 -01 -0049 0 an d NW O 047-008-009 . The autho r i s grateful t o V. R. Nigmatulli n fo r hi s help during the proofreadin g of th e text .

D. Zhelobenk o

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Part 1

Introduction

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10.1090/mmono/228/01

CHAPTER 1

Basic Notion s In this chapter w e collect basi c facts abou t algebrai c structures, includin g linea r operators o n vecto r spaces . Followin g th e classica l tradition , w e reserv e th e ter m linear algebra for th e context o f finite-dimensional vecto r spaces . Not e however tha t methods o f linea r algebr a ar e als o use d i n th e stud y o f infinite-dimensiona l vecto r spaces. A s a n example , w e mentio n Sectio n 5 , where , fo r referenc e purposes , w e discuss linea r operator s o n Hilber t spaces . 1. Algebrai c structure s 1.1. Semigroup s an d groups . A semigroup i s a n abstrac t se t S togethe r with a given binar y operatio n S x S —> 5 , (x,y) — i > xy, calle d multiplication, whic h satisfies th e associativity axiom (1.1) x(yz)

= (xy)z

for al l x,y,z G S. Th e elemen t (1 .1 ) i s denoted xyz. Similarl y (usin g inductio n o n n) on e define s associative words (or monomials) x\ • • - x n. I n particular , fo r eac h x G S th e associative power x n i s defined . The semigrou p S i s sai d t o b e commutative i f xy — yx fo r al l x,y G 5, an d S is calle d a semigroup with identity i f ther e i s a n elemen t e G S (a n identity ) suc h that (1.2) ex

= xe = x

for eac h x G S. Th e identit y e is uniquely determine d b y axio m (1 .2) . Indeed , i f e' is anothe r identity , the n e — ee f = e'. An elemen t x G S i s said t o b e invertible i f there i s a n elemen t y £ S suc h tha t xy — yx — e. I f w e als o hav e xy' = y'x = e fo r som e y' ', then th e equalit y y = ye = y{xy) = (yx)y f = ey f = y' implies th e uniquenes s o f y. Suc h a n elemen t i s denote d x~ l an d i s calle d th e inverse o f x. Not e tha t {xy)'1 =y~ 11 x~ for al l invertibl e x,y G S. A semigrou p G i s calle d a group i f G i s a semigrou p wit h identit y i n whic h every elemen t i s invertible . A commutativ e grou p i s als o calle d a n abelian group. Sometimes multiplicatio n i n a n abelia n grou p i s written additively : (x , y)— i > x 4 - y. In tha t cas e th e identit y elemen t i s referre d t o a s th e zero (o r neutral) elemen t OGG. Examples.1 . Fo r eac h se t M le t EndA f b e th e se t o f al l endomorphisms (i.e., transformation s a: M -* M) o f M. I t i s clea r tha t E n d M i s a semigrou p 3

4

1. BASI C N O T I O N S

with respec t t o th e compositio n o f endomorphism s (1.3) (ab)x

= a(bx)

where a,b E EndM, x G M. I n thi s example , th e identit y transformatio n ex = x (x G M) i s the identity . 2. Th e subse t A u t M c E n d M , consistin g o f automorphisms (i.e. , invertibl e endomorphisms) o f M , i s a grou p (calle d th e automorphism group o f M). The associativit y axio m (1 .1 ) ca n b e viewe d a s a n abstractio n o f th e corre sponding propert y o f E n d M . Examples o f abelian group s include: Z (th e set of integers), Z p (th e cyclic grou p of integer s modul o p) , Q (th e se t o f rationa l numbers) . A s example s o f abelia n semigroups w e mention Z + (th e se t o f nonnegativ e integers ) an d N = Z + \ {0 } (th e natural numbers) . 1.2. Ring s an d fields . A se t R wit h tw o associativ e binar y operations , addition (x , y) \-+ x + y an d multiplication (x, y)— i > xy, i s called a ring i f R i s a grou p with respec t t o additio n an d th e distributivity condition s hold : (1.4) x(y

+ z) — xy + xz, (x

+ y)z = xz + yz

for al l x,y,z G R. I n particular , R ha s a zer o elemen t 0 such tha t 0 • x = x • 0 = 0 for eac h x G R. The rin g R i s sai d t o b e commutative (respectively , a ring with identity) i f R is commutativ e (respectively , ha s a n identity ) wit h respec t t o multiplication . A ring wit h identit y i s calle d a skew field i f ever y nonzer o elemen t i s invertible . A commutative ske w fiel d i s calle d a field. A subse t G\ C G (respectively , R\ C R) i s calle d a subgroup o f th e grou p G (respectively, a subring o f R) i f G\ i s close d unde r th e grou p operation s (x,y) — i » xy~1 (respectively , i f R\ i s close d unde r th e rin g operations) . I n tha t cas e G (respectively, R) i s calle d a n extension o f G\ (respectively , o f R\). The identit y elemen t o f th e rin g R i s usuall y denote d 1 . W e shal l als o assum e that a rin g wit h identit y i s nontrivial , i.e. , 0 ^ 1 . Examples. 1 . I f R i s a rin g wit h identity , the n R contain s th e subrin g R\ o f multiples o f th e identity , i.e. , 0 , ±n, wher e n = 1 + • • • + 1 (n summands) . 2. An y commutativ e rin g R give s ris e t o a n extensio n R[x], consistin g o f al l polynomials i n x. Here , b y a polynomia l w e understan d a n elemen t o f th e for m f{x) = a 0 + a\x H h

a nxn,

with coefficient s ai G R (i — 0 , 1, . . ., n) . A mor e detaile d analysi s o f thi s notio n will b e give n i n 4.1. We als o recal l th e standar d notatio n R , C , H I (respectively) fo r th e field o f rea l numbers, th e field o f comple x numbers , an d th e ske w field o f rea l quaternions . 1.3. Vecto r spaces . Give n a field F , a se t X wit h operation s o f additio n X x X — > X, (x , y) H ^ x -f - y an d multiplicatio n F x X — > X, (A , x) i- » Ax, i s calle d a vector space over F i f X i s a n abelia n grou p wit h respec t t o additio n an d (1.5) \{x

+ y) = \x + Ay, ( A + ji)x = \x + fix

for al l A , \i G F an d x , y G X. I n particular , 0 • x = 0 fo r eac h x G X, wher e 0 i n the left-han d (right-hand ) sid e i s th e zer o elemen t o f F {X). Th e element s x G X are calle d vectors, an d th e element s A G F scalars. W e shal l cal l th e operation s o f addition an d multiplicatio n th e vector operations o n X.

1. A L G E B R A I C S T R U C T U R E S

5

We shal l als o us e th e symbo l 0 t o denot e a trivia l (zero ) vecto r spac e consistin g of a singl e elemen t 0 . E x a m p l e s . 1 . Th e Cartesia n powe r F n o f an y field F ha s a n obviou s structur e of a vecto r spac e ove r F. Indeed , an y x G Fn ca n b e writte n a s a n ordere d n-tupl e X = [X\ , .

.. , X

n),

where Xi G F (i = 1 , . . . , n ) an d th e vecto r operation s o n X n ar e define d compo nentwise (coordinatewise) . T h e latte r mean s t h a t th e vecto r x + y (respectively , Xx) ha s coordinate s xi + yi (respectively , Xxi), wher e i = 1 , . . . , n. 2. Fo r an y se t M , th e se t F(M) o f al l F- valued function s / : M — » F i s als o a vector spac e ove r F i f th e vecto r operation s ar e define d pointwise . 1.4. Linea r o p e r a t o r s . Le t X an d Y b e vecto r space s ove r a field F. A m a p a: X — > Y i s sai d t o b e linear i f (1.6) a(A

x + iiy) = Xax + \iay

for al l A,/ i G F an d x,y e X. I n particular , a(0 ) = 0 (wher e th e sam e symbo l 0 denotes th e zer o element s i n X an d Y). Linear map s a : X — » Y ar e als o calle d linear operators (fro m X t o Y ) o r homomorphisms (fro m X t o Y ) . T h e latte r ter m emphasize s th e fac t t h a t operator s (1 .6 ) respect th e vecto r operation s i n X an d Y . The se t o f al l homomorphism s a: X — » Y i s denote d Hor n (X , Y ). I t i s clea r t h a t Hor n (X , Y ) i s a vecto r spac e ove r F wit h respec t t o th e operation s (1.7) (Xa

+[ib)x =

Xax + fibx,

where A,^ x G F an d a,b • X ar e called endomorphisms o f X , an d thei r totalit y i s denote d E n d X = Hor n (X , X ) . We remar k t h a t th e symbo l E n d X i s onl y use d fo r th e se t o f linear map s (1 .6) . Sometimes (t o avoi d possibl e confusion ) instea d o f E n d X on e use s th e symbo l L(X). There i s a n importan t generalizatio n o f th e notio n o f linea r m a p . Namely , give n any collectio n o f vecto r space s Xi (i = 0 , 1 , . . . , n ) , a m a p a : X\ x • • • x X n —* X Q is sai d t o b e multilinear (o r n-linear ) i f i t i s linea r i n eac h argumen t Xi G X % [i — 1, .. . , n). I f n = 2 , 3 on e use s th e term s (respectively ) bilinea r an d trilinear . 1.5. A l g e b r a s . A vecto r spac e A ove r a field F i s calle d a n algebra (ove r F) if i t carrie s a n F-bilinea r operation , calle d multiplication, A x A —> • A: (x , y)— i > xy. The bilinearit y conditio n i s expresse d b y th e distributivit y condition s (1 .6 ) an d (1.7) wher e w e assum e t h a t a , b,x,y G A. The algebr a A i s sai d t o b e associative (respectively , commutative) i f th e mul tiplication o n A i s associativ e (respectively , commutative) . T h e algebr a A i s sai d to b e unital (o r an algebra with identity) i f i t ha s a multiplicativ e identity , whic h is usuall y denote d 1 . A subse t X\ C X (respectively , A\ C A) i s calle d a subspace o f X (respectively , a subalgebra o f A) i f X i (respectively , A{) i s close d wit h respec t t o th e vecto r oper ations i n X (respectively , algebrai c operation s o n A). I n t h a t cas e X (respectively , A) i s calle d a n extension o f X i (respectively , o f A\). If A i s a unita l algebra , the n th e m a p F —> A, A — i > A • 1 i s a n embeddin g o f F as a subalgebr a o f A. Thu s w e ca n writ e A G A instea d o f A • 1 G A.

6

1. BASI C N O T I O N S

Examples. 1 . Fo r each vector spac e X ove r F th e se t L(X ) = E n d X (se e 1 .4 ) is a n associativ e algebr a ove r F. 2. Th e algebr a L(n ) = Ma t (n,F ) consistin g o f n x n-matrice s wit h entrie s i n F i s a n associativ e algebr a ove r F. 3. Th e subalgebra D(n) C L(n ) o f all diagonal matrices in L(n) i s commutative. 1.6. Algebrai c structures . A n algebraic structure i s a se t S togethe r wit h a collectio n o f n-ary relation s (i n the Cartesia n power s S n) an d a system o f axiom s for thos e relations . Usuall y thos e relation s ar e writte n a s functions , i.e. , map s Sn— > 5m . A s example s o f suc h map s w e mentio n th e operation s o f additio n an d multiplication considere d above . In tha t sense , al l example s considere d s o fa r (semigroups , groups , rings , etc. ) are specia l case s of algebrai c structures . Tw o suc h structure s Si an d 5 2 ar e sai d t o be o f th e same kind i f they ar e define d b y th e sam e relation s an d axioms . The convenienc e o f using th e sam e terminology fo r variou s algebrai c structure s leads u s to th e concep t o f a morphism o f algebraic structures . Namely , a morphis m from a structur e Si t o a structur e 5 2 o f th e sam e kin d i s an y ma p (p: Si— » 52 respecting thes e structure s (thi s mean s tha t ip maps th e structura l relation s i n Si to th e correspondin g relation s i n 52) . For example , a morphis m (o r homomorphism ) betwee n semigroup s Si an d 5 2 is an y ma p

: Si — » 52 preservin g th e operatio n o f multiplication , i.e. , (1.8) (p(xy)

= _1 (^>(A)) « A fo r al l object s A o f if i an d ^(^~ 1 (B)) « B for al l object s B o f if2) - A covarian t functo r : ifi— » if 2 i s calle d a n embedding of if i i n if 2 i f $ give s ris e t o a n isomorphis m betwee n if i an d som e subcategor y of i f 2.

8

1. BASI C N O T I O N S

We shal l occasionall y us e th e categorica l vocabular y t o shorte n som e state ments. Sometime s (i n Par t III ) suc h shortening s wil l b e substantial . Example. Fo r eac h morphis m a: M — » N i n SE T w e ca n defin e th e dua l morphism a * : F(N) - • F(M) (se e 1 .3 ) b y (1.12) (a*f)(x)

= f(ax),

where / G F(N) an d x G M. I t i s eas y t o chec k tha t (ab)* = 6*a * fo r composi tions (1 .1 1 ) . Thus th e ma p M — i > F(M) give s ris e t o a contravarian t functo r fro m SE T t o the categor y o f algebra s ove r F.

2. Vecto r space s 2.1. Notation . Le t X b e a vecto r spac e ove r a field F. W e shal l conside r systems o f elements e = (e^)^ / o f X (wher e / i s an arbitrar y se t o f indices). Finit e sums

(2.1) x

= J2 X^

where A ^ G F (wit h onl y finitely man y A ; ^ 0 ) ar e calle d linear combinations o f th e ei (i e I). Th e se t Fe o f al l suc h linea r combination s i s calle d th e linear hull (or span) o f e . Clearly, Fe i s a subspac e o f X. I f X\ — Fe, the n w e shal l sa y tha t X\ is spanned b y e . A syste m e i s sai d t o b e linearly independent (respectively , spanning) i f x = 0 in (2.1 ) implie s tha t A ; = 0 fo r eac h i G I (respectively , i f F e = X). Linea r inde pendence i s equivalent t o th e uniquenes s o f decomposition (2.1 ) . I n tha t cas e (2.1 ) is written a s yz.z) x

— y x^e^, i

where th e Xi (i G I) ar e uniquel y determine d b y x. Her e w e se t Xi — Si(x). Th e coefficients xi (i G I) ar e calle d th e coordinates o f x (wit h respec t t o e) . Each linearl y independen t spannin g syste m e i n X i s calle d a basis of X. An y vector x G X i s the n uniquel y represente d i n th e for m (2.2) , wit h coordinate s %i ~ &i \%) •

Examples. 1 . Th e coordinat e spac e F n (se e 1 .3 ) ha s a basi s consistin g o f th e elements e ^ = ( 0 , . . . , 0,1 , 0 , . . ., 0 ) wit h 1 a t it h place . I n thi s cas e th e notatio n x — ( x i , . . ., x n) identifie s th e coordinate s X{ — £i(x) o f x G Fn wit h respec t t o th e basis (ei), wher e i — 1, . . ., n. 2. Le t F[M] C F(M) b e th e subse t o f al l finite function s f:M->F, i.e. , th e functions whic h ar e no t zer o onl y a t a finite numbe r o f point s o f M. Clearl y F[M] is a vecto r subspac e o f F(M) wit h "delta-functions " S a (a G M) a s a basis , wher e da(x) = 0 when x ^ a an d S a(a) = 1 . Henceforth w e shal l identif y a G M wit h th e delta-functio n 5 a G F[M\. A s a result, M embed s (a s a basis ) i n F[M\. Th e spac e F[M] i s called th e formal linear hull (ove r F) o f M.

2. V E C T O R SPACE S

9

2.2. Bases . Accordin g t o the well-known theore m o f Hamel (see , for example , [1]), every vector space X over a field F has a basis. Th e proo f o f thi s theore m i s based o n th e Zor n lemma , whic h implie s th e existenc e o f a maxima l (wit h respec t to inclusion ) linearl y independen t syste m i n X. A s a n eas y exercise , th e reade r should chec k tha t eac h suc h syste m i s a basi s o f X. A substantia l refinemen t o f Hamel' s theore m i s that any two bases of X are of the same cardinality. W e shal l sketc h a proo f o f thi s statement . If X admit s a finite basis , the n th e proo f i s an eas y exercise . Assum e no w tha t X ha s base s A an d B wit h infinit e cardina l number s a = card A an d (3 — card B. Note that A i s a disjoint unio n of subsets A n ( n G N) , where A n consist s of element s a G A whic h ca n b e represente d a s a linea r combination , wit h nonzer o coefficients , of exactl y n element s b G B. I t i s eas y t o chec k tha t car d .An < nf3 n. Therefor e oo

(2.3) a

< ]Tn/T. 71=1

It i s well known (see , fo r example , [65] ) tha t (3 2 — (3 fo r eac h infinit e cardina l f3, so that j3 n — (3 fo r eac h n G N. Moreover , th e right-han d sid e o f (2.3 ) coincide s wit h j3 and therefor e a < (3. Similarly, (3 < a an d therefor e a — j3. Thus ever y vector spac e X give s rise to a unique cardina l numbe r dimX , calle d the dimension o f X, an d define d a s th e cardinalit y o f a n arbitrar y basi s o f X. Th e space X i s said to he finite-dimensional (respectively , infinite-dimensional) i f dim X is finite (respectively , infinite) . Examples. 1 . d i m F n = n. 2. Ther e ar e vecto r space s o f arbitrar y dimension . Fo r example , dimF[M ] = cardM. Exercise. Chec k (similarl y to the proof of Hamel's theorem) tha t ever y linearl y independent syste m i n a vecto r spac e X i s containe d i n som e basi s o f X. Fo r example, eac h nonzer o vecto r i s part o f a basi s o f X. 2.3. Proposition . Let a G Hor n (X, Y) (znVECT F ). Then (a) For each basis e^ (i G /) of X, the operator a is uniquely determined by its values aci. (/?) For each collection f % G Y (i G I), there is an operator a G Horn (X, Y) such that aei — fi for all i G /. PROOF. Fo r the first assertion , i t suffices t o notice that, i n the notation o f (2.2), (2.4) ax

— y^

jXja(el).

i

For th e secon d assertion , on e quickl y check s tha t th e sam e formul a define s a a s a linear map . • As a consequence , w e have th e rul e (2.5) H o m ( X , Y ) ^ y

a

,

where a = dimX . 2.4. Corollary . The equality dim X = d i m F for X and Y in VECT^ ? is equivalent to X « Y.

10

1. BASI C N O T I O N S

Indeed, i f d i m X = di m Y, the n th e formul a ae % — fi, wher e a an d fa ar e base s (respectively) i n X an d Y , define s th e desire d isomorphis m X « Y . E x a m p l e s . 1 . Eac h n-dimensiona l vecto r spac e X (n G Z + ) i s isomorphi c t o Fn. 2. Eac h vecto r spac e X ove r a field F i s isomorphi c t o F [ M ] , wher e d i m X = cardM. 2.5. M a t r i c e s . Le t e = (cj) jej an d / = {fi)iei b e base s i n vecto r space s X and (respectively ) Y . Expressin g ae 3 (j G J) i n term s o f th e element s o f / , w e hav e (2.6) ae

3

= ^Pa^/i * i

with a tJ G F. T h e collectio n a/ 5e = ( a ^ ) , wher e i G / an d j G J, i s calle d th e matrix o f a (i n t h e base s / an d e) . T h e element s a ^ wit h fixed i (respectively , j) ar e calle d th e rows (respectively , columns) o f th e matri x a/ }€>. Accordin g t o (2.6) , th e column s aej o f a j 5 e satisf y th e followin g finiteness condition : ( $ ) Fo r eac/ i j E J only finitely many numbers a^ are different from zero. Conversely, i t follow s fro m 2. 3 (/3 ) t h a t eac h matri x a ^ e satisfyin g conditio n (e (i n othe r words, w e shal l identif y th e operato r a G Hor n (X , Y) wit h it s matri x (2.6)) . A s an exercise , th e reade r shoul d chec k t h a t th e compositio n o f operator s a — be (whenever i t i s defined ) correspond s t o th e standar d matri x multiplicatio n (2.7) a,ij

= }^b

ikckj.

k

In particular , th e algebr a E n d X i s isomorphi c t o th e matri x algebr a M a t ( n , F ) , where n = d i m X , consistin g o f al l squar e matrice s o f orde r n satisfyin g condi tion () . Normally, w e shal l us e th e symbo l Ma t (n , F) whe n n < oo . I n t h a t cas e Mat (n , F) consist s o f al l n x n matrice s ove r F. 2.6. D u a l s p a c e s . A vecto r spac e X * consistin g o f al l F- valued linea r func tions (als o calle d functionals ) o n X i s calle d th e (algebraically ) dual space o f X (X* = H o m ( X , F ) ) . As i n (2.5) , th e value s

(2.8) f(x)

= Y,fi x*>

where Xi — £i(x) ar e th e coordinate s o f x i n th e basi s e (se e 2.1 ) , o f eac h functiona l / G X* ar e uniquel y determine d b y th e coefficient s f t G F. I n t h a t sense , (2.8 ) ca n be writte n a s (2.9) X * ^ F where a = d i m X .

a

,

2. V E C T O R SPACE S

11

Let e — (ei)iei b e a linearl y independen t se t i n X. W e wan t t o sho w that , in X* , ther e exist s a dual system Si (i G I) give n b y (2.10) £i(e

j)

= 5 ij,

where 6ij i s Kronecker' s delt a (5 i3 — 0 whe n i ^ j , and Su = 1 ) . Indeed , e ca n be extende d t o a basi s o f X (se e 2.2) . No w defin e Si(x) = Xi a s th e coordinate s o f x G X i n tha t basi s (se e 2.1 ) . In particular , fo r eac h nonzer o x G X ther e i s a n / G X * suc h tha t f(x) ^ 0 . The dua l system s allo w us to find explici t expression s fo r matri x element s (2.6 ) of a. Namely , (2.11) dij

= ipi(aej),

where ipi (i G I) i s the syste m dua l t o th e basi s fi (i G I) o f Y. Examples. 1 . Th e vecto r spac e F n ca n b e identifie d wit h it s dua l vi a (2.8) , where i = 1 , . . . , n. 2. Th e spac e dua l t o F[M] ca n b e identifie d wit h F(M) usin g th e formul a

9(f) = £/(*)