Principal Structure s and Method s o f Representation Theor y

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Translations o f

MATHEMATICAL MONOGRAPHS

Volume 22 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^*

10.1090/mmono/228

E D I T O R I A L C O M M I T T E E

A M S S u b c o m m i t t e e

Robert D . MacPherso n Grigori i A . Marguli s Jame s D . Stashef f (Chair )

A S L S u b c o m m i t t e e Steffe n Lemp 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

O C H O B H b l E C T P Y K T Y P b l H M E T O H b l

T E O P M M r i P E H C T A B J I E H M M

MIIHMO, MOCKBA , 200 4

This wor k wa s originally 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 17B10 , 20G05 , 20G42 .

For additiona l informatio n an d update s o n thi s book , visi t w w w . a m s . o r g / b o o k p a g e s / m m o n o - 2 2 8

Library o f Congres s Cataloging-in-Publicat io 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 Russian b y MTSNMO 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 0-8218-3731- 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

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10 9 8 7 6 5 4 3 2 1 1 1 10 09 08 07 06

Contents

Preface i x

Part 1 . Introductio n 1

Chapter 1 . Basi c Notion s 3 1. Algebrai c structure s 3 2. Vecto r space s 8 3. Element s o f linea r algebr a 1 4 4. Functiona l calculu s 2 0 5. Unitar y space s 2 6 6. Tenso r product s 3 5 7. 5-module s 4 0 Comments t o Chapte r 1 4 6

Part 2 . Genera l Theor y 4 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 9 16. Genera l question s 9 9 17. Solvabl e Li e algebra s 10 5 18. Bilinea r form s 10 9 19. Th e algebra. /7(g) 11 5 20. Semisimpl e Li e algebra s 12 0 21. Fre e Li e algebra s 12 5 22. Example s o f Li e algebra s 13 0 Comments t o Chapte r 3 13 7

Chapter 4 . Topologica l Group s 13 9 23. Topologica l group s 13 9 24. Topologica l vecto r space s 14 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

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 Lie group s 37. Example s an d exercise s Comments t o Chapte r 5

152 157 164 170 175 181 185

187 187 192 198 203 209 214 219 224

Part 3 . Specia l Topic s

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

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

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

225

227 227 233 238 244 250 256 262

263 263 268 272 277 283 289 294 300

301 301 307 312 317 323 329 335 340 346

CONTENTS

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

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

Appendix B . Banac h Space s

Appendix C . Conve x Set s

Appendix D . Th e Algebr a B(H)

Bibliography

Index

vii

347 347 353 359 365 370 377 383 388

391 402

403

407

413

421

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 way 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 have suitabl e linearizations (linea r hull s o f semigroups , tangen t Li e algebra s o f Lie groups , etc.) .

Here we come to the question o f the role representation theor y play s in moder n mathematics. Originall y (i n the beginning o f the 20t h century) tha t rol e was rathe r modest an d wa s confined 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 with problems of symmetry i n algebra and geometry, including 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 foundations o f quantum 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 specifically 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 feature o f harmonic analysi s is that numerica l functions o n G ca n b e recovered fro m thei r (operator ) "Fourie r images" , wher e th e role of elementary harmonic s i s played b y irreducible representations 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 the 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 harmonic 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 1940s) . Beginnin g wit h the 1950s , the theory 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 structure algebra s of manifolds an d Lie groups, universa l enveloping alge-bras o f Lie algebras, group (convolution ) algebras , Hop f algebras , quantum groups , etc. Notic e that th e theory o f Lie groups, born within the contex t o f differential ge -ometry, i s now included in 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 representation 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 the entir e mathematics .

It ma y b e tha t th e desir e t o systematiz e mathematic s i n the 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 original 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 theory o f Lie groups an d Li e algebras , etc .

At present , ther e i s a large number o f monographs dealin g with various aspect s of representation theory , includin g Li e groups an d Li e algebras ([4] , [10] , [14] , [31], [35], [61]) . Banac h algebra s ([6] , [8] , [13] , [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 no monograph whic h woul d pu t togethe r al l of those aspect s of representation 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 .

P R E F A C E x i

The content s o f the 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 those 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 the theore m o n the conjugac y o f Car t an subal -gebras (i n complex Li e algebras) bu t omi t a similar 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 representation theory .

The divers e nature o f the compiled 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 End 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, are designed a s tests for beginners . Sometimes (i n moderation ) th e result s o f the 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 the developmen t o f mathematics , a period whic h i s prob -ably no t ye t full y appreciated .

The content s o f the boo k are , t o a larg e extent , base d o n tw o electiv e course s the author gav e at the Independent Universit y of Moscow in 1996-1998. Th e lectur e notes o f one o f those course s were published i n 200 1 ([78]) . Th e wor k o n thi s boo k was partially supporte d b y th e RFF I Gran t 01-01-0049 0 an d NW O 047-008-009 .

The author i s grateful t o V. R. Nigmatullin fo r hi s help during the proofreadin g of the text .

D. Zhelobenk o

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Bibliography

Yu. A . Bakhturin , Basic structures of modern algebra. Nauka , Moscow , 1990 ; Englis h transl. , Kluwer, Dordrecht , 1993 . I. N . Bernstein , I . M . Gelfand , an d S . I . Gelfand , Structure of representations that are gen-erated by vectors of higher weight. Funktsional . Anal , i Pril . 5 (1971) , no . 1 , 1-9 . (Russian ) A. Borel , Linear algebraic groups. Secon d edition . Graduat e Text s i n Mathematics , vol . 126 . Springer-Verlag, Ne w York , 1991 . N. Bourbaki , Lie groups and Lie algebras. Chapter s 1-3 , 4-6 , 7-9 . Springer-Verlag , Berlin , 1998, 2002 , 2005 .

, Integration. Chapter s 1-6 , 7-9 . Springer-Verlag , Berlin , 2004 . , Theories spectrales. Chapitr e I : Algebre s normees . Chapitr e II : Groupe s localemen t

compacts commutatifs . Actualite s Scientifique s e t Industrielles , No . 1332 , Hermann , Pari s 1967.

, Topological vector spaces. Springer-Verlag , Berlin , 1987 . 0 . Brattel i an d D . W. Robinson , Operator algebras and quantum statistical mechanics. Vol . 1 . C*- an d Vy*-algebras , algebras , symmetr y groups , decompositio n o f states . Springer-Verlag , New York-Heidelberg , 1979 . V. Char i an d A . Pressley , A guide to quantum groups. Cambridg e Univ . Press , Cambridge , 1995. C. Chevalley , Theorie des groupes de Lie I , II , III . Vol . I i s availabl e i n English : Princeto n University Press , Princeton , NJ , 1999 . Vols . I I an d III : Hermann , Paris , 1951 , 1955 . A. Connes , Noncommutative geometry. Academi c Press , Sa n Diego , CA , 1994 . E. Demidov , Quantum groups. Factorial , Moscow , 1998 . (Russian ) J. Dixmier , Enveloping algebras. Amer . Math . Soc , Providence , RI , 1996 .

, C*-algebras. North-Holland , Amsterdam-Ne w York-Oxford , 1977 . V. G . Drinfeld , Quantum groups. Proc . Intern . Congres s Math. , Berkeley , 1986 . Amer . Math . Soc , Providence , RI , 1987 . M. Enoc k an d J.-M . Schwartz , Kac algebras and duality of locally compact groups. Springer -Verlag, 1992 . L. D . Faddee v an d A . O . Yakubovskii , Lectures on quantum mechanics for students of math-ematics. Leningra d Stat e University , 1980 . (Russian ) G. Gaspe r an d M . Rahman , Basic hypergeometric series. Wit h a forewor d b y Richar d Askey . Encyclopedia o f Mathematic s an d it s Applications , vol . 35 . Cambridg e Univ . Press , Cam -bridge, 1990 . L. Garding , Vecteurs analytiques dans les representations des groups de Lie. Bull . Soc . Math . France 8 8 (1960) , 73-93 . 1. M . Gelfan d an d A . A . Kirillov , Structure of the Lie sfield connected with a semisimple decomposable Lie algebra. Funktsional . Anal , i Pril . 3 (1969) , no . 1 , 7-26 . (Russian ) I. M . Gelfan d an d M . A . Naimark , Unitary representations of the classical groups. Trudy -Mat. Inst . Steklov. , vol . 36 , Izdat . Akad . Nau k SSSR , Moscow-Leningrad , 1950 . (Russian ) I. Gelfand , D . Raikov , an d G . Shilov , Commutative normed rings. Chelsea , Ne w York , 1964 . M. Got o an d F . D . Grosshans , Semisimple Lie algebras. Lectur e Note s Pur e Appl . Math. , vol. 38 , Marce l Dekker , Ne w York-Basel , 1978 . P. R . Halmos , Measure theory. D . Va n Nostrand , Ne w York , 1950 .

, A Hilbert space problem book. 2n d ed. , Springer-Verlag , Ne w York-Berlin , 1982 . S. Helgaso n Groups and geometric analysis. Integral geometry, invariant differential opera-tors, and spherical functions. Mathematica l Survey s an d Monographs , vol . 83 , Amer . Math . Soc. Society , Providence , RI , 2000 .

421

422 BIBLIOGRAPH Y

[27] E . Hewit t an d K . A . Ross , Abstract harmonic analysis. I , II . Springer-Verlag , Berlin-Ne w York, 1979 , 1970 .

[28] A . S . Kholevo , Introduction to quantum information theory, MCCME , Moscow , 2002 . (Rus -sian)

[29] J . E . Humphreys , Linear algebraic groups. Graduat e Text s i n Mathematics , No . 21 . Springer -Verlag, Ne w York-Heidelberg , 1975 .

[30] , Introduction to Lie algebras and representation theory. Springer-Verlag , Ne w York -Berlin, 1978 .

[31] N . Jacobson , Lie algebras. Dover , Ne w York , 1979 . [32] A . Joseph , Quantum groups and their primitive ideals. Springer-Verlag , Berlin , 1995 . [33] V . G . Kac , Infinite-dimensional Lie algebras. Thir d edition , Cambridg e Univ . Press , Cam -

bridge, 1990 . [34] V . G . Ka c an d D . A . Kazhdan , Structure of representations with highest weight of infinite-

dimensional Lie algebras, Adv . Math . 2 4 (1979) , 97-108 . [35] I . Kaplansky , Lie algebras and locally compact groups. Univ . Chicag o Press , Chicago , IL ,

1995. [36] M . Kashiwara , The universal Verma module and the b-function. In : Algebrai c group s an d

related topic s (Kyoto-Nagoy a 1983) , North-Holland , 1985 , pp . 69-81 . [37] , Crystallyzing the q-analogue of the universal enveloping algebras. Comm . Math .

Phys. 12 2 (1990) , 249-260 . [38] Ch . Kassel , Quantum groups. Springer-Verlag , Berlin-Ne w York-Heidelberg , 1995 . [39] J . Kelley , General topology. Springer-Verlag , Ne w York-Berlin , 1975 . [40] A . A . Kirillov , Elements of the theory of representations. Springer-Verlag , Berlin-Ne w York ,

1976. [41] A . I . Kostriki n an d Yu . I . Manin , Linear algebra and geometry. Gordo n an d Breach , Ams -

terdam, 1997 . [42] A . W. Knapp , Representation theory of semisimple groups. Princeto n Univ . Press , Princeton ,

NJ, 2001 . [43] A . N . Kolmogoro v an d S . V . Fomin , Elements of representation theory. Nauka , Moscow ,

1978. (Russian ) [44] S . Lang , Algebra. Thir d Edition , Springer-Verlag , Ne w York , 2002 . [45] L . H . Loomis , An introduction to abstract harmonic analysis. Va n Nostrand , Toronto-Ne w

York-London, 1953 . [46] Yu . I . Manin , Quantum groups and noncommutative geometry. Montreal , 1988 . [47] A . Molev , A basis for representations of symplectic Lie algebras. Comm , Math . Phys . 20 1

(1999), 591-618 . [48] , Gelfand-Tsetlin bases for classical Lie algebras. Univ . o f Sydney , 2002 , 120-181 . [49] G . J . Murphy , C*-algebras and operator theory. Academi c Press , Boston , MA , 1990 . [50] M . A . Naimark , Normed rings. Wolters-Noordhoff , Groningen , 1970 . [51] M . A . Naimar k an d A . I . Stern , Theory of group representations. Springer-Verlag , Ne w York ,

1982. [52] R . Narasimhan , Analysis on real and complex manifolds. North-Holland , Amsterdam , 1985 . [53] Yu . A . Neretin , Categories of symmetries and infinite-dimensional groups. Oxfor d Univ .

Press, Ne w York , 1996 . [54] S . P . Noviko v an d A . T . Fomenko , Basic elements of differential geometry and topology.

Kluwer Academi c Publishers , Dordrecht , 1990 . [55] R . S . Pierce , Associative algebras. Springer-Verlag , Ne w York , 1982 . [56] L . S . Pontryagin , Continuous groups. Fourt h ed. , Nauka , Moscow , 1984 ; Englis h transl . o f

the secon d ed . Topological groups. Gordo n an d Breach , London , 1966 . [57] W . Rudin , Functional analysis. Secon d ed. , McGraw-Hill , Ne w York , 1991 . [58] S . Sakai , C* -algebras and W* -algebras, Springer-Verlag , Berlin , 1998 . [59] H . H . Schaefe r an d M . P . Wolff , Topological vector spaces. Secon d edition . Springer-Verlag ,

New York , 1999 . [60] H . Seifer t an d W . Threlfall , Seifert and Threlfall: a textbook of topology. Academi c Press ,

New York-London , 1980 . [61] J.-P . Serre , Lie algebras and Lie groups. Secon d edition . Lectur e Note s i n Mathematics , vol .

1500. Springer-Verlag , Berlin , 1992 . [62] , Linear representations of finite groups. Springer-Verlag , Ne w York-Heidelberg, 1977 .

BIBLIOGRAPHY 423

[63] N . N . Shapovalov , On a bilinear form on the universal enveloping algebra of a complex semisimple Lie algebra, Funktsional . Anal , i Prilozh . 6 (1972) , no . 4 , 65-70 ; Englis h transl. , Functional Anal . Appl . 6 (1972) , 307-312 . I. R . Shafarevich , Basic algebraic geometry. 1 , 2 . Springer-Verlag , Berlin , 1994 . A. She n an d N . K . Vereshchagin , Elements of set theory. MCCM E Publishers , Moscow , 1999 ; English transl. , Amer . Math . Soc , Providence , RI , 2002 . G. E . Shilov , Mathematical analysis. A special course. Fizmatgiz , Moscow , 1960 ; Englis h transl., Pergamo n Press , Oxford-Ne w York-Paris , 1965 .

, Mathematical analysis. A second special course. Nauka , Moscow , 1965 . (Russian ) E. M . Stei n an d G . Weiss , Introduction to Fourier analysis on Euclidean spaces. Princeto n Univ. Press , Princeton , NJ , 1971 . S. Sternberg , Lectures on differential geometry. Prentic e Hall , 1964 . A. Weil , L'integration dans les groupes topologiques et ses applications. Actual . Sci . Ind. , no . 869. Herman n e t Cie. , Paris , 1940 . H. Weyl , Classical groups. Their invariants and representations. Princeto n Universit y Press , Princeton, NJ , 1997 . E. B . Vinberg , A course in algebra. Faktorial , Moscow , 1999 ; Englis h transl. , Amer . Math . Soc , Providence , RI , 2003 . E. B . Vinber g an d A . L . Onishchik , Seminar on Lie groups and algebraic groups, URSS , Moscow, 1995 . (Russian ) F. Warner , Foundations of differentiable manifolds and Lie groups. Springer-Verlag , 1983 . D. P . Zhelobenko , Compact Lie groups and their representations. Nauka , 1970 ; Englis h transl., Translatio n o f Mathematica l Monographs , vol . 40 . Amer . Math . Soc , Providence , RI, 1973 .

, Harmonic analysis on complex semisimple Lie groups. Nauka , Moscow , 1974 . (Rus -sian)

, Representations of reductive Lie algebras. Nauka , Moscow , 1992 . (Russian ) , Introduction to representation theory. Faktorial , Moscow , 2001 . (Russian ) , Classical groups. Spectral analysis of finite-dimensional representations. Russia n

Math. Survey s 1 7 (1962) , 27-120 . , Constructive modules and extremal projectors over Chevalley algebras. Funktsional .

Anal, i Prilozh. 2 7 (1993) , no . 3 , 5-14 , 95 ; English transl. , Funct . Anal . Appl . 2 7 (1993) , no . 3, 158-165 .

, On Weyl algebras over quantum groups. Teoret . Mat . Fiz . 11 8 (1999) , no . 2 , 190-204 ; English transl. , Theoret . Math . Phys . 11 8 (1999) , no . 2 , 152-163 . O. Zarisk i an d P . Samuel , Commutative algebra. I , II , Springer-Verlag , 1975 .

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Index

A-module Artinian, 9 3 diagonal, 5 7 free, 5 5 induced, 9 7 Noetherian, 9 4 primitive, 5 6 semisimple, 5 2 simple, 5 2

action adjoint, 35 2 transitive, 4 2

admissible VK-module , 8 8 algebra, 5

A(G), 270 , 28 3 Az(fl), 37 0 Do(C), 16 9 F[[x]], 8 0 Li(G), 16 5 M0(G), 16 8 5q(f l), 36 0 C/(fl), 11 5 tfg(fl), 35 9 associative, 5 Banach, 30 1 Calkin, 33 2 Clifford, 79 , 22 2 commutative, 5 contragredient, 37 8 Drinfeld-Jimbo, 36 0 exterior, 7 8 free, 10 3 Gaussian, 377 , 37 9 graded, 7 2 Grassmann, 7 8 normed, 30 1 of bounde d operators , 25 , 30 2 of Carta n type , 38 1 of continuou s functions , 30 1 of differentia l operators , 74 , 224 , 35 3 of polynomials , 18 1 of rationa l functions , 18 1 opposite, 5 1 quotient, 5 4

semisimple, 54 , 5 9 simple, 5 4 structure, 18 2 symmetric, 7 7 tensor, 38 , 7 5 unital, 5 universal enveloping , 11 5 Weyl, 86 , 35 5 Wiener, 30 1 with involution , 30 4 with unit , 34 7 Witt, 13 2

algebraic structure , 6 annihilator, 5 4 antipode, 35 0 approximate identity , 32 0

strict, 32 0 associativity axiom , 3 atlas, 18 7 average, 6 4

basis Cartan-Weyl, 23 1 Gelfand-Tsetlin, 29 7 orthonormal, 2 9

basis o f topology , 14 0 bialgebra, 34 9 bicommutant, 4 6 bilinear for m

canonical, 1 1 invariant, 10 9 nondegenerate, 1 1

bimodule, 5 2 Borel extension , 31 7 Borel measur e

invariant, 16 3 quasiinvariant, 16 3

Borel subalgebra , 23 2 Borel subgroup , 27 9 boundary, 13 9 bounded functional , 32 0 branching rules , 29 7

425

426 INDEX

Burnside identit y first, 6 7 second, 6 8

canonical filtration , 11 6 canonical projection , 1 3 Cartan decomposition , 23 8 Cartan lattice , 28 2 Cartan matrix , 39 8

irreducible, 40 0 reducible, 40 0 symmetrizable, 39 9

Cartan subalgebra , 111 , 22 7 Cartan subgroup , 26 7 Casimir element , 12 0 category, 7

monomial, 5 2 of groups , 7 of sets , 7 of vecto r spaces , 7 semisimple, 5 2

category O, 24 7 Cayley-Hamilton equation , 1 6 cell, 39 6 center, 5 4

of a Li e algebra , 10 1 central series , 10 2 centralizer, 26 7 character

central, 24 2 of a representation , 15 6 primitive, 6 8

Chevalley basis , 23 2 Chevalley generators , 23 2 Clebsch-Gordan rule , 29 9 closure, 13 9 coaction, 34 9 coalgebra, 34 8

cocommutative, 34 8 opposite, 34 8

coassociativity, 19 8 commutator, 52 , 102 , 17 5 commutator subgroup , 17 5 comodule, 34 9 composition series , 9 5 comultiplication, 198 , 34 8

coassociative, 34 8 connected component , 14 3 constant term , 8 0 convex hull , 14 6 convolution, 43 , 16 4

of measures , 16 8 coordinates, 2 8

exponential, 20 7 cotangent space , 18 9 counit, 198 , 34 8 countability axio m

first, 14 1

second, 14 1 Coxeter graph , 40 0

decomposition reduced , 39 5 derivation, 53 , 8 3

inner, 10 0 diffeomorphism, 18 8 differential, 18 9 dimension

Hilbert, 3 1 of a vecto r space , 9

direct sum , 1 2 orthogonal, 3 1 topological, 17 4

directed ordere d set , 14 0 distributivity, 4 Dynkin diagram , 40 0

eigenvalue, 1 4 eigenvector, 1 4 element

central, 10 1 even, 7 8 generalized nilpotent , 30 4 group-like, 12 7 homogeneous, 7 2 integral, 28 2 invertible, 3 normal, 30 5 odd, 7 8 positive, 31 7 regular, 27 6

equivalent seminorms , 2 6 extremal point , 328 , 41 0

field, 4 algebraically closed , 1 4 skew, 4

field o f fractions , 9 6 flag o f subspaces , 1 5

invariant, 1 5 free structure , 7 0 formal powe r series , 8 0

convergent, 8 5 Fourier transform , 34 1 function

Mobius, 12 6 modular, 16 3 positive definite , 15 5

functional calculus , 2 1 functor, 7 fundamental group , 21 0 fundamental sequence , 15 0 fundamental weight , 24 6

G-module admissible, 16 9 differentiable, 21 8 holomorphic, 21 8

INDEX 427

topological, 15 2 Garding subspace , 21 8 Gauss decomposition , 178 , 26 5

binary, 29 1 Gaussian binomia l coefficients , 35 9 Gelfand transform , 30 9 Gelfand-Naimark-Segal constructio n

(GNS), 32 4 genetics, 51 , 71 group, 3

abelian, 3 algebraic, 182 , 18 3 formal, 19 4 Gaussian, 26 8 general linear , 14 2 fundamental, 21 0 linearly reductive , 26 3 local, 19 3 nilpotent, 17 6 one-parameter, 15 4 orthogonal, 22 0 solvable, 17 6 spinor, 22 2 symplectic, 22 1 topological, 14 2 unimodular, 16 3 unitary, 142 , 22 1 universal covering , 21 3

group algebra , 64 , 16 9

Hermitian form , 2 6 homeomorphism, 14 0 homomorphism

diagonal, 11 9 Harish-Chandra, 25 2

homotopy, 20 9 homotopy group , 20 9 Hopf algebra , 35 0

dual, 35 2

ideal, 5 4 derived (i n a Li e algebra) , 10 2 left, 5 4 maximal, 96 , 30 7 of a Li e algebra , 10 0 right, 5 4 two-sided, 5 4

identity, 3 inequality

Bessel, 2 8 Cauchy-Bunyakovsky, 2 7 Holder, 16 4 Schwarz, 2 7

interior, 13 9 irreducible variety , 18 3

Jacobi identity , 54 , 9 9 Jacobson radical , 9 6 Jordan block , 1 8

Jordan norma l form , 1 9 Jordan-Holder series , 9 5

Killing form , 11 0 Kostant function , 24 2

LCS, 14 6 lemma

Dini, 15 8 Quillen, 4 6 Schur, 4 5 Urysohn, 14 4

Lie algebra , 9 9 sl(2), 10 3 associated wit h a forma l group , 19 5 associated wit h a Li e group , 19 5 free, 10 4 Kac-Moody, 13 6 linear, 13 0 nilpotent, 10 5 orthogonal, 13 0 quotient, 10 2 reductive, 11 4 semisimple, 10 6 simple, 11 4 solvable, 10 5 symplectic, 13 1 tangent, 19 5

Lie group , 19 2 local, 19 4 nilpotent, 26 3 reductive, 26 3 semisimple, 26 3 simple, 26 3 solvable, 26 3

local coordinates , 187 local isomorphism , 19 0

manifold, 18 7 /c-smooth, 18 8 analytic, 18 8

map /-smooth, 18 8 continuous, 14 0 covering, 21 1 exponential, 20 7 open, 14 0 regular, 19 1

matrix elements , 63 , 15 5 measure

cr-fmite, 15 8 Borel, 15 8

left-invariant, 15 9 right-invariant, 15 9 two-side invariant , 15 9

finite, 15 8 projection, 31 3 regular, 15 8 spectral, 31 3

428 INDEX

morphism, 7

neighborhood, 13 9 net, 14 0 nilradical, 10 6 normalizer, 111 , 26 7

objects, 7 one-parameter group , 15 4 operator

anti-Hermitian, 3 3 bounded, 2 5 compact, 3 3 finite-dimensional, 3 8 Fredholm, 3 4 Hermitian, 3 3 Hilbert-Schmidt, 41 4 linear, 5 multilinear, 5 of trac e class , 41 6 one-dimensional, 3 8 positive, 3 3 projection, 1 9 unitary, 3 3

Ore condition , 9 6

parity function , 7 8 Pfaffian, 29 4 Poincare series , 12 5 Poisson bracket , 13 2 Pontryagin dualit y theorem , 34 5 prebasis o f topology , 14 1 precompact set , 14 5 primitive exponents , 25 1 principal affin e variety , 28 5

quantum deformation , 37 2 quantum determinant , 37 3 quantum factorial , 35 9 quantum integer , 35 9 quantum Serr e algebra , 36 0 quantum Serr e conditions , 36 0 quotient manifold , 19 2 quotient module , 4 3 quotient space , 1 3

radical, 10 6 rank o f a roo t system , 39 1 reflection, 39 7 representation, 4 0

absolutely continuous , 15 3 adjoint, 10 0 coadjoint, 12 0 completely reducible , 4 4 Gelfand-Naimark, 32 4 irreducible, 4 4 of a C*-algebra , 32 3

nondegenerate, 32 3 of a Li e algebra , 10 0

regular, 6 5 topologically irreducible , 15 7 unitary, 15 5 weakly differentiable , 16 9

resolvent, 30 2 resolvent set , 30 2 ring, 4

Artinian, 9 3 commutative, 4 Jacobson-semisimple, 9 6 Noetherian, 9 4 of forma l exponents , 24 1 of fractions , 9 6 semisimple, 9 3 simple, 9 3 with identity , 4

root, 108 , 39 1 negative, 39 3 positive, 39 3 simple, 39 3

root subspace , 10 8 root system , 39 1

reduced, 39 1 root vector , 10 8

semidirect product , 164 , 21 7 semigroup, 3

commutative, 3 one-parameter, 15 4 with identity , 3

semigroup algebra , 4 3 seminorm, 2 3

countably additive , 40 3 series

Campbell-Hausdorff, 12 8 convergent, 8 5

Serre relations , 23 2 Shapovalov form , 25 3 similar matrices , 1 2 space

Banach, 2 4 complete, 15 0 double dual , 15 0 dual, 14 9 Euclidean, 2 6 Hilbert, 2 9 homogeneous, 4 2 locally Euclidean , 14 5 quasicomplete, 15 0 sequentially complete , 15 0 unitary, 2 6

spectral radius , 30 4 spectrum, 30 2

joint, 31 1 spinor modules , 29 3 state, 32 5

pure, 32 7 structure constants , 10 2

INDEX 429

subgroup, 4 Lie, 19 2 local, 20 3 one-parameter, 20 6 virtual, 20 8

submanifold, 19 1 submodule, 4 3 subring, 4 subset

bounded, 14 9 closed, 13 9 connected, 14 3 convex, 14 6 everywhere dense , 14 0 multiplicatively closed , 9 6 open, 13 9

subspace graded, 7 2 invariant, 1 3

substitution rule , 8 2 superalgebra, 7 8

commutative, 7 9 symmetric subalgebra , 30 6 system o f generators , 51 , 70

tangent map, 18 9 space, 18 8 vector, 18 8

Taylor formula , 2 1 tensor

antisymmetric, 7 5 symmetric, 7 5

tensor product , 35 , 3 6 theorem

Artin, 5 7 Baire, 40 3 Banach

on close d graph , 40 4 on invers e operator , 40 5

Banach-Steinhaus, 40 5 Burnside, 6 2 Calkin, 33 2 Campbell-Hausdorff, 12 7 Cartan, 113 , 20 5 Chevalley, 25 1 Engel, 107 Fitting, 10 8 Frobenius, 1 4

on reciprocity , 9 7 Gelfand-Mazur, 30 4 Gelfand-Naimark, 309 , 32 5 Godement, 17 9 Haar, 15 9 Hahn-Banach, 14 8 Hamel, 9 Harish-Chandra, 25 2 Hilbert, 34 , 9 5

Hilbert-Schmidt, 3 5 Jacobson, 6 2 Kadison, 33 4 Kaplansky, 33 3 Kolchin, 17 6 Krein-Milman, 41 0 Levi, 12 2 Levi-Maltsev, 12 3 Lie, 106 , 177 , 20 4 Maschke, 6 9 Peter-Weil, 17 0 Poincare-Birkhoff-Witt (PBW) , 11 7 Pontryagin, 34 5 Raikov, 34 1 Stone-Weierstrass, 40 6 Tikhonov, 14 4 von Neuman n o n bicommutant , 33 0 Wedderburn, 6 1 Wedderburn-Artin, 9 3 Weil, 16 2 Weyl, 274 , 27 9 Weyl o n semisimpl e representations , 12 1

topological module , 15 2 topological space , 13 9

compact, 14 4 connected, 14 3 Hausdorff, 13 9 linearly connected , 14 3 locally compact , 14 5 separable, 13 9 simply connected , 21 0

topological vecto r spac e (TVS) , 14 5 locally conve x (LCS) , 14 6

topology bounded convergence , 15 0 discrete, 14 0 quotient, 14 0 reflexive, 15 0 semireflexive, 15 0 simple convergence , 15 0 strong, 149 , 32 9 Tikhonov, 14 1 trivial, 14 0 uniform, 32 9 weak, 149 , 32 9 Zariski, 18 2

torus, 27 2 transition function , 187 TVS, 14 5

uniform norm , 2 4 unital norme d algebra , 30 1

vector /c-smooth, 21 8 analytic, 21 8 cyclic, 18 , 32 3 differentiable, 21 8 infinitely differen t iable, 21 8

430 INDEX

vector field, 132 , 18 9 left-invariant, 19 6 smooth, 13 2

vector space , 4 dual, 1 0 graded, 7 2

Verma module , 366 , 38 2 universal, 239 , 36 4 with highes t weigh t A , 23 9

Weyl chamber, 39 6 character formula , 24 8 group, 27 4

Yang-Baxter equation, 37 3 matrix, 37 3

Young product , 28 6

weight lattice , 24 4 zerodivisor, 9 6

Titles i n Thi s Serie s

228 D . Zhelobenko , Principa l structure s an d method s o f representatio n theory , 200 6

227 Takahir o Kawa i an d Yoshitsug u Takei , Algebrai c analysi s o f singula r perturbatio n

theory, 200 5

226 V . M . Manuilo v an d E . V . Troitsky , Hilber t C*-modules , 200 5

225 S . M . Natanzon , Modul i o f Rieman n surfaces , rea l algebrai c curves , an d thei r

superanaloges, 200 4

224 Ichir o Shigekawa , Stochasti c analysis , 200 4

223 Masatosh i Noumi , Painlev e equation s throug h symmetry , 200 4

222 G . G . Magaril-IPyae v an d V . M . Tikhomirov , Conve x analysis : Theor y an d

applications, 200 3

221 Katsue i Kenmotsu , Surface s wit h constan t mea n curvature , 200 3

220 I . M . Gelfand , S . G . Gindikin , an d M . I . Graev , Selecte d topic s i n integra l

geometry, 200 3

219 S . V . Kerov , Asymptoti c representatio n theor y o f th e symmetri c grou p an d it s

applications t o analysis , 200 3

218 Kenj i U e n o , Algebrai c geometr y 3 : Further stud y o f schemes , 200 3

217 Masak i Kashiwara , D-module s an d microloca l calculus , 200 3

216 G . V . Badalyan , Quasipowe r serie s an d quasianalyti c classe s o f functions , 200 2

215 Tatsu o Kimura , Introductio n t o prehomogeneou s vecto r spaces , 200 3

214 L . S . Grinblat , Algebra s o f set s an d combinatorics , 200 2

213 V . N . Sachko v an d V . E . Tarakanov , Combinatoric s o f nonnegativ e matrices , 200 2

212 A . V . Mel'nikov , S . N . Volkov , an d M . L . Nechaev , Mathematic s o f financial

obligations, 200 2

211 Take o Ohsawa , Analysi s o f severa l comple x variables , 200 2

210 Toshitak e Kohno , Conforma l fiel d theor y an d topology , 200 2

209 Yasumas a Nishiura , Far-from-equilibriu m dynamics , 200 2

208 Yuki o Matsumoto , A n introductio n t o Mors e theory , 200 2

207 Ken'ich i Ohshika , Discret e groups , 200 2

206 Yuj i Shimiz u an d Kenj i U e n o , Advance s i n modul i theory , 200 2

205 Seik i Nishikawa , Variationa l problem s i n geometry , 200 1

204 A . M . Vinogradov , Cohomologica l analysi s o f partia l differentia l equation s an d

Secondary Calculus , 200 1

203 T e Su n Ha n an d King o Kobayashi , Mathematic s o f informatio n an d coding , 200 2

202 V . P . Maslo v an d G . A . Omel'yanov ? Geometri c asymptotic s fo r nonlinea r PDE . I ,

2001

201 Shigeyuk i Morita , Geometr y o f differentia l forms , 200 1

200 V . V . Prasolo v an d V . M . Tikhomirov , Geometry , 200 1

199 Shigeyuk i Morita , Geometr y o f characteristi c classes , 200 1

198 V . A . Smirnov , Simplicia l an d opera d method s i n algebrai c topology , 200 1

197 Kenj i Ueno , Algebrai c geometr y 2 : Sheave s an d cohomology , 200 1

196 Yu . N . Lin'kov , Asymptoti c statistica l method s fo r stochasti c processes , 200 1

195 Minor u Wakimoto , Infinite-dimensiona l Li e algebras , 200 1

194 Valer y B . Nevzorov , Records : Mathematica l theory , 200 1

193 Toshi o Nishino , Functio n theor y i n severa l comple x variables , 200 1

192 Yu . P . Solovyo v an d E . V . Troitsky , C*-algebra s an d ellipti c operator s i n differentia l

topology, 200 1

191 Shun-ich i Amar i an d Hirosh i Nagaoka , Method s o f informatio n geometry , 200 0

TITLES I N THI S SERIE S

190 Alexande r N . Starkov , Dynamica l system s o n homogeneou s spaces , 200 0

189 Mitsur u Ikawa , Hyperboli c partia l differentia l equation s an d wav e phenomena , 200 0

188 V . V . Buldygi n an d Yu . V . Kozachenko , Metri c characterizatio n o f rando m variable s

and rando m processes , 200 0

187 A . V . Fursikov , Optima l contro l o f distribute d systems . Theor y an d applications , 200 0

186 Kazuy a Kato , Nobushig e Kurokawa , an d Takesh i Saito , Numbe r theor y 1 :

Fermat's dream , 200 0

185 Kenj i Ueno , Algebrai c Geometr y 1 : Fro m algebrai c varietie s t o schemes , 199 9

184 A . V . Mel'nikov , Financia l markets , 199 9

183 Haj im e Sato , Algebrai c topology : a n intuitiv e approach , 199 9

182 I . S . Krasil'shchi k an d A . M . Vinogradov , Editors , Symmetrie s an d conservatio n

laws fo r differentia l equation s o f mathematica l physics , 199 9

181 Ya . G . Berkovic h an d E . M . Zhmud' , Character s o f finit e groups . Par t 2 , 199 9

180 A . A . Milyut i n an d N . P . Osmolovskii , Calculu s o f variation s an d optima l control ,

1998

179 V . E . Voskresenski i , Algebrai c group s an d thei r birationa l invariants , 199 8

178 Mi t su o Morimoto , Analyti c functional s o n th e sphere , 199 8

177 Sator u Igari , Rea l analysis—wit h a n introductio n t o wavele t theory , 199 8

176 L . M . Lerma n an d Ya . L . Umanskiy , Four-dimensiona l integrabl e Hamiltonia n

systems wit h simpl e singula r point s (topologica l aspects) , 199 8

175 S . K . Godunov , Moder n aspect s o f linea r algebra , 199 8

174 Ya-Zh e Che n an d Lan-Chen g W u , Secon d orde r ellipti c equation s an d ellipti c

systems, 199 8

173 Yu . A . Davydov , M . A . Lifshits , an d N . V . Smorodina , Loca l propertie s o f

distributions o f stochasti c functionals , 199 8

172 Ya . G . Berkovic h an d E . M . Zhmud 7, Character s o f finite groups . Par t 1 , 199 8

171 E . M . Landis , Secon d orde r equation s o f ellipti c an d paraboli c type , 199 8

170 Vikto r Prasolo v an d Yur i Solovyev , Ellipti c function s an d ellipti c integrals , 199 7

169 S . K . Godunov , Ordinar y differentia l equation s wit h constan t coefficient , 199 7

168 Junjir o Noguchi , Introductio n t o comple x analysis , 199 8

167 Masay a Yamaguti , Masayosh i Hata , an d Ju n Kigami , Mathematic s o f fractals , 199 7

166 Kenj i U e n o , A n introductio n t o algebrai c geometry , 199 7

165 V . V . Ishkhanov , B . B . Lur'e , an d D . K . Faddeev , Th e embeddin g proble m i n

Galois theory , 199 7

164 E . I . Gordon , Nonstandar d method s i n commutativ e harmoni c analysis , 199 7

163 A . Ya . Dorogovtsev , D . S . Silvestrov , A . V . Skorokhod , an d M . I . Yadrenko ,

Probability theory : Collectio n o f problems , 199 7

162 M . V . Boldin , G . I . Simonova , an d Yu . N . Tyurin , Sign-base d method s i n linea r

statistical models , 199 7

161 Michae l Blank , Discretenes s an d continuit y i n problem s o f chaoti c dynamics , 199 7

160 V . G . Osmolovskii , Linea r an d nonlinea r perturbation s o f th e operato r div , 199 7

159 S . Ya . Khavinson , Bes t approximatio n b y linea r superposition s (approximat e

nomography), 199 7

158 Hidek i Omori , Infinite-dimensiona l Li e groups , 199 7

157 V . B . Kolmanovski i an d L . E . Shaikhet , Contro l o f system s wit h aftereffect , 199 6

For a complet e lis t o f t i t le s i n thi s series , visi t t h e AMS Bookstor e a t w w w . a m s . o r g / b o o k s t o r e / .