Other Title s in This Series

148 Vladimi r I. Piterbarg, Asymptotic methods i n the theory of Gaussia n processe s an d fields, 199 5 147 S . G . Gindiki n and L. R . Volevich , Mixe d problem s for partia l differentia l equation s with quasihomogeneous principal part, 199 5 146 L . Ya. Adrianora , Introduction to linea r systems of differential equations , 199 5 145 A . N. Andriano v an d V, G. Zhuravlev , Modular forms an d Hecke operators, 199 5 144 O . V . Troshkin, Nontraditional methods in mathematical hydrodynamics , 199 5 143 V . A. Malyshe v an d R. A. Minlos, Linea r infinite-particle operators , 199 5 142 N . V . Krylov, Introduction to th e theory of diffusio n processes , 199 5 141 A . A. Davydov , Qualitative theory of control systems , 199 4 140 Aizfl c I. Volpert , Vitaly A. Volpert , an d Vladimir A. Volpert , Traveling wave solutions of paraboli c systems, 199 4 139 I . V . Skrypnlk, Methods for analysis of nonlinear elliptic boundary value problems, 199 4 138 Yu . E Razmyslov , Identities of algebra s and thei r representations, 199 4 137 F . I. Karpelevich an d A. Ya. Kreinin , Heavy traffi c limits for multiphase queues, 199 4 136 Masayosh i Miyanishi, Algebraic geometry, 199 4 135 Masar u Takeuchi, Modern spherical functions , 199 4 134 V . V. Prasolov, Problems and theorem s in linear algebra, 199 4 133 P . I. Naumki n an d I. A. Shishmarev, Nonlinear nonlocal equations in the theory of waves, 199 4 132 Hajim e Urakawa, Calculus of variation s and harmonic maps, 199 3 131 V . V. Sharko, Functions on manifolds: Algebrai c and topological aspects , 199 3 130 V . V. Vershinin, Cobordisms and spectral sequences, 199 3 129 Mitsu o Morimoto, An introductio n to Sato's hyperfunctions, 199 3 125 V . P. Orevkov, Complexity of proofs and their transformations in axiomatic theories , 199 3 127 F . L. Zak, Tangents and secants of algebrai c varieties, 199 3 126 M . L. Agranovsku , Invariant functio n space s on homogeneou s manifolds of Li e group s and applications, 199 3 125 Masayosh i Nagata, Theory of commutative fields, 199 3 124 Masahis a Adachi, Embeddings and immersions, 199 3 123 M . A. Akivis and B. A. Rosenfeld , felie Cartan (1869-1951) , 199 3 122 Zhan g Guan-Hoa, Theory of entire and meromorphic functions: Deficient an d asymptotic values and singular directions, 199 3 121 I . B. Fesenk o and S. V . Vostokov, Loca l fields and their extensions: A constructiv e approach , 199 3 120 Takeyuk i Hida an d Masuyuki Hitsoda, Gaussian processes, 199 3 119 M . V . Karasev and V . P. Maslov, Nonlinear Poisson brackets. Geometry and quantization, 199 3 118 Kenkkh i Iwasawa , Algebraic functions, 199 3 117 Bori s Zilber, Uncountably categorical theories , 199 3 116 G . M. Fel'dman , Arithmetic of probabOit y distributions, and characterization problems on abelia n groups, 199 3 115 Nikola i V . IV&DOY , Subgroups of Teichmuller modular groups, 199 2 114 Seiz d ltd, Diffusion equations , 199 2 113 Michai l Zhitomirskil , Typica l singularitie s of differentia l i-form s an d Pfaffia n equations , 199 2 112 S . A. Lomov , Introduction to th e general theory of singular perturbations, 199 2 111 Simo n Gindikin , Tube domains and the Cauchy problem, 199 2 110 B . V. Shabat, Introductio n to complex analysi s Part II. Functions of severa l variables, 199 2 109 Isa o Miyadera, Nonlinear semigroups, 199 2 108 Take o Yokonmna, Tensor spaces and exterior algebra, 199 2 107 B . M. Makarov , M. G . Goluzina , A. A. Lodkin, an d A. N. Podkorytov , Selecte d problem s in rea l analysis, 199 2 (Continued in the back of this publication) This page intentionally left blank 10.1090/mmono/040

TRANSLATIONS OF MATHEMATICAL MONOGRAPHS

VOLUME 40

D. P. Zelobenko

Compact Lie Groups and Their Representations

||f^l]| American Mathematical Society KOMF1AKTHME TPynribl JI M H M X nPE^CTABJIEHHfl

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M3joiaTejibCTBO ^HayKa ^ nnaBHaa peaaKtm a <|)H3HKO-MaTeMaTHHecKOH JiMTepaTyp w MocKBa 197 0

Translated fro m th e Russia n b y Israel Progra m fo r Scientifi c Translation s

2000 Mathematics Subject Classification. Primar y 22-02 ; Secondary 17-02 .

Library o f Congres s Cataloging-in-PubHcatio n Dat a Zhelobenko, Dmitri ! Petrovich. Compact Li e groups an d thei r representations . (Translations of mathematica l monographs , v . 40) Translation o f Kompaktny e grupp y L i i ikh predstavleniia . Bibliography: p . 1. Li e groups. 2 . Representation s o f groups . I . Title. II . Series. QA387.Z443I3 5I2'.5 5 73-1718 5 ISBN 0-8218-1590- 3 ISSN 0065-928 2

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© Copyrigh t 197 3 b y th e America n Mathematica l Society . A H rights reserved . Second printin g with corrections , 197 8 The America n Mathematica l Societ y retain s al l rights except thos e grante d t o th e Unite d State s Government . Printed i n th e Unite d State s o f America. @ Th e pape r use d i n thi s boo k i s acid-fre e an d fall s withi n th e guideline s established t o ensur e permanenc e an d durability . Visit the AMS home page at URL : http://www.aias.org/ 12 1 1 1 0 9 8 7 6 5 0 6 0 5 04 03 02 0 1 TABLE O F CONTENT S

PREFACE 1

PART I. INTRODUCTION 5 CHAPTER I . TOPOLOGICA L GROUPS. LIE GROUPS 6 1. Definitio n o f a group 6 2. Topologica l group s 8 3. Parametri c groups and Li e groups 1 2 4. Li e theory 1 4 5. Locall y isomorphic Li e groups 1 8 6. Invarian t forms on Li e groups 2 1 7. Metrics . Haa r measure 2 3

CHAPTER II . LINEA R GROUPS 2 6 8. Genera l linear group. Exponential 2 6 9. Genera l linea r group. Fundamental decomposition s 2 7 10. Linea r groups associated wit h second-order form s 3 0 11. Quaternion s 3 3 12. Simpl y connected group s 3 6 13. Complexificatio n 3 9 14. Transformation s i n the class of tensors 4 0

CHAPTER III . FUNDAMENTA L PROBLEMS OF REPRESENTATIO N THEORY 4 4 15. Function s o n a homogeneous spac e 4 4 16. Terminolog y o f representatio n theory 4 6 17. Reductio n o f the fundamental proble m 5 0 18. Elementar y harmonic s 5 1 19. Algebra s and groups of an equation 5 4 20. Schur' s Lemm a 5 6 21. Burnside' s Theorem 5 9 22. Grou p algebras and thei r representations 6 2 23. Statemen t o f the fundamental problem s 6 3

v VI TABLE O F CONTENT S

PART II. ELEMENTARY THEOR Y 6 7 CHAPTER IV . COMPAC T LIE GROUPS. GLOBA L THEORE M 6 8 24. Definitio n o f compact groups 6 8 25. Statemen t of the global theore m 7 0 26. The averaging technique 7 1 27. Orthogonalit y propert y 7 3 28. Approximatio n lemma for a linea r group G 7 4 29. Fourie r series on a linear group G 7 4 30. Completio n of the proof for a linear group G 7 6 3L Completio n of the proof in the general case 7 8 32. Harmoni c analysis on a homogeneous manifold 7 9 33. Character s 8 1 34. Representatio n theory of finite groups 8 2 35. Universalit y of the group U(n) 8 4

CHAPTER V . TH E INFINITESIMA L METHO D I N REPRESENTATIO N THEOR Y 8 7 36. Differentia l o f a representation 8 7 37. Irreducibl e representations of the group SU(2) 9 0 38. Matri x elements of the group SU(2) 9 6 39. Som e groups related to SU(2) 10 1 40. Problem s arising in applications of the infinitesimal metho d 10 4

CHAPTER VI. ANALYTI C CONTINUATIO N 10 9 41. Genera l principle of analytic continuation 10 9 42. Hypercompac t Li e groups. Weyl's "unitarian trick" 11 3 43. Bicomple x Li e groups and Lie algebras 11 5 44. Complexificatio n o f U(n). Weights and roots 11 8 45. Mode l of irreducible representations of SU(3) 12 1

CHAPTER VII. IRREDUCIBL E REPRESENTATION S O F THE GROUP U(/I) 12 6 46. Existenc e of highest weight 12 6 47. Uniquenes s of the highest-weight vecto r 12 8 48. Differen t model s of d(a) 13 0 49. Inductiv e weights 13 2 50. Youn g product 13 4

CHAPTER VIII. TENSOR S AND YOUN G DIAGRAM S 13 7 51. Descriptio n of Z-invariants 13 7 52. Youn g diagrams 14 0 53. Youn g symmetrizers 14 2 54. Characterizatio n of irreducible tensors in terms of symmetry 14 4 55. Dualit y principle 14 8 TABLE O F CONTENT S vn

56. Realizatio n o f d(a) o n rectangula r matrices 15 0 57. Harmoni c oscillato r 15 2

CHAPTER IX . CASIMI R OPERATOR S 15 6 58. Universa l envelopin g algebras 15 6 59. Casimi r operators for the group GL(w ) 15 9 60. Eigenvalue s of the operators C k 16 1 61. Separatio n o f the points of the spectrum and algebraic proof of complete reducibilit y 16 5 62. Complet e descriptio n of th e center for GL(n) 16 7 63. Th e cycle rul e 17 0

CHAPTER X . INDICATO R SYSTEM S AND TH E GEL'FAND-CETUN BASI S 17 6 64. Left-translatio n operator s on the group Z 17 6 65. Indicato r systems 18 0 66. The algebra of Z-multipliers; restrictio n to a subgroup 18 3 67. The Gel'fand-Cetlin basi s 18 7 68. Lowerin g operators i n infinitesimal for m 19 0 69. Normalizatio n o f the basis vectors 19 8 70. Differentia l o f d(a) 20 2 71. Matri x elements of d(a) 20 5

CHAPTER X L CHARACTER S 21 0 72. Invarian t measur e on the group U(n ) 21 0 73. Primitiv e characters of U(/i) 21 2 74. Weigh t diagra m of d(a) 21 4 75. WeyF s second formul a 21 9 76. Concludin g remar k 22 2

CHAPTER XI L TENSO R PRODUC T O F TWO IRREDUCIBL E REPRESENTATIONS O F V(n) 22 4 77. Metho d o f characters 22 4 78. Th e method of Z-invariants 22 7 79. Specia l cases 23 0 80. Wey l determinants 23 4

PART III. GENERAL THEOR Y 23 7 CHAPTER XIII. BASI C TYPES O F LIE ALGEBRA S AND LI E GROUPS 23 8 81. Adjoin t representatio n o f a Li e algebra 23 8 82. Idea l an d norma l diviso r 23 9 83. Basi c types of Li e algebras 24 1 84. Solvabl e Li e algebras 24 3 85. Nilpoten t Li e algebras 24 6 Vlll TABLE O F CONTENT S

86. Fittin g decompositio n 248 87. Th e Killing-Cartan bilinea r for m 252 88. Basi c types of Li e groups 254 89. Th e Levi-Mal'cev Theore m 256

CHAPTER XIV . CLASSIFICATIO N O F COMPACT AND REDUCTIV E LI E ALGEBRAS 259 90. Compac t Li e algebras 259 91. Carta n subalgebras 261 92. Cartan-Wey l basi s 264 93. Simpl e roots 266 94. Carta n structure matrix 268 95. Simpl e complex Li e algebras 271 96. Rea l forms of semisimple complex Li e algebra s 275 97. Completio n of the classificatio n 277

CHAPTER XV. COMPAC T LI E GROUPS IN TH E LARG E 280 98. Invarian t polynomials 280 99. Algebrai c groups 282 100. Gaus s decomposition 284 101. Iwasaw a decompositio n 288 102. Maxima l toroid s 291 103. Fundamenta l group and center 294 104. Ever y semisimple complex Li e group is linear 297 105. The Weyl group 299 106. Existenc e of complexificatio n 303 107. Additiona l result s 306

CHAPTER XV L DESCRIPTIO N O F IRREDUCIBLE FINITE-DIMENSIONA L REPRESENTATIONS 311 108. Fundamenta l theore m 311 109. Highes t weights and signatures 314 110. Normall y embedded subgroup s 316 111. Polynomial s on the group Z 318 112. Fina l classificatio n 321 113. Symplecti c group 324 114. Orthogona l group 330 115. Theor y of spinors 335 116. Rea l forms 339 117. Arbitrar y connected Li e groups 341 118. Remark s 343 TABLE O F CONTENT S IX

CHAPTER XVII . INFINITESIMA L THEOR Y (CHARACTERS , WEIGHTS , CASIMIR OPERATORS ) 34 8 119. Cartan-Wey l decompositio n i n the unversal enveloping algebra 34 8 120. Representation s wit h highest-weight vecto r 35 0 121. Classificatio n o f finite-dimensional irreducible representations o f the algebra X 35 2 122. FreudenthaF s formula SS S 123. WeyF s character formula 35 9 124. Consequence s of Weyl's formula 36 4 125. Polynomial s on th e Cartan subalgebra whic h are invariant under the Weyl group 36 5 126. Casimi r operators 36 8 127. Computatio n o f the eigenvalues of Casimir operators 37 1

CHAPTER XVIII . SOM E PROBLEM S O F SPECTRAL ANALYSI S FO R FINITE - DIMENSIONAL REPRESENTATION S 37 5 128. Genera l schem e of restrictio n fro m grou p to subgroup 37 5 129. Restrictio n SO(/x)/SO(/i-l) 37 7 130. Restrictio n Sp(/?)/Sp(/f- 2) 38 0 131. Tenso r product of two irreducible representations 38 3 132. Restrictio n SU( w + n)/SU(m) x SU(*) an d SU(m/i)/SU(m) x $U(n) 38 5 133. Restrictio n SU(w)/SO(/i ) 38 7 134. Spherica l harmonic s i n euclidean w-spac e 39 0 135. Representation s of th e group of motions in euclidean it-space 39 4

APPENDIX I . O N INFINITE-DIMENSIONA L REPRESENTATION S O F SEMISIMPLE COMPLEX LI E GROUP S 39 8 1. Elementar y representation s 39 8 2. Th e spac e of an elementary representatio n 40 0 3. Th e differential o f an elementary representatio n 40 1 4. Questio n of irreducibilit y 40 2 5. Analo g of the Planchere l formula 40 4 6. Paley-Wiene r theorem s 40 5 7. Minima l representation s 40 6 8. Classificatio n o f irreducible representation s 40 7 9. O n semireducible representation s 40 8

APPENDIX II . ELEMENT S O F TH E GENERAL THEOR Y O F UNITAR Y REPRESENTATIONS O F LOCALL Y COMPAC T GROUPS 40 9 1. Commutativ e groups 40 9 2. Stone-vo n Neuman n theore m 41 1 3. Induce d representation s 41 3 X TABLE O F CONTENT S

4. Semidirec t products 41 5 5. Nilpoten t Li e groups 41 8 6. Decompositio n of unitary representations into irreducible representations 42 0

APPENDIX HI . UNITAR Y SYMMETR Y I N TH E CLASS OF ELEMENTARY 42 2 PARTICLES 1. Invarianc e and conservation laws 42 2 2. Elementar y particles. Isotopic spin 42 4 3. Unitar y symmetry in the class of hadrons 42 7 4. Th e discovery of the ^-particle 43 0 5. Problem s 43 1

REFERENCES 433 SUBJECT INDEX 444 APPENDIX I

ON INFINITE-DIMENSIONA L REPRESENTATION S O F

SEMISIMPLE COMPLE X LI E GROUP S

In 1943, Gel'fand and Raikov [102] showed that every locally compact group (with Haar measure) possesses a fairly "rich" supply of irreducible unitary representations in Hilbert spaces. If the group G is not compact, these representations are generally infinite dimensional. Th e article [102] is essentially th e beginning of th e theor y of infinite - dimensional representations. Since the fifties, the theory of nonunitary representations has also seen progress.1 > This appendix contains a brief survey of the theory for semisim- ple complex connected Lie groups. Also considered are some problems of harmonic analysis of functions on (7.

§ 1 . Elementar y representations Let G be a semisimple connected complex Lie group, and X its Lie algebra. A function/(g) on G is said to be differentiate i f it is in the domain of the infinitesi- mal left- and right-translation operator s on G. Let E be the space of all infinitel y differentiable function s o n (? . The right-translation operator s Rgf(x) = f(xg\ JC, g 6 G, constitut e a representatio n o f the group G in the space E. Thi s rep- resentation is of course reducible. In particular, a s we have see n throughou t thi s book, every irreducible finite-dimensional representation of the group G is contain- ed in an invariant subspace of E. Following the analogy with this construction, let us try to use the representation Eg to construct all irreducible representations of G. Let Z_, Dand Z+ be the components of the Gauss decomposition i n the group G. The maximal solvabl e subgrou p B = Z-D i s also sometime s calle d a Borel subgroup of G.® Le t a b e an arbitrary character (one-dimensional representation ) of the group 2? , and Ea the set of all functions i n E such that f(bx) = a(b)f(x% b € S, x eG. W e topologiz e E by unifor m convergenc e o f function s an d all their derivatives on every compact subse t o f G. Then E a is a closed subspac e of E. Ea is obviously invariant under Rg. *> For more details on this subject, see for example, [121] or [134]. 2> As we know from the main text of the book, the group B is unique up to an inner automor- phism of G.

398 1. ELEMENTAR Y REPRESENTATION S 399

DEFINITION. The restriction of R g to Ea is called the elementary representation of the group G with signature a. We denote the elementary representatio n qf G with signature a b y e(a). Le t us find th e parameter s tha t determin e thi s representation . First , a(z) = 1 , zeZ- (since Z- i s the derived subgrou p of B). Consequentl y to determine a{b) w e need 7 only compute a{S) y 5 e D. Further , a(5) « a(e)a(f) fo r d = ey 9 eeE9 re/ , where r i s a maximal toroid and E a simply connected subgroup (isomorphic to a vector space). If r is the rank of the group <7 , then dim E = di m /*= r. The character a(f) is defined b y r integers , th e character a(e) b y r arbitrary complex numbers . Con- sequently ever y elementary representatio n of G is defined b y a set of 2r numbers: r integers and r complex numbers. We now consider more closely the case in which the group G is simply connected. Let 5, i « l , .-,r, b e multiplicative coordinate s i n the group D (§ 112) . Then the character a(5) ma y also be expressed as

a (5)= 115?$*, 1=1 where Xi and /*, are arbitrary comple x number s whose difference i s an integer. I t is readil y seen that thi s condition is necessary and sufficien t fo r the function a(6) to be single valued on the group D. Recall that dt- = ex p v, wher e /'are coordinates in a Carta n algebr a H relativ e t o the basi s e , « 2a>,7(t» „ &d, i » l»— > r. Conse- quently we also have

log a (8) = (X, x) + (p, x), d = ex p x, where A « Xtf an d ^ « ^^ wit h e th e basi s dua l t o ev , an d th e ba r denote s complex conjugation of the coordinates t\ Thu s the character a(S) is defined by a pair of vectors X, p. e H whose difference is integral relative to the basis e*.z) We find it convenien t t o modif y th e above parameter s slightly . Se t X = p — d and ft — q -~ dy wher e rf is hal f th e su m o f th e positiv e root s i n th e algebr a H. This is equivalent t o th e substitutio n Xi = p t• — 1, /*, = q t; — 1 for all * * = I,--, r. The pair of vector s a ~ (/> , q\ p,qeH, whos e difference i s integral, will be called a signature. Thus e(a) = e(/> , r^/7, « e u,

3> I n the genera] case (if G is not simply connected), the last condition must be replaced by the condition that the difference A—/ / be a weight of the group /\ 400 APPENDI X I . INFINITE-DIMENSIONA L REPRESENTATION S

defines a closed subspace of 3). Sinc e th e character a{f)y a = (/? , q\ depend s onl y on th e integra l differenc e v = p — q, w e denote this subspace b y 2) w and call th e vector \> the index of the signature a. Th e representation e(a) ca n be realized in the space 3)v by the familiar formul a

Tg

where the elements e' and ug are defined by the Iwasawa decomposition ug = e'^Ug, C € Z_. Note that the character

§ 2. Th e space of an elementary representatio n The space 3) is a topological vector space with respect to the topology of uniform convergence o f functions an d all thei r derivative s on U. I t is readily seen that 3 ) is a complete metrizabl e space , i.e. a Freche t space. Mor e profoun d informatio n on the topological nature of 3) will be obtained if we decompose the representation e{a) int o multiplet s relative to the compact subgrou p U. Note that the restrction of e(a) to U reduces to right translations R*

k(X) = W + c(A )

(summation over i from 1 t o r), where X, and X1 are dual coordinates in the algebra H and c(X) is a linear form in X (see § 126) . Thus ifc(A)^C|m| fo r sufficientl y larg e X. Since the function

XX{k(X)Y\c>Xo>(X)(X) the squared norm of the matrix element. Hence, as in classical Fourier analysis, it follows tha t th e Fourie r serie s o f p(i/ ) converge s t o

\4\£Au(k(X))»9 n = 0,l,....

Hence i t follows tha t the space 3) is isomorphic to the Kothe space of rapidly de- creasing sequences [132] . Using the estimates in [132], one easily shows that 3) is a Montel and nuclear space. As we know, the multiplet dx of the group U with highest weight X is contained in the space S > with multiplicity nx = di m dx. It is not hard to see that the number of multiplet s correspondin g t o th e subspac e 2) v is nx(v\ wher e nx(v) is th e mul - tiplicity of the weight y in the representation dx. This determines the multiplicity of occurrence of dx in the representation e(a).

In particular, th e minima l weigh t X for whic h dx is contained in S) v is Xo = \v\> where |JC | is a dominant vector in the orbit of the vector J C relative to the Weyl group (see § 105). Th e correspondin g multiplicit y i s n^(v ) = 1 , i.e. dx, is contained in

3)y once.

% 3. Th e differentia] of an elementary representation As follow s fro m th e definitio n o f a n elementar y representatio n (§1) , th e infinitesimal operator s of the group G are applicable to any vector in the space of the representation e(a). Let e(a , JC ) be the infinitesimal operato r of the representa- tion e(a) correspondin g t o the element J C e X. Settin g e(a, xy)=e(cr , x)e(a, y), w e extend the differential e(a, x) to a representation of the universal enveloping algebra X. The elements e(a, JC) , X G X, are polynomials in the infinitesimal operators of the group G.

Since differentiation o f th e operator T 8 ( § 2) involves differentiation o f th e ex- ponential a(e) = ex p (p — 2d, /), s = ex p / , th e infinitesima l operator s e(a 9 JC), xeXy depen d linearly on the exponent p of the signature a. Consequentl y the oper- ators e(a, JC) , J C € 3£ , are also polynomials in the vector p. Let j£ ? be the linear span (the set of all finite linear combinations) of the matrix elements of U. Le t j£ f v be the intersection of jS f and 3)„ . We shall call the space if v the fundamental lineal in th e spac e 3) v. W e clai m tha t th e fundamenta l linea l i s invariant under the differential e(a 9 JC) , J C e X. TO this end, it will suffice to consider only elements xe l x Indeed, let V be the linear span of all matrix elements in &v wit h fixed highest x x weight X. Let #be th e Lie algebra of the subgroup U. If J C e K 9 then e{ay x)V c z V by the definition of VK On the other hand, consider the bilinear function/(JC, £) = e(a, JC)£ , J C e X, f e V x, as an element of the tensor product X 0 VK Th e formula e(a, z)f(x, f ) = /([z, JC] , £) + /(JC, e(a9 z) f), z eK, show s that/(jc, £) transforms the algebra K accordin g t o th e la w o f th e tenso r produc t % ® dx, wher e 7 r i s th e representation o fA T in A " generated b y th e adjoin t representation . Thu s 402 APPENDI X I. INFINITE-DIMENSIONA L REPRESENTATION S

x (a, x) V c: 2 V* 9 where X' ranges over a finite family of irreducible components of the tensor product x 0 dx. In defining the family of elementary representations e(a\ w e saw that, formall y speaking, this family i s obtained b y "analytic continuation" (wit h respec t to the signature) fro m th e famil y o f irreducibl e finite-dimensional representations d Xtt of the group G. We can now give this statement a more rigorous meaning. Consider- ing the indicator systems of the representations dxp> on e readily shows (see [166]) that th e spac e Vx? o f thi s representatio n i s define d b y the followin g syste m o f equations: jritip=*0, X$t x

where X ±i ar e the infinitesimal left-translatio n operator s on U corresponding t o the root vectors e«, w « ±

&»= E V Xfi

f Fix two arbitrary matri x elements e, e e J?v and conside r the function /(a)= 2 {e{ay x)e9 e')9 where (p, X— fi~v. I n this sense the differential o f e(a) is an analytic continuation of the family of differentials dx^ We mention a simple but important corollary, relating to the Casimir operators e(a,z), 2 € 3» where 3 i s the center of the algebra 3£ . I f e and e' are two matrix elements in &„ the n (e(a, z)e, e') — x(a)5(e, e'\ wher e $(

§4. Questio n of irreducibility The irreducibility of the elementary representations has been investigated by many authors, beginning with the pioneering work of Gel'fand an d Naimark [100], who studied unitary representations of the "fundamental series " for the classical groups. A definitive solution to the problem has been obtained by the present author [166]. Since the proof involves a complicated polynomial technique, we shall only state the results. Let us say that a signature a — (p>q) has a degeneracy in the direction of the root co if the numbers pw ~ 2(/> , , o>) and qm = 2 (qy a>)/(a>, o> ) are nonzero integers with the same sign (i.e. pm q„ > 0). We shall say that the signature a i s degenerate 4. IRREDUCIBILIT Y 403

if it has a degeneracy in the direction of at least one root. The result of our irreduci- bility study of e(a) is summarized in the following theorem: An elementary representation e(a) is topologically irreducible if and only if its signature a is nondegenerate. It also turns out that topological irreducibility of e(a) is equivalent to algebraic irreducibility o f the differentia l e(a 9 JC), X e X, i n th e linea l jSf ^ I n addition , a n analog of Burnside's Theorem i s valid for irreducibl e representations e(a): Every continuous linear operator in the space 3D V can be weakly approximated by linear combinations of operators Tg = e(a, g)9 geG. An y representation with this property is said to be completely irreducible [166J. Consequently e(a) is completely irreducible if and only if its signature a i s nondegenerate. Let W be the Weyl group of the algebra X. W e define the action o f the Weyl group on the signature a — (p, q) by or, = (sp 9 sq\ s e W. It can be shown that if the signature a is nondegenerate4* the representations e(a) and e(a,)are equivalent. In th e genera l cas e there ar e certai n relation s o f "partia l equivalence " betwee n e(a) and e(a s). As an example, consider the Lorentz group G » SL(2 , C). Here the signature is a = (/> , q)y where p and q are complex numbers such that v « p — q is an integer. The realization of e(a) on the group Z is

Tgf(z) -Q3 z + ay-* 0FTd^f(~^} Here z i s a complex number. The class of functions/"(z) i s described in the book [19]. The representation e{a) is irreducible if and only if the numbers p and q are not nonzero integers of the same sign. When this condition holds, the representa- tions e(a) and e(—a) are equivalent. For brevity' s sake, let us call a degenerate signature a an integral point. Call a point a positive {negative) i f p,q > 0 (p,q < 0). I f a i s a positive integra l point , then e(a) contains th e finite-dimensional irreducible representation d(a) i n an in- variant subspace. It then turns out that the operator D « (d/dzy i s an intertwining operator for "partia l equivalence" of e(a) and e(jS) , /3 = (—p , q): De(a,g) = e(fra)D. The operato r D annihilate s a finite-dimensional subspace o f th e representatio n d(a) an d i s an intertwinin g operator fo r equivalenc e of the factor representatio n e(a)ld(a) an d the representation e(/3) . Similarly, the operator 3 = (3/3z)« demon- strates "partial equivalence" of e(j}) and e(—a), and it can be shown that e(fi) acts in an invariant subspace of e(—ct), while e(—a)le(ji) is equivalent to d(a). Equivalence relation s o f thi s typ e wer e first discovered b y the presen t autho r [1G3]. See also [19] and [159].

4> I n this case the signature at i s also nondegenerate. 404 APPENDIX I . INFINITE-DIMENSIONA L REPRESENTATION S

§ 5. Analo g of the Plancherel formula By analogy with the Peter-Weyl theory, it is natural to conjecture that the ele- mentary representations e(a) play the role of "elementary harmonics" in harmonic analysis o f function s o n G. The fundamenta l resul t i n thi s directio n i s du e t o Gel'fand an d Naimar k [100] , wh o constructe d a n analo g o f th e ZMheor y o f Fourier transforms for the classical groups (?.5) These results were then generalized to arbitrary semisimple complex connected Lie groups by Harish-Chandra [110]. We first observe [166] that the operators Tg = e(a 9g) ar e bounded in the norm of the Hilbert space JP « L?Q1). Hence the representation e(a) can also be extended to a representatio n in the Hilber t space JP„ defined a s the completion of 2> v in the metric of Mf. Further , given a signature a — (p>q)9 set a* = (—q,—p). It can be shown [166] that any pair of functions

faff, g) 9* *(<**. g) ) - («> > <1>) where ($,) is the inner product in jfv(nott tha t the signature a* has the same index as a). In particular, ifp~/> + ?isa purel y imaginary number, then a* = a. I n this case the operators e(ayg) are unitary. Let A0 be the set of all such signatures. The corresponding family of unitar y representation s e(a) is known as ihe funda- mental series. Investigatio n of th e fundamental serie s was the first step toward a theory of infinite-dimensional representations of (?. We now consider functions on the group G. If a function x(g) is locally integrable with respect to Haa r measure dg and fall s sufficiently rapidl y to zer o at infinit y (e.g., it has compact support), then one can define the operator integral

*(ff)-JxfrMff.*)*. 0 )

Here X{a\ lik e e(a9 g)f i s a linear operator in the Hilbert space jf*. Th e operator valued function X(a) will be called the Fourier transform of the function x(g). If the function x(g) has compact support and is infinitely differentiate, the n reasoning analogous to that in § 2 shows that the matrix elements of the operator X(a) are rapidly decreasing in the Peter-Weyl basis. The rate of decrease is so rapid that one can meaningfully defin e th e trace trX(a) a s the sum of the absolutely convergent series whose terms are the diagonal matri x elements. It turns out that there is a formula inverse to (I): x00 - J*t r {X(a)e(a, g)*} dfi{a\ (2 ) where the star denotes Hermitian conjugation and d[i{a) is a measure on the set of signatures A& know n as Plancherel measure. Thu s the invers e formul a involve s only representations e(a) of the fundamental series. To describe the measure djx{a\

5> A simpler version of the theory was presented by Gel'fand and Graev {9Q. 6. PALEY-WIENE R THEOREM S 405

we set a = (p,$) « Integration with respect to dji(a) is performed b y summing over v and integrating with respect to pure imaginary values of p with density

o («, o>)2 where a > is an arbitrary positive root and d half the sum of all positive roots (the constant c 0 is independent of a). In particular, consider formula (2) for g = e> and replace the function x(g) by a convolution of type $x(a)y(ag)da. It is readily seen that this substitution transforms X(a) int o X(a)Y(a)* 9 where F(a ) i s th e Fourie r transfor m o f th e function^) - Hence

Jx(g)J&)dg - J^t r {X(a)Y(aT} dtfa). In particular , settin g x(g) = y(g) we get an expressio n fo r th e square d nor m o f x(g) in the space LHG), as th e integra l of the trace tr X(a)X(a)* over A0 with re- spect to the measure djAcc). This formula is meaningful eve n for arbitrary function s in IHG); i t i s the analog of the classical Plancherel formula in harmonic analysis on the straight line. Note that, b y the results of § 4, the elementary representations in the fundamen- tal series are irreducible.6) Formul a (2) may be interpreted as a decomposition of the regular representation i n LHG) into irreducible unitary representations.

§6. Analogs of Paley-Wiener theorems Let N be the space of all infinitel y differentiabl e compact-supporte d function s x(g) on the group G. A particularly interesting problem is to describe the set 31 of Fourier transforms X(a) of the functions x(g) e N. Note that Wis an algebra with respect to convolution on G. The Fourier transform o f a convolution of function s on G is the product o f their transform s X(a). Thu s the set 9 ? is an algebr a wit h respect to multiplication of the functions X(a). For any linear operator A in the space &» w e set \A |j = {trAA*} 1/2. The opera- tor A is known as a Hilbert-Schmidt operator if \A\ < oo . In particular, all oper- ators X(a) € 9i are Hilbert-Schmidt operators. Moreover, X(a) is a weakly analytic operator value d function , satisfyin g estimate s o f the typ e

\\fA*X (a) J<|| £ C(« , K /) e»»**», «,*, / = 0,1, 2, .... (1 ) where p is the exponent of the signature a and A the Laplace-Beltrami operator on the group U. In addition , X(a) 9 like e{ag)9 satisfie s the "partial equivalence " relations mentioned i n § 4. If r is the rank o f the group G, there exist 2r operators

$ > Fo r the classical groups G 9 this result was proved by Gerfand and Naimark [100] . 406 APPENDI X I. INFINITE-DIMENSIONA L REPRESENTATION S

Wi and Su i = 1 . •••» r, dependin g o n th e signatur e a , fo r whic h the followin g "symmetry relations " hold :

WiX(a) = X(a t) W u SiX(a) = X(aJS<; i = I , 2-, r. (2 )

The operators W { are integral operators, and th e 5 f are differential operator s in 3DV. All other equivalence and "partial equivalence" relations among the operators X{a) are consequences o f relations (2). It ca n b e show n tha t th e abov e propertie s characteriz e th e operato r value d function X(a) e 31 (this result is due to the author [167]). These properties make it possible to expres s the fact s tha t th e functio n x(g) ha s compact suppor t an d is infinitely difFerentiabl e i n terms of its Fourier transform . An analogous description can be obtained [167 ] for square-integrable compact- supported function s x(g), "rapidl y decreasing " function s x{g)P an d eve n al l generalized functions x(g) with compact support on G. Specia l interest attaches to the family of generalized functions with support at a single point of G. This family is isomorphic to the universal enveloping algebra X of the Lie algebra of G. All the results listed above are analogs of th e classical Paley-Wiene r theorems* on th e straigh t lin e (see , for example , [40]) . They are essential fo r th e theory of infinite-dimensional representation s of the group G.

% 7. Minima l representations The elementary representations o f a group G with nondegenerate signature s a furnish importan t example s o f irreducibl e infinite-dimensiona l (no t necessaril y unitary) representations. Other examples can be obtained by studying the structure of the invariant subspaces of the representations e(a) at degenerate points. Fix some nonzero vecto r e0 e V\ wher e 1$ is the minimal highes t weight o f th e Weyl compact form i n the lineal i? v (§ 2). We call e0 a minimal vector of class v. Let jS?(a ) be the cyclic module generated b y e0 relative to e(a, JC) , J C 6 £ (wher e a is an arbitrary signatur e wit h inde x v). If the signature a i s nondegenerate, then 3f(a)=&v I t i s interesting to investigat e the structure of the cyclic module &(a) in the general case. First observe that if /3=cr„ 5 € Wy then the cyclic modules &(a) and j£?(|8) satisfy a relatio n of type j£f(/3)=4(/3, a) J2?(a), where A(fiya) is some "symmetry operator " generated b y the operators W iy i = I,—,r , of § 6 (see [164]). It can be shown that the orbit {a„ s € W} always contains two points a- an d a+ with the following prop- erties: l)j£f(a- ) = JS?„ ; 2 ) £f(a+) is irreducible. Th e representatio n i n th e spac e J?(a+) is called a minimal representation of the algebra X with signature a+. Let 3>(a ) be the closure of J?(a) in the topology of the space 3). Then $(a) is

7> A simila r resul t ha s bee n announce d fo r "rapidl y decreasing " functions b y Gel'fan d an d Graev [99J. 8. CLASSIFICATIO N O F IRREDUCIBL E REPRESENTATION S 40 7 invariant unde r e(a< g). Se t a=er+. The restriction of e(,#) , coincides with the finite- dimensional representation dXft considered in §3. I n this case, as we saw in §3, the space Vip of th e representation dip is defined in 2) by a certain system of equations involving infinitesimal left-translatio n operator s on the group XL As yet, no adequate description is known in the general case for the space $Ka) of th e minima l representation s /*(a). It i s onl y known tha t $>(a+ ) is the image of <$>„ wher e v is the index of a~, unde r the symmetry operator A***A(a+jx-)*

§8. Classificatio n o f irreducibl e representations

The classification o f infinite-dimensional irreducibl e representations of a group G depends essentially on the axioms defining the term "irreducible representation"- Indeed, even if we replace G by the additive group J ? of real numbers, the problem of classifyin g it s topological^ irreducibl e representation s i s equivalen t t o th e as yet unsolved problem of classifying the continuous linear operators (up to equiva- lence) in a topological vecto r space. The situation is radically simplified if , instea d of topological^ irreducibl e representation s o f JR , we conside r completely irredu - cible representations (see § 4). In this case, thanks to the availability of Burnside's Theorem, one can state [166] that any completely irreducible representation of the group R i s one dimensional, so that it is an exponential e** , J C € JR . However, th e passag e fro m R t o th e genera l cas e G raise s anothe r difficulty . As we have already seen, the operators of the representation e(a) ma y be defined both in the Hilbert space jf „ and in the Montel space SD„ . Now %, is dense in jf^ and the representation e(a) i s completely irreducible in both spaces. It is natural to assume that such representation s are in some sens e "identical," i.e. to introduce a suitable definitio n o f equivalence . A definitio n o f this typ e wa s suggeste d b y Naimark in [134]. We present a somewhat modified version of his definition. Let T g be a representation of G in a topological vecto r space L, and let LQ be an invariant subspace in L. W e shall say that the representation T g in LQ is a contrac- tion of the representation T g in L i f LQ can be endowed with a stronger topology in which LQ is complete and the operators T g continuous. Tw o representation s o f G are said to b e weakly equivalent if they have equivalent contractions. We can now formulate one version of the problem of classifying th e irreducible representations o f G: to classif y al l completel y irreducibl e representation s o f th e group up to weak equivalence. A complete solution of this problem has recently been achieved by Naimark and 408 APPENDI X I . INFINITE-DIMENSIONA L REPRESENTATION S

the author [169] (see also [168]) :8) Any completely irreducible representation of the group G is weakly equivalent to one of its minimal representations fi(a).

§ 9. O n semireducible representation s Investigation o f elementar y representation s reveal s another peculia r featur e o f infinite-dimensional nonunitar y representation s of a group G: WeyVs theorem on complete reducibility fails to hold for such representations. In fact, w e saw in § 4 that the representation e(a) o f the Lorentz group with a positive integra l point a contain s the finite-dimensional representation d(a) i n an invariant subspace, and the factor representation e{a)/d(a) i s equivalent to e(fi) for some £. It can be shown that e(a) is not expressibl e as the direct sum of th e rep- resentations d(a) an d e(fi). This result is a corollary o f a Paley-Wiene r theore m first proved fo r the Lorentz group by the author ([163]; see also [159 ] and [164]). Similarly, if G is a n arbitrar y semisimpl e comple x group , the elementar y rep- resentation e(a) a t degenerat e point s ha s a fairl y complicate d "step " structure. The representation space contains only finitely many invariant subspaces. If (0) = I0czL1c:--c:Lw = L i s a maxima l chai n o f invarian t subspace s (generall y not uniquel y defined) , the n i n eac h facto r spac e Z*/I*- i th e representatio n i s equivalent to a minimal representatio n of th e group G, In [164] we made an attempt to investigate all semireducible representations o f the Lorentz group which have "finite stepwise structure," i.e. contain only finitely many irreducibl e components . Usin g a Paley-Wiene r theorem , thi s proble m wa s reduced to a certain proble m o f linea r algebra, recentl y solved b y Gel'fan d an d Ponomarev [101]. The theory of infinite-dimensional (eve n unitary) representations of a semisimple real Lie group is far more complicated. No survey of this theory is possible within the scope of the present book.

8> A simila r (but weaker) result was obtained i n an earlier paper of Berezi n [59]. APPENDIX I I

ELEMENTS O F TH E GENERA L THEOR Y O F UNITAR Y

REPRESENTATIONS O F LOCALL Y COMPACT GROUP S

The problems of representatio n theory become far more complicated in relation to general locally compact groups; this is true in particular of harmoni c analysis on the group. Indeed, we saw in Appendix I with regard to semisimple complex groups that the irreducible representations of a locally compact group need not be finite dimension- al. Instea d of the generalized theory of Fourie r series (Peter-Weyl theory), we need a generalized theory of Fourie r integrals, and moreover the latter are usually operator valued. At present the only well-developed part of the theory is the theory of unitary representations (i n whic h a significan t rol e i s allotte d t o th e fundamenta l spectra l theorem of functional analysis) . A brie f survey of this theory is presented in this ap- pendix.

§ 1 . Commutativ e groups We first consider the additive group R of real numbers. A unitary representation of this group is any one-parameter group Uh — c o < t < oo , of unitary operators in a Hilbert space HS> B y the well-known theorem of Stone (see, for example, [55]), the famil y o f operator s U t can b e expresse d a s U t = e** , — o o < t < oo, where A is a selfadjoint operato r in H. The operator A is known as the generating operator of th e one-paramete r grou p U t. Applicatio n o f th e classica l spectra l theorem of functional analysi s ([39], [44], [55]) to this operator yields the following information o n the structure of the representation U t. First , the operator value d function U t can be expressed as an integral:

where P(X) is a family o f projections, known as a resolution of the identity, which has the following properties: 1) P(X)P(ft) = P(X) for X ^ p; 2 ) P(X) commutes with every continuous linear operato r which commutes with A; 3) P(X)E is left contin - uous in X for an y £ e H; lim*—o o P(X)£ = 0 and lim;- +0o P(X)£ = £ for an y £ e H.

2> By the definition of a representation , this operator valued function i s assumed to b e con- tinuous in t. However, see the remark in § 6, below.

409 410 APPENDIX II. LOCALL Y COMPAC T GROUP S

Formula (*) furnishes the most important information regarding th e representa - tion Uu It is clear that U t is irreducible only if the spectral measure dP(X) is con- centrated at a single point X$ and the space His one-dimensional . I n this case, U t = 2 e*^. * Formul a (•) thu s represents a decomposition o f U t into irreducible repre - sentations. Formula (* ) als o yield s a n alternativ e interpretatio n o f th e spectra l theore m for U t. W e first assume that the space H contains a vector $0 which is cyclic for the n generating operator A (i.e. H is the closure of the linear span of the vectors A £0y n = 0, 1 ,•••)• Se t a(X) = (P#)£o » £<>) • The n th e spac e H i s isomorphi c t o th e Hilbert spac e o f vecto r value d functions/(A) , — co

while the action of the operator A reduce s to multiplication by X: Af(X) « Xf(X). 3) The action of the projection P{JJ) is then simply multiplication by the characteristic function of the half-axis — co< X < fi. I n the general case, th e representatio n U t splits int o cyclic representations (i.e. representation s fo r whic h a cyclic vecto r £ 0 exists). As before, formula (*•) remains valid, except that/(^) is now a vector valued function (generall y infinite dimensional) ofXS> Note tha t i n th e realizatio n (•* ) ever y delt a functio n 8{X — XQ) is a forma l eigenvector of th e famil y U t9 with eigenvalu e e*^ . Thoug h the delta functio n i s not an element of H, this statement can be made rigorous if one uses the technique of generalize d eigenvector s due to Gel'fan d [18 ] and Kostjucenko . Finally, conside r a n arbitrar y locall y compac t commutativ e grou p G . I n thi s 5) case w e hav e Nalmark's theorem [39], whic h generalize s formul a (*) . Le t U t be a unitary representation of G in a Hilbert space H, and JTthe group of all unitary characters of the group G, i.e. one-dimensional unitar y representations X: g -+ X(g). Then, b y Naimark's theorem,

Ug= lX(g)dP(X\

where the integral i s taken over the group JTand dP(X) is a spectral measur e de- fined on Bore l sets in X Fo r details, see [39], Russian p. 487. Henc e one readil y

*> Consequently ever y unitar y irreducibl e reprsentatio n o f th e grou p R i s on e dimensiona l (this ca n als o b e prove d independently). As regard s nonunitar y topologicall y irreducibl e rep- resentations of R, the problem is still open (see, for example, (104]). However , th e resul t remain s valid for completely irreducible representations of rt. 3) Not e that the operator A is not in general bounded. Its domain of definition is dense in H. 4> The vector dimensionality of the function/(A) generally depends on X. On this question, see the general definition of the "direct integral*' for Hilbert spaces ( § 6 below). See also [39]. 5) Somewha t later , this theorem was proved independentl y b y other authors (Stone, Ambrose, and von Neumann). 2. STONE-VO N NEUMAN N THEORE M 411

derives a generalization o f formula (** ) as well. The space H has an isomorphic realization as the Hibert space of functions /(X), X e X, which are square integrable with respect to a certain measure. The representation U 8 is defined b y the explicit formula

As in the special case of th e group R, these result s imply a decomposition of Ug into irreducible representations. In particular, every irreducible unitar y representa- tion of the group G is one dimensional and is defined by one of the characters X(g). Harmonic analysis on a locally compact commutative group G is at present the most throughly developed generalization of classical Fourier analysis. (In particu- lar, harmoni c analysi s on G yields th e Pontrjagi n dualit y theory , mentione d in § 107. ) We cannot presen t a more detailed surve y of this theory here. Individua l problems are treated in the monographs [21] , [36] an d [39].

§ 2. Stone-vo n Neumann theorem Hitherto we have dealt only with commutative families of unitary or Hermitian operators in a Hibert space //. One of the simplest examples of a noncommutative system is the system of Hermitian operators P and Q such that [P, Q] = // , where / = V^ T and / i s the identity operator in H (the coordinate and momentum opera- tors i n quantum mechanics). Thi s example also play s a significant rol e in the de- velopment of the general theory. First observe that [P n, Q] = inP»~ l for all natural numbers /? , whence follow s th e general formul a

for arbitrary polynomials/(z) wit h P substituted for z, where the prime stands for differentiation wit h respec t to z. It is natural to conjecture tha t (*) remains valid for a broade r clas s o f function s /(z). I n particular, conside r th e resolvent Rx = (P — A/)" 1 at points where it is defined. Then

[Rx, Q] = Rx [Q, P-U] Rx = - IRl corresponding t o formula (* ) for/(z) = ( z — A)"" 1. But then formula (*) remain s valid for all polynomials i n P and R^ an d for strong limits of such polynomial s (provided f\P) i s defined). In particular, the last-named class of functions contains 6) the one-paramete r grou p o f operator s U t = d* [55]. Consequentl y [U h Q] = 7 - tV u an d so >

6 > The existence of the unitary group Ut follows from the spectral theorem for the operator P. Were the operator P bounded, we could express Ut directly as a power series in P. 7) Thi s formula may be derived in a far simpler way if we observe that the function U t & differ- l entiable wit h respec t to /, and U\ = iPU t. Setting F (/) = U~ QUty w e see that the n F (tY = - '[/* > Q] = / • Consequently F{tY = 0 , and so F(/) = F(0 ) 4- tl = Q + // . 412 APPENDIX II. LOCALL Y COMPAC T GROUPS

1 u; Qu t = Q + ti. (** ) It follows from (**) that the spectrum of the operator Q fills out the entire real axis. Using the spectral theore m fo r g, i.e . expressing Q as multiplication b y x in the space of vector valued functions/(JC) which are square integrabl e with respect to a measure do(x\we hav e Qf(x) = xf(x). I t follows from(**) tha t the vector value d function f(x) ha s the same dimensionality a t all points x and the measure da(x) is ordinary Lebesgu e measure . Consequentl y w e may assume that/(;c) take s values in some fixed Hilbert space H0. Th e space H then consists of all functons/(;c) suc h where that J-oo||/(*)||o <* * (*) < °°> ||/|| o denotes the #0-norm. Consider the differ- entiation oprator P=id/dx i n //(defined on a dense subset of H). As is easily seen, we have [P, Q] = U; that is, the operators P and Q satisfy th e same commutatio n relations as P and Q. Consequently the operator P— P commutes with Q. But then, as is well known from spectral theory, P— P is multiplication by a certain functio n of JC , and so

P = P +

**. We have SQS~ = Q and 5P5- = id/dx. Thu s th e unitary trans - formation S in the space H map s th e pair P,Q ont o th e pair o f operator s P 0 = W/d!x, go = ^ - Consequently, up to unitary equivalence, any two Hermitian operators P and Q satisfying the commutation relation [P,Q] = H reduce to the pai r P 0, Q0. This is the celebrated Stone-von Neumann theorem. Of course, the proof has been somewhat abbreviated. ® REMARK. The equation [P, Q] = // has no solutions (Hermitian or non-Hermitian) in the class of finite-dimensional linear operators, for the trace of the left-hand sid e of this commutation relation is zero while that of the right-hand side is not. We now consider the group Z(3) of all triangular 3x3 matrice s wit h one s along the principal diagonal. It is easy to see that the Lie algebra of this group is spanned by thre e basi s elements satisfyin g on e nontrivial commutatio n relatio n [/?, q] = r . The element r is central. Now let P, Q and R be the images of/?, q and r under some unitary representation of Z(3). Since the representation is unitary, w e may assume (via multiplication b y i = V^-T ) that P and Q are Hermitian operators. If the rep- resentation is irreducible, th e operator R i s a multiple of the identity, and we get the commutation relation [P, Q] = iXI, where / i s the identity operator in H. If X = 0, then [P, Q] = 0 and the representation is commutative. (Hence, since it is irreduci- ble, it must be one dimensional.) But if X / 0 , then, as before , P = id/dx an d Q = Xx (up to unitar y equivalence) . Th e corresponding one-paramete r subgroup s U t = e^ and V t = e* s then satisfy the following formula :

») See also [lid], [136] and [151]. 3. INDUCE D REPRESENTATION S 413

Utfix) - f{x-t\ V tf(x) - **/(*) . (••• ) Thus the Stone-von Neumann theorem yields a description of all irreducible unitary representations o f th e group Z(3) (up to unitar y equivalence) . (Indeed, the one- parameter subgroups Ut and V t generate thes e representations.) We see that every such representation is either one dimensional or infinite dimensional. In the first case the representation is defined by two real scalars a and $ , each of the operators P and Q being multiplication b y one of these scalars (R = 0) , and Ut a * e** and V t = e** . I n the second case th e representation i s define d b y th e explicit formul a (***), and so the resul t o f th e classification depends on th e rea l number X # 0 . It i s readil y see n tha t representation s wit h differen t A-value s are inequivalent. In the geometric interpretation, the set of all irreducible unitary representations of the group Z(3) can be identified wit h the set of all planes X~ const #0 and the set of all points on the plane A = 0 in the three-dimensional euclidean space X,a,fi.

§ 3. Induce d representations Before proceeding to our exposition of the theory, we devote some attention to a special type of representation, the induced representations. Let A" be a homogeneous space with group of motions G, where the action of an element g e G on a point x e X is denoted by xg. Consider the linear operator Tg acting in the space of func- tions on Xy defined by

Tgf(x)~a(x,g)f(xg).

A necessary and sufficien t conditio n fo r the family o f operators T g to be a repre- sentation of the group G is that the following equalities hold:

a (x, gtf2) = a (x, ft) a (xgu g2\ a (x, e) = 1 . (* ) Let us determine the general solution of this functional equation . Fi x some point x0 e X&nd set a(g) = a(xo,g). Then, by (*), the function a(g) satisfies the functional equation

cc(gog) = oc(Zo)a(g), a(e)=h (** ) for arbitrar y element s g0 e G0 and g e G, wher e G 0 is the stable subgroup o f the point Xo. For every point x e X, let gs e G be an element mapping x0 onto x. Then any element g e G has a unique expression as gogX9 go e G, and we see from (*) that

a (xo,gsg) = a (xo, gs) a (x, g). Hence it is clear that the function a(x,g) can be expressed in terms of a(g): 414 APPENDIX II. LOCALL Y COMPAC T GROUP S

i cc(x*g) = a(g sy a(gsg).

Express th e elemen t gxg a s gog» where g0 e G 0 and y e X. I t then follows from the last relation that*> cc(*,g) ~

This formula give s the general form of the solution a(x fg). Usin g the equivalence transformation Af(x) = a(g x)f(x)9 w e ca n replac e the representation T g by the l equivalent representation fg = AT gA~ . The n

fgf(x) = a(x,g)f(xg),

and th e functio n d(x,g ) satisfie s th e conditio n a(g x) = L Consequentl y i n thi s case we have a(x,g) = a(g 0), where , as before, g0 i s defined b y the decomposition gxg « £ogy . By formul a (**) , th e functio n a(go) w a representatio n o f th e grou p C 0. The representation 7 ^ of G is said to be induced by the representation a(g0) of the group 10) <70- Al l the above constructions remain valid when/(x) is a vector valued func - tion with values in some vector space L and a(x,g) ar e linear operators in L. The representation T g is also said to be induced in this case. If dim L = 1 , the induced representation i s sometimes called a monomial representation. The irreducibility of a(go) is a necessar y but not sufficient conditio n for irredu- cibility of T g. Indeed, in Appendix I we discussed the elementary representation s of a semisimple complex group G. They are all monomial representations , i.e . th e representation a(go) is on e dimensiona l an d therefor e irreducible . W e have seen , however, tha t a n elementar y representatio n ma y b e reducible . Incidentally , th e group G0 in this case is a maximal solvable subgroup of G. As we know from the main text ( § 17) , a homogeneous space A " can be identified with the factor space C/Co, where C0 is the stable subgroup of a fixed point x 0 e X. We may also identify X with a set of representatives gx$ xeX. Usin g the decomposi- tion g s = g^gX9 le t u s replace the functions/(JC ) b y the functions/(g) = g{fog x) = a(go)f(x). A H these functions clearly satisfy th e relation figog) = a (go) fig) for arbitrar y go e G 0 and g e G. Le t Fa b e the set of all such functions on (7 . The right translatio n Rgf(a) = f(ag) generate s a representatio n i n F a which i s readil y seen to be equivalent to the original representation T v Consequentl y we have yet another realizatio n o f th e induce d representation s (whic h ha s alread y bee n re - peatedly used i n the main text).

*> This follows from the functional equation (**). 10 > Recal l that G 9 is the stable subgroup of an arbitrary fixed point x 9 G. X I f we replace x9 1 by x9gt th e subgroup G9 is replaced by the conjugate subgroup g" G^g. 4. SEMIDIREC T PRODUCTS 415

We have not yet imposed any restrictions on the space of functions in which the induced representation acts. In applications, however, such restrictions are always necessary, though the range of possible spaces is quite broad (for example, differ- entiable functions o r Inunctions unde r the assumptions of Appendix I). If there is a measur e dx o n X suc h tha t d(xg) =

cc (*, g) a (x, g) = a> (x, g), (*** ) where the ba r denote s comple x conjugation , enable s on e to construct a unitar y representation o f the group G using the formula fo r a n induce d representation . To this end we must confine ourselve s to the class of all functions/(jt) whic h are square integrable with respect to the measure dx. (The easy verification i s left t o the reader.) We assume here that/(x) is a scalar function, but if L is a Hilbert space then condition (***) remain s valid, with complex conjugation replace d by Hermi- tian conjugation. In this case a(gQ) = «Gro)/3(gb X where j3(go) is a normalizing facto r and u(g 0) is a unitary representation o f the group GQ. With regard to unitary rep- resentations, on e usuall y say s that th e representation T g is induced b y the rep- resentation u(g0) (and not by a(go\ as in our text). However , i f the measur e dx i s invariant (d(xg) = dx), then $go) = I - The techniqu e o f induce d representation s is often utilize d to construct explicit models of irreducible representations for variou s groups G. n) Th e most familia r irreducible representations of the classical Lie groups are induced and even mono- mial representations.

§ 4. Semidirect products We now consider the important case in which the group G can be expressed as G = ST (uniqu e decomposition), wher e S is a subgroup and T a normal divisor. Such a produc t o f tw o subgroup s i s known a s a semidirec t product . (Example : the group of motions G with S the rotation subgroup and T the translation sub - group.) In this section w e consider only the special case in which the subgroup T is commutative. We shall show that a knowledge of the irreducible unitary repre- sentations of the subgroup S yields a description of all irreducible unitary represen- tations of the group 71 Let Ug be an irreducible unitary representation of G in a Hilbert space H. B y the results of § 1 , the restriction of this representation t o th e subgroup 7 * can be de- scribed as a direct integral [see § 6 of this appendix] of one-dimensional representa- tions over the set X7 where X i s the group of all unitary characters of the group G.

11 > Th e theor y o f induce d representation s date s bac k t o Frobenius , wh o considere d finit e groups. The modern development of this theor y bega n with th e wor k of Gel'fand and Naimark [100] and has been made systematic by Mackey [122], [123] and Bruhat [69]. 416 APPENDIX II . LOCALL Y COMPAC T GROUP S

In other words, the space H can be realized as a space of vector valued functions f(x),J C e X, an d moreover UJ(x) - x(t)f(x), (* )

where x(t) is the value of the character x € X on the element / € 7* . Recal l that the function/(jc) is square integrable with respect to some measure do(x\ A knowledge of this measure is essential for description of the representation. Now conside r a transformation £/ „ $ € S. Sinc e T is a normal diviso r in G, it follows that s*1 ts e T. Setting t(s) = s~ l ts> we have

UtU, « U,U« $y

On th e othe r hand , conside r th e transformatio n V sf(x) = f(xs) i n th e spac e H9 where the character JC , is defined b y the formula xs(t(s)) « x(t). I t is then readily seen that

UtVs « K,l/, (,>.

l Comparing these two formulas, we see that the transformation U sV~ commutes with all operators Vu t € 71 But any such transformation is known [39] to coincide l with multiplication by some function of JC . Hence U, Vj = a(x 9 s\ s o that

Utf(x) = a{x ys)f{xs). (*• ) Formulas (*) and (**) uniquely determine the representation of the group G. We see that this representation may be expressed in terms of the formula for an induced representation i s a suitabl e clas s o f function s f(x)9 x e X However , w e cannot usually state in general that the space Xis homogeneous. ^ We have not yet utilized the irreducibility of the representation Ue. It is obvious that this property depends essentially on the structure of the transformation x -* xs in the space X, and also on the measure d

12> Tha t is to say, that the transformation x — JC , is transitive. 4. SEMIDIREC T PRODUCTS 417

position gxg = gogy. G0 is the stable subgroup of som e fixed point JC 0 € X. Henc e it follow s tha t G 0 contains T (since T is commutative). Consequentl y G 0 = S0T> where S 0 i s a subgrou p o f S. I t i s als o obviou s tha t a(g 0) = a(s 0t) = a(s0)x<£t)9 since a(f) = x 0(t). Usin g a slight abuse of language , on e sometime s say s that the 1Z) representation U g is induce d b y th e representatio n a(s 0) o f th e subgrou p S 0. In actual fact, however, the representation Ug is characterized by the following data: 1) a n irreducible representation a(s0) of the subgroup S0; 2) a character x0(t) o f the subgroup Tsuch that Xo(t(s)) = *<>(/) ; and 3) a measure da(x) o n the orbit xo(t(s))> s e S. 14) EXAMPLE. Le t G b e th e grou p of motion s o f euclidea n /i-space , wit h rotatio n subgroup S an d translatio n subgrou p T. Symbolicall y on e represent s element s g € G by matrices of order it + I of type Hi .1- where s e S and f is an n-vector corresponding to the translation t e T. The elements seSand / e Tar e then identified wit h the matrices Ho ill- H o i|- where e is the n x / * identity matrix. Computing s^ts, w e see that the action of this transformation i s to replace the vector / b y 5"" 1/. Further, every character x(t) ca n be writte n as x(t) = ex p i(a,t), wher e (a,t) i s the inner produc t of th e /i-vectors a and / . Henc e th e action o f th e transformatio n x - > xs i s to replac e th e vecto r a by J-1 a . Indeed ,

-1 -1 xs(t(s)) = ex p ifa" a, J /) = ex p i(a, /) = x(f) .

Every orbit of the transformation a -> s*1 a is a sphere in /i-space, of radius r, say. Let Xbt on e such sphere. We can define a measure dx on A " which is invariant under rotations x - * sx. Th e require d spac e o f function s f(x) consist s o f al l function s /(JC), x e Xy which are square integrable with respect to dx. It remains to determine the subgroup G 0. Let x0 e X b e an arbitrary fixed point. The subgrou p o f rotation s S 0 preservin g thi s poin t i s obviousl y isomorphi c t o 0(n — 1). Hence G 0 = S 0T, wher e 7 * is the translation subgroup . B y the genera l theory set forth above, it is not hard to show that every irreducible unitary repre- sentation of G is defined by the following data:

13> Physicist s call this subgroup a "little group * or "small group." 14> I t can be shown tha t the measure da(x) mus t be quasi-invariant unde r the transformatio n JC -• JC, . If d

1) a n irreducible (finite-dimensional) representation o f the group 0(/i — 1) ; and 2) a nonnegativ e numbe r r ^ 0 (the radiu s o f th e orbit) . The measur e dx i s then defined up to a constant factor. 15) The results presented i n this sectio n are due to Macke y [122] , [123] . W e have omitted all complicated questions relating to measure theory, including the consid- eration of ergodic motions.

§ 5. Nilpotent Lie groups Now let G be a nilpotent connected Lie group. In this section we shall prove the following theorem of Kirillov: Every irreducible unitary representation of the group G is monomial The proof wil l proceed by induction on the dimension o f the group (7 , i.e. the dimension of its Lie algebra A. This will enable us to assume, without loss of gener- ality, tha t th e representatio n o f th e algebra A i s faithful. (Indeed , otherwis e we need dea l onl y wit h a representatio n o f a suitabl e facto r algebr a o f A o f lowe r dimension.) Now let Z be the center of the algebra A. Thanks to irreducibility, the operators i n the center are scalar. 1® Sinc e the representation i s faithful, w e may thus conclud e tha t dimZ= 1 . (Indeed, ever y nilpoten t algebr a ha s a nontrivia l center (§ 85); thus, dim Z # 0 . On the other hand, the field of scalars has dimen- sion 1. ) Consequently , w e may assum e that th e algebr a A ha s one-dimensiona l center. If Aim A = 1, 2 the n A i s a commutativ e algebr a (Exercis e 1 in § 85) and th e theorem hold s (ever y irreducibl e representatio n i s on e dimensional , henc e als o monomial). If dim A = 3> then, apart from the commutative case, we have only one Lie algebra: the Lie algebra of the group Z(3) (Exercise 2 in § 85). In this case the statement of the theorem follows from the results of § 2 in this appendix. Thus the theorem is proved for dim A = 1,2,3 . Now assume that dim A > 3. The rest of the proof splits into several steps. 1. By the structure theory of nilpotent Lie algebras ( § 85) we can extend Z to a two-dimensional subalgebr a Y suc h that [A, Y] cz Z. Let y9z be a basis in Y, where z e Z. Then

[a,y] = tp(a)z (•) for all a e A (since Z is one dimensional). Here

15> Indeed , any measure which is quasi-invariant unde r a compact group (in our case, 0(n — I) ) is equivalent to an invariant measure. The proof is left to the reader. 16> This follows fro m th e generalized Schur' s lemm a [39], which wil l be discussed in detail in §6. 5. NILPOTEN T LI E GROUPS 419 algebra of th e group Z(3), an d its center coincides with the cente r of th e entir e algebra A. 2. Let A0 denote the hyperplane defined in A by the condition p(a) « 0 . We claim that A0 is an ideal in A. Indeed, A0 consists of all elements a € A such that D ey * * 0, where Dey = [a yy\ I n particular,

in view of (*) and the fact that 2 is in the center. Consequently every commutator is in A& In particular, [A,A , and so AQ is an ideal. Obviously A ** AQ+ {x}* Since i40 » a n ideal, this in turn implies a decomposition of the group: G — G0T (semidirect product), where G 0 « ex p ^40 and T = {ex p tx). W e could now use the results of the preceding section, but we prefer a modified construction which shows that the representation of the group G is induced b y a representation of the sub- group G 0. 3. Thus, let T g be an irreducible unitary representation of G in a Hilbert space //. B y the Stone-von Neumann theorem we may assume that H is a space of vector valued functions/(/), — 0 0 < / < 00 , square integrable with respect to Lebesgu e measure, so that the image s T x, Ty and T z of the elements x9y an d z are given by

Tn-dldt, T,-tM, T t**M.

Here X # 0 , since the representation is faithful. Now let g0 € G0. By the definition of Go, the operator T^ commutes with Tr Consequentl y for any gQ the latter opera- tor is simply multiplication by a suitable function of the variable t:

TgJ(t)~A(t,g0)f(t).

On the other hand , T txpsxf(t) ~f(t + x). Henc e i t follows tha t the straigh t line X0 ~ { — 0 0 < / < 00 } i s a homogeneou s spac e fo r G . B y th e decompositio n G « GoT, the formulas for Tgtt an d T^** uniquel y determin e the representatio n Tr W e see that this representation is induced by the representation U t9 - A(t 0f gQ) of the subgroup G0 (where /0 is an arbitrary fixed point in XQ). 4. I t remains to us e the following "transitivity rule" from the theory of induced representations [116] : Le t G 2 c G i c r G be an ascendin g chai n of subgroups ; i f a representatio n r of the group G is induced by a representation rx of the subgroup Gi, which is in turn induced by a representation r 2 of G 2, then r is also induced by tz* This rule makes it possible to use induction: progressively lowering the dimen- sion, we finally reach the case dim A « 3 considered above. Q.E.D . Kirillov [116 ] refine s thi s constructio n further . A monomia l representatio n i s obviously define d b y some linear function/over th e algebra A (the Lie algebra of the grou p G) . I t turns ou t tha t representation s induce d b y functional f x an d f z are equivalent if and only if/J = flig)/i>ge G , where p(g) is the representation of G 420 APPENDIX II. LOCALL Y COMPAC T GROUP S

dual to the adjoint representatio n o f G in the algebra A. We thus discover a one- to-one correspondence between the orbits p(g) and the irreducible unitary representa- tions of G. Using various operations of tensor algebra (tensor products, restriction to subgroups) , on e ca n als o revea l remarkabl e informatio n i n term s o f orbits . It i s worth notin g that th e "languag e o f orbits' * is independent o f the structur e theory (whic h i s in fac t poorl y develope d fo r nilpoten t groups). We may expect that, to a certain degree, this language will prove to be universal in regard to arbi- trary Lie groups (for description of all irreducible unitary representations).

§ 6. Decomposition of unitary representations into irreducible representations Hitherto we have been considering only individual types of groups and, generally speaking, their irreducible representations. We now discuss a fe w general results, on the assumption that G is an arbitrary locally compact group satisfying the first axiom of countability. (The latter is true, at any rate, for every lie group.) Let U g be a unitary representation o f G in a separable Hilbert spac e H Not e that for the operator valued function U g to be strongly continuous in this case it is sufficient that it be weakly continuous ([39], Russian p. 443). W e have already seen in the main text that unitar y representations satisf y th e principle of complete re- ducibility. Successive application of this principle shows that the representation Ug can be decomposed into irreducible representations. However, the set of irreducible components need not be denumerable, and therefore a rigorous description of this situation requires the special concept of a direct integral. Let A be a compact space with measure dc(X). Suppose that almost every point X € A ca n b e associated wit h a Hilbert spac e Hx of dimension n(X) » 1,2 , •••, oo, where n(X) is measurable with respect toda(X). Sinc e the function n(X) can assume only denumerably many values, the space A splits into a denumerable set-theoretic union of disjoint subspaces Ak> on each of whic h n(X) = const . Consequentl y al l spaces Hx fo r X e A ca n be identified with a fixed space Hk. Le t us say that a vector valued functio n f(X) eH* XeAk,i$ measurabl e if the scalar functio n (f(X) 9y>) i s measurable for every - A Correspondingly w e defin e th e inne r produc t o f tw o function s j[X), g(X) € H. Functions/^) such that |/|| = 0 are identified with 0. The space H is thus made into a Hilber t space , known a s the direct integral of the space s Hi an d denote d b y //= U»ida{X) Note that in general Hx is not a subspace of H (an exception is the case in which the measure of the point X is finite). If an operator Ax is defined i n almost every 6. DECOMPOSITIO N O F IRREDUCIBLE REPRESENTATION S 42 1 space H%, th e operator Af(X) = Axf\X) is called the direct integral of the operators

We now retur n to ou r unitary representation U g of th e group G. The principal result is as follows: The representation Ug is the direct integral of irreducible unitary representations Ug(X) with respect to a measure da(X). It is essential to note that, in general, this decomposition i s not unique . More - over, there exist group s for whic h the spectrum, i.e..the "list " of irreducibl e rep- resentations appearing in the decomposition, i s not unique . The situation i s sim- pler if instea d of irreducibl e representations one considers representation s whic h are "multiples of irreducible representations," more precisely, the so-called facto r representations.

The representation U g is the direct integral of pairwise inequivalent factor-rep- resentations [39] . Thi s decompositio n is unique . Conditions ar e know n unde r which every bounde d operato r in H commuting with U g is th e direct integra l o f scalar operators /*(A)ii, where h i s the identity operator on Hx and p(X) a number (this is the continuous analog of Schur's Lemma; see [39], Russian p. 413). In particular, ever y bounde d operato r commutin g wit h al l operator s o f a n ir- reducible unitary representation is a multiple of the identity (but this result can of course be proved independently and by a far simpler technique). If U g is a regula r (left or right ) representation o f the group G , then th e above decomposition lead s to the so-called "abstrac t Plancherel formula " (Segal [147]). However, for each given class of groups, the problem is to describe explicitly the measure figuring in the decomposition (th e Plancherel measure) . The investigatio n o f nonunitar y representation s involve s considerabl y greate r complications. Sinc e the principle of complete reducibility is no longer available, there is not eve n a "well-posed " formulation o f th e decomposition problem . Ir - reducible representation s hav e bee n considere d i n comparativ e detai l onl y fo r semisimple complex Lie groups. APPENDIX H I

UNITARY SYMMETR Y I N TH E CLAS S O F

ELEMENTARY PARTICLE S

As we mentioned in the Preface to this book, the methods of the theory of linear rep- resentations of Lie groups have given rise to remarkable dicoverie s i n certai n fields of theoretical physics, chief among these being the classification of elementary particles. The purpose of this appendix is to trace in brief the basic logical reasoning underlying these discoveries.

§ 1 . Invariance and conservation laws We first recall the basic postulates of quantum mechanics, which remain valid to a certain degree in quantum field theory. 1. Wave function. According to the fundamenta l assumptio n o f nonrelativisti c quantum mechanics , the evolution of any physical system in time is described by the Schrodinger equation 3^/3 / = iT^ , wher e r i s the energy operator (Hamil- tonian, mas s operator) , an d ^ i s an elemen t of a certain Hilber t spac e H, th e state space. Al l information concernin g the physica l system i s contained i n the vector ^, whic h is known as the state vector or wave function. An important role is assigned in the theory to quadratic forms o f typ e (A. The space H is usually realized in some functional form . An y such realization involves the selection of a suitable system of Hermitian operators Aw»9 A m, which are "diagonalized" in the common basis in the sense of Appendix II, § I. In other words, the elements of the space /fare vecto r valued functions/(JC) = /(xi,---, xm) which are square integrable with respect to some measure da(x% the action of each operator At being described by a formula Aif(x) = x tf(x% i = l,---,m . The role of these operators is sometimes played b y coordinate operators , which describe the spatial position of the individual particles in the system. I n this case, if Aw i s the operator o f multiplication b y the characteristic function o f some region m in the

422 1. INVARIANC E AND CONSERVATIO N LAW S 423

m-dimensional spac e of vectors JC , then the form (Amf,f) define s the probability o f finding th e system in the region a> . The fundamental role is that of states with fixed energy A. For such states we have r = X = p$ 9 then the state (]> corresponds to a definite value p of the physical quantity A. 2. Conservation laws. Assume that the operator A commutes with the Hamiltoni- an. Then th e equality A$ = p = e* r $>, where and $. A physical quantity is said to be additive if its value in the mixed system is the sum of its values in states

Here we hav e a physical applicatio n of the tensor product of two group repre- sentations (the group in this case being the rotation group). If the wave functions

transfor m lik e irreducibl e representation s a an d / 3 of th e group , the n the tensor produc t

1> I t i s noteworth y tha t charge-conservatio n law s ar e relate d t o th e invarianc e o f th e wav e equation unde r transformation s

where the right-hand side is a direct sum of irreducible representations. This means that with corresponding probabilities, the mixed system may be in any of the states —»$» which transform like Yu ~%rm>The probabilities themselves are calculated by suitable formulas which involve the Clebsch-Gordan coefficients. Thus the group structure has an essential part to play. If the physical quantity is additive, it s eigenvalue s are generally known as addi- tive quantum numbers. 4. Breaking of symmetry. I t i s essential t o note tha t any symmetr y i n quantu m mechanics is as a rule approximate, or is attained only under ideal conditions. For example, symmetr y with respect to the group SO(3) is possible only in a centrally symmetric force field. Spatia l symmetry of the molecules of an ideal gas can hold only in the absence of a force field. When a longitudinal magneti c field i s applied, the for m o f th e Hamiltonia n i s modifie d an d it s symmetr y break s dow n (onl y rotational symmetr y about a certain axis is conserved). Let us consider the last example more closely. When there is no field, the states of the gas molecules ma y be characterized b y an irreducible representatio n of the group SO(3) with highest weight L. As we know, ther e are all in all 2L + 1 such states, all correspondin g to the sam e eigenvalu e of th e operato r /* , whic h i n this case is a Casimir operator of the group SO(3). Since r i s a mass-energy operator, 3) this also implies that all 2L + 1 states have the same energy level (the same mass). The physicist will then speak of "degeneracy " (i.e. multiplicity) of the eigenvalue. Application of a magnetic field causes the 2L + 1 states to become differentiated . In particular, they fall on different energ y levels. (The physicist will then speak of "removal of degeneracy.") This is known as the Zeeman effect; it corresponds to an easily detectable physi- cal phenomenon (splitting of spectral lines in a magnetic field).

§ 2. Elementary particles. Isotopic spin The theory of elementary particles deals with the simplest of the currently known states of matter, which serve as building blocks for the more complicated structure s (atomic nucleus, atom, molecule). Som e o f these particles hav e a stable existenc e (proton, neutron); others have vanishingly small life times. The states of elementary particles underg o incessan t change s i n nature ; collision s lea d t o destructio n o f particles and creation of new ones. 1. Basic types of elementary particles. Th e elementary particles which are known at present fall into the following basi c categories, depending on their rest masses. 1) Photons, with zero rest mass. 2) Leptons (light particles), including the electrons, muons (/i-mesons) , th e neutrino , an d th e correspondin g antiparticles . 3 ) Mesons (medium-weight particles). A fundamental rol e in classification of mesons is played by an additive physical quantity 5, know a as strangeness. I f suitably normalized ,

3> B y virtue of the equivalence of mass and energy (Einstein). 2. ISOTOPI C SPI N 425

this quantity assume s only integer values ; for mesons , 5 = 0 or ±1.5 = 0 cor- responds t o 7r-mesons , ^mesons, p-mesons, the /°-meson, etc . Th e valu e 5 = 1 corresponds to th e A-meson . 4 ) Baryons (heav y particles). Fo r baryons , 5=0, — 1,-2. Include d i n thi s categor y ar e nucleon s (proto n p an d neutro n «) , the ^-hyperon, J-hyperons, £-hyperons, etc. 5) Antibaryons. There is also a set of unstable particles, known as resonances, which are usually interpreted as excited states of stable particles. These particles have as yet been in- vestigated only experimentally. 2. Basic types of interactions. Together with the rest mass, it is important to con- sider the different type s of interaction (nuclea r reactions ) in whic h th e particle s can take part. The physicist distinguishes between electromagnetic, weak and strong interactions. Electromagnetic interaction s necessaril y involv e photons . Wea k interactions are characterized b y at least one of the following tw o indications: a) participation o f a neutrino , b ) nonconservatio n o f strangenes s (breakdow n o f 5-symmetry). All other interactions are said to be strong. They are characteristic, in the main, fo r processe s i n the atomic nucleus. The differen t type s o f interactio n differ i n their strengt h (as evidenced by their names; electromagnetic interaction s are the weakest). The best known o f the above types of interaction i s the electromagnetic inter - action. This is the realm of the classical part of quantum field theory (including the theory o f divergences). The weak interaction i s the least investigated. (Under this heading one can include the research of Fermi, and also of Lee and Yang on parity nonconservation.) In the theory of nuclear forces (strong interactions), the energy carriers are jr-mesons (or other particles) instead o f photons. In the sequel we shall consider only particles taking part i n strong interactions. These particles are known as hadrons. 3. Isotopic spin. Much importanc e attaches i n the theory o f the nucleus to the so-called isotopic invariance between the basic components of the nucleus, protons and neutrons. Both particles have the same spin (with respect to the space rotation group) and almost the same mass. Except for electromagnetic interactions, in which the charge is significant (4- 1 for the proton, 0 for the neutron), these particles are essentially identical . The nucleu s i s thus symmetric unde r the substitution p <-+ n (proton—neutron).4) Instead of symmetry under substitutions, we can consider more general, unitary transformations i n 2-space, with "proton"—"neutron" degrees of freedom. Le t p be the wave function o f a "proton" state, and n that of a "neutron" state. Consider the wave function

9 = (* )

4> Two nucle i derive d from eac h othe r b y this substitution ar e indeed simila r in thei r energy properties. They are known as mirror nuclei or mirror-conjugate nuclei. 426 APPENDIX III. ELEMENTAR Y PARTICLES

where (p(p) and

u~\ Si 9 m * ** *' detw= = l ' By th e genera l symmetr y principle s ( § 1), th e state s (•) hav e th e sam e energ y properties as the proton—neutron pair. We thus get the simplest (two-dimensional) representation of the group SU(2). In reality , the theory of the nucleus operates with many nucleon s (i.e. proton s and neutrons). Treating the corresponding wave functions a s the tensor product s of th e necessar y numbe r of factor s p an d n (suitabl y indexed) , w e get reducibl e representations of the group SU(2). Decomposing them into irreducible representa- tions, we get the wave functions of the irreducible components, which are charac- terized with respect to SU(2) by their highest weight and by the weight of th e basi s vector relative to the diagonal subgroup of SU(2). These characteristics are known as total isotopic spin [or isospin] and its projection. I n particular, the projection is an additive quantu m number . Note that antinucleons (antiproton p an d antineutron ft) transform like the dual representation o f the isotopi c group SU(2). However, we know fro m th e genera l representation theor y of SU(2) that these two representations are equivalent. This theor y become s meaningfu l onl y afte r a n isotopi c spi n i s assigne d t o al l other hadrons (mesons and baryons). Once this has been done, one can formulat e the principle of total isotopic spin6) i n strong interactions. This gives real selectio n rules for nuclear reactions. Moreover, the Clebsch-Gordan coefficients of the isoto- pic group SU(2) determine the probability of the various processes in nuclear reac- tions. Together wit h isotopi c spin , ther e ar e severa l othe r discret e characteristic s o f elementary particles . Amon g thes e ar e strangenes s S (mentione d above ) an d baryon charge B, which is simply the number of baryons in the system. I t is clear that B is additive. I n strong interactions bot h B and S ar e conserved. Thei r su m Y = B + S i s known as the hypercharge. *> These transformations are related to conservation of charge (see footnote on p. 423). 6> I t should be clear from the definition that isotopic spin has no connection with spatial spin. However, ther e is a forma l analog y betwee n thes e two quantities, since the y are related to th e locally isomorphic groups SO(3 ) and SU(2). Th e operato r of th e magnitud e of isotopi c spin is a Casimir operator of the group SU(2) (whence it is clear that this quantity is not additive). 3. UNITAR Y SYMMETR Y FO R HADRON S 427

Note tha t the presenc e o f discret e invariants of motio n enables one to explai n the stability of hadrons in strong interactions. For example, the law of hypercharge conservation implie s tha t a proto n canno t b e converte d int o a positro n wit h emission of a photon. From th e historical viewpoint , th e discovery of isotopic spin was the first step toward establishmen t o f mor e complicated symmetrie s i n th e class of elementar y particles. The next section is devoted to these generalizations.

§ 3. Unitary symmetry in the class of hadrons After man y attempts to generalize the concept of isotopic spin, Gell-Mann and (independently) Ne'eman suggeste d i n 196 1 what appears to be the mos t satisfac - tory model, base d on consideration o f the unitary group SU(3). In this model the additive quantum numbers are the projections of isotopic spin and the hypercharge. In addition, they considered the total isotopic spin (highest weight of the subgroup SU(2)) an d the so-called unitar y spin , define d b y two integer s (highes t weigh t o f the group SU(3)). 1. W e first consider baryons . Amon g th e baryons , w e distinguish certai n sub- systems of particles: I) the yl-hyperon (invariant of the isotopic group); 2) nucleons p and n (isotopic spin 1/2) ; i?-particle s 5" an d £T°(isotopic spin 1/2); 3 ) 1-particles J~, 2° and J+ (isotopi c spin 1) . In the accepte d physical terminology , w e have a singlet, tw o doublets , an d on e triplet . I n othe r words , w e hav e th e followin g representation of the group SU(2):

a =

m(/l) = 1115 , w(A 0 = 939 , m(5*) = 1318 , m(2)=119 3

(in arbitrary units), where the symbol N denotes the nucleon subsystem. We no w adop t a n importan t assumption . W e assum e th e existenc e o f som e idealized "ultrastron g interaction " [171] i n whic h th e masse s o f al l thes e eigh t particles ar e equal. The n th e representatio n a belong s to th e same eigenvalue o f some hypothetica l Hamiltonia n jf 0« No w not e tha t formul a (• ) coincide s wit h the restrictio n to SU(2) of the irreducible representatio n did2 o f the group SU(3), i.e. th e adjoin t representatio n o f thi s group . Can one conclude that this octet of particles has the property of SU(3)-symmetry ? 2. Le t u s analyz e th e representatio n d\dz formally . Thi s representatio n ha s a natura l realizatio n i n th e clas s o f matrice s x = ||x,>|*/=1.2, 3 wit h zer o trace . 428 APPENDIX III. ELEMENTAR Y PARTICLE S

The formul a (x fy) = trxy* define s th e inne r produc t o f th e vector s (matrices ) 7) x and y9 which is invariant under SU(3). Restrictin g the representation to SU(2) and acting on the indexes ij = 1,2 , we get the following invariant subspaces: 1x 0 0 1) matrice s o f type o X 0 — singlet ; 1 o 0 -2X ! °0 0 | 0 0 a 2) matrice s o f type 0 0 0 and 0 0 — tw o doublets; I a b o 0 0 0 x y 0 3) matrice s o f type Z -J C 0 triplet. 0 0 0 These subspace s ar e pairwis e orthogonal . I n addition , i f al l th e coordinate s (X9 a9 b, a, j8 , x9 y9 z) excep t one ar e equate d t o zero, w e get an orthogona l basi s for the entire representation space. This basi s is easily seen to be orthonormal if we set X = ( I /*/~6)o an d x = ( l j«/~2) t, an d leave al l other coordinate s as before. Hence we obtain the following parametrization of an arbitrary matrix x:8)

T/«/2 + a!

C = H£I I + £ 3 + £B X r 0 = i(£b~£n), r = £33- C Using tensor calculus relative to SU(3) (see § 45), the reader should have no diffi- culty in determining the weights of all basis functions in the space of matrices x, i.e. the eigenvalues of the operators C , TQ and Y. In particular, the operator TQ determines the weights for the subgroup SU(2) which acts on the indexes 1 an d 2. C is a scalar operator (Casimir operator for U(3)). The operator Y takes the value 0 for the singlet and the triplet, and the respective values ± 1 fo r the first and the second doublet.

7> A transformation u e SU(3 ) acts on the matrix x according to the formula x -» war1. *> If the wave function of the octet is expressed as T* ( n)f 9

3. We have now constructed a formal schem e whereby the wave function of the octet can be identified wit h the vector JC . The operator T 0 is identified wit h the pro- jection of isotopic spin. Y is identified with hypercharge (in agreement with experi- mental data) . The operato r Q = C — E n i s identifie d wit h the electric charge o f the particle. We have thus obtained an interpretation of the most important addi- tive physical quantities within the group SU(3). We now adopt th e assumption o f SU(3)-invarianc e (i n actua l fact , eve n U(3) - invariance, but this difference is inessential). One result of this assumption i s con- servation law s fo r th e Casimi r operator s o f th e grou p SU(3) . I n particular , th e quadratic Casimir operator can be identified (i n a suitable normalization) with the Hamiltonian /V I t is, however, essential tha t /Q b e invariant for SU(3)-symmetr y only, i.e. for the idealized "ultrastrong" interaction. 4. W e no w conside r a n importan t question : removal of degenerac y relativ e t o mass, i.e. determination o f the true difference betwee n the baryo n masses. Le t us look fo r th e true Hamiltonia n a s f = f 0 + ^F . The quantit y Ar i s determine d by the formula

3 3 AT = B 2 X&XQ + S Z I *3 * **3 > (** ) i=l /= i where the ba r denotes comple x conjugation . Thu s i t i s assumed that , unlik e th e ultrastrong interaction, a strong interaction mainly affects the index/ = 3 . Setting all coordinates except one in the matrix x equal to zero, we get the corresponding mass increment Am for each elementary particle. A simple computation yields the following result s :9)

Jm(A)=i(e + 8), Am(N) = e 9 Am(E) = d, Am(Z)~0. This i s th e so-calle d "Gell-Mann—Okub o mas s formula. " Usin g th e tabl e o f masses given before, the reader will easily verify that this formula furnishes a good approximation wit h e = 12 5 and 5 = —254 . We hav e thus classifie d th e octe t o f baryon s (A,N,3,2) relativ e to th e grou p SU(3). Together with this octet, we have the corresponding antiparticle octet. Both octets are equivalent relative to SU(3).10) 5. Th e classificatio n o f meson s i s analogous . Her e w e distinguis h tw o octets : I. Octet of pseudoscalar mesons. Relative t o th e isotopi c grou p SU(2), thi s octe t splits into the following subsystems : 1 ) ^-meson (singlet) , 2 ) AT-meson s and thei r antiparticles (tw o doublets) , 3 ) zr~meson s (triplet) . II . Octet of vector mesons. Relative t o SU(2) , thi s octe t split s int o th e followin g subsystems : 1 ) ^>-meso n

9 > The coefficient 2/3 is the square of the normalizing factor in the equality x3Z = (—2 / v'T) o. 10> I n view of th e special rol e of th e octe t (th e adjoin t representatio n o f SU(3)) , th e entir e theory of unitary symmetry has been named the Gell-Mann Eightfol d Wa y (Buddha's Eightfol d Path to cessation of pain). 430 APPENDIX III. ELEMENTAR Y PARTICLES

(singlet), 2 ) if *-mesons an d their antiparticles (two doublets), 3 ) p-mesons (trip- let).") For mesons, as before, the Gell-Mann—Okubo formul a holds , except tha t the expression (**) now gives the correction to the squared mass, Ail*2). Note that the mesons of th e octet ar e symmetric unde r transformation to antiparticles . Hence e = 5 . The Gell-Mann—Okubo formula is in good agreement with experiment.

§ 4. Hie discovery of the 0-particle Apart from the octets described above, physicists kne w of the existence of the following subsystems of baryons, classified with the aid of the isotopic group SU(2): 1) i?*-hyperons (doublet), 2 ) JFMiyperon s (triplet), 3 ) J-hyperons (quadruplet) . These form a reducible representation of SU(2):

8' =

The attempt to embed this representation in a representation of the group SU(3) led to the study t>f the latter's irreducible representation 8 = df (symmetrize d cube of the vector d{). However, the restriction of 8 to SU(2) does not coincide with d':

8 =

It was natural to conjecture that the singlet e2,e3 be an orthonormal basis in unitary three- space. The n th e symbol s (ijk) = eseje^ ij,k = 1,2,3 , wher e th e factor s e s ar e assumed to commute , for m a basi s in the space of th e representation d. Le t us arrange these symbols as in the following diagram, known as an "Egyptian pyra- mid":

(333) (332) (331) (322) (321 ) (311 ) (222) (221 ) (211 ) (111 )

The row s o f th e diagra m correspon d t o th e irreducibl e representation s o f th e

n> Th e octet of vector mesons is usually completed to a none t by attaching an o>-meso n (see, for example, 1183]). 5. PROBLEMS 431 subgroup SU(2). Its apex corresponds to a singlet, denoted by 0. Th e mass correc- tion is given b y the formula

where X& ar e th e coordinate s o f a symmetri c tenso r relativ e t o th e basi s (i/Jfc) . Hence we get the following formula:

4»t(i)~0, Jm(2*)» 4 dm(8*)^2d, dm(Q)**3d 9 giving an "equidistance law" for the rows of the Egyptian pyramid. In other words, the mass correction is proportional t o the number of indexe s 3 in th e symbol (jjk). This equidistanc e la w wa s adequately satisfie d fo r previousl y know n particles . The experimental discovery of the £?-hyperon in 1964 [174] also corroborated all the theoretical assumptions. Thus the classification of elementary particle s relative to SU(3) ha s played th e same rol e as the periodic table in the discovery of ne w ele- ments. The SU(3)~symmetr y hypothesi s ha s also le d to th e discovery o f severa l othe r properties of hadron s (relations betwee n magneti c moments of baryons , strengt h of meson-baryo n interactions , probabilitie s of nuclea r reactions). Fo r details w e refer the reader to the specialized literature. See, for example, [171], [177] [182] and [183],

§ 5. Problems SU(3)~symmetry does not account for such important characteristics of particles as baryon number (only Y « B - f S is defined), spatial spin (the group SO(3, JR)), and relativisti c invarianc e (iahomogeneou s Lorent z group). Thi s ha d le d t o th e search for larger symmetry groups.l2) I n particular, the compact group SU(6) and the noncompac t grou p SL(2,C) x SL(2,C ) have been investigated. Fo r th e diffi - culties involved here , the reader can consult, for example, [182]. However, even in the framework of SU(3) there are some ambiguities. The tensor algebra of the group SU(3) has two generators, d\ an d dz (vector and bivector). Ar e thes e representation s physicall y meaningful ? I f the y are , on e ca n hypothesize a "protomatter " fro m whic h th e know n elementar y particle s can b e built u p (followin g th e law s o f tenso r products) . Th e physicist s hav e name d d x and d z "quark " and "antiquarie s respectively . B y the general formula s in § 3 of this appendix, these particles must have fractional charges and hypercharges (with denominator 3) relative to the nucleons. Such particles have not yet been detected

l2 > Apar t from the group SU(3), i.e. the Lie algebra A Zf the algebras Bz an d Gt hav e also been considered {182]. However, A z yields better results. 432 APPENDIX III. ELEMENTAR Y PARTICLE S

experimentally. From the mathematical viewpoint , adoption of SU(3 ) unavoidably implies adoption of d\ an d d2» but it is possible that only the group SU(3)/Z, where Z i s the center of SU(3), is physically meaningful . Th e representations d\ an d cf e are then automatically eliminated. 13) The theor y of unitar y symmetr y came int o existenc e o n th e basi s of scan t ex - perimental data, which as yet provide no idea of the dynamics of th e elementary particles. However , eve n thes e ambiguou s indications hav e bee n successfull y interpreted within the framework of the theory of groups and their representations. This gives good reaso n to suppos e tha t th e law s of grou p theor y ar e intimatel y connected with the laws of the structure of matter.

13> An argument in favor of the actual existence of quarks and antiquaries is the special role of the third tensor index, underlying the derivation of the Gell-Mann—Okubo formulas. It would appear that if, and dt ar e real vector fields whose components experience different perturbations when one goes from ultrastrong to strong interactions. Note, moreover, that the generators E, 7 of the group SU(3) may be expressed in terms of creation and destruction operators for quarks and antiquaries (see § 57). BIBLIOGRAPHY*

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B. PAPERS ON THE THEORY OF GROUPS AND THHR REPRESENTATION S

1. D. Ado 58. On representations of finite continuous groups by linear substitutions, Izv. Fiz.-Mat. Obsc . Naucn.-Issled. Inst. Mat. Meh. Kazan. Univ (3) 7 (1934/35), 3-43. (Russian). F. A. Berezin •59. Laplace operators on semisimple Lie groups, Trudy Moskov. Mat. Obsc. 6 (1957), 371- 463; Englis h transl , Araer . Math . Soc. Trans l (2 ) 21 (1962) , 239-339. M R 19,867 ; 27 #245. 60. Letter to the editors, Trudy Moskov. Mat. Obsc. 12 (1963), 453-466= Trans . Moscow Math. Soc. 1963, 510-522. MR 28 #3335. 61. Several remarks on the associative hull of a Lie algebra, Funkcional Anal , i Prilozen . 1 (1967), no. 2,1-14 = Functiona l Anal. Appl 1 (1967), 91-102. MR 36 #2750. F. A. Berezin and I. M. Gd'fand 62. Some remarks on the theory of spherical functions on symmetric Riemannian manifolds, Trudy Moskov . Mat . Ob§5 . 5 (1956) , 311-351 ; Englis h transl , Amer . Math . Soc . Transl. (2) 21 (1960), 193-238. MR 19,152; 27 #1910. BIBLIOGRAPHY 437

F. A. Berezin, I. M . Gel'fand, M. I. Graev and M. A. Naimark 63. Group representations, Uspeh i Mat . Nauk 1 1 (1956), no. 6 (72), 13-40; English trans!., Amer. Math. Soc. Trans!. (2) 16 (1960), 325-353. MR 19,662; 22 #8367. G. Birkhof f 64. Analytical groups, Trans . Amer. Math. Soc. 43 (1938), 61-101. 65. Representability of Lie algebras and Lie groups by matrices, Ann. of Math. (2) 38 (1937), 526-533. 66. Lie groups simply isomorphic with no linear group,.Bull. Amer. Math . Soc. 42 (1936), 833-888. A. Borel and C. Chevalley 67. The Betti numbers of the exceptional groups, Mem. Amer. Math . Soc. No. 1 4 (1955), 1—9. M R 16,996. F. Bruhat •68. Travaux de Harish-Chandra, Seminair e Bourbak i 1956/57 , Expos e 143 , Secretaria t mathematique, Paris , 1959 ; Russian transL , Mathematik a 6 (1962), no. 5,43-50. M R 2811090. 69. Sur les representations induites des groupes de Lie, Bull . Soc. Math. France 84 (1956), 97-205. M R 18,907. fi. Carta n 70. Sur la structure des groupes de transformations finis et continus, These, Univ. de Paris, 1894; 2nd ed., Librairie Vuibert, Paris, 1933. 71. Les groupes projectifs qui ne laissent invariante aucune multiplicity plane. Bull. Soc. Math. France 41 (1931), 53-96. 72. Les groupes reels simples et continus, Ann. Sci.«£cole Norm. Sup. 31 (1914), 263-355. 73. Les tenseurs irreductibles et les groupes simples et semisimples, Bull. Sci. Math. 49 (1925), 130-152. P. Cartier •74. Representations des groupes de Lie {d'apres Harish-Chandra), Seminaire Bourbak i 1953/54, Expose 96, Secretariat mathematique, Paris, 1959; Russian transl., Matematik a 6 (1962), no. 5, 33-41. M R 28 #1087. 75. On H. WeyVs character formula, Bull. Amer . Math . Soc. 67 (1961), 228-230; Russian transl., Matematika 6 (1962), no. 5,139-141. MR 26 #3828. P. Cartier and J. Dixmier •76. Vecteurs analytiques dans les representations des groupes de Lie, Amer. J. Math. 80 (1958), 131-145. MR 20 #924. H. Casimir and B. L. van der Waerden 77. Algebraischer Beweis der vollstandigen Reduzibilitdt der Darstellungen halbeinfacher LieschenGruppen, Math . Ann. Ill (1935) , 1-12. C. Chevalley 78. The Betti numbers of the exceptional Lie groups, Proc . Internat. Congress Math. (Cam- bridge, Mass. , 1950) , vol. 2 , Amer . Math . Soc. , Providence , R . L , 1952 , pp. 21-24. MR 13,432. 79. Invariants of finite groups generated by reflections, Amer. J. Math. 77 (1955), 778-782. MR 17,345, 1436. 80. Sur certaines groupes simples, Tohoku Math. J. (2) 7 (1955), 14-16. MR 17,457. C. Chevalley and S. Eilenberg 81. Cohomology theory of Lie groups and Lie algebras. Trans. Amer. Math. Soc. 63 (1948), 85-124. M R 9,567. J. Dixmier •82. Quelques resultats d'Harish-Chandra, Seminair e Bourbak i 1951/52 , Expose s 50-58 , 438 BIBLIOGRAPHY

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K. R. Parthasarathy, R. Ranga Rao and V. S. Varadaraja n •139. Representations of complex semi-simple Lie groups and Lie algebras, Ann . of Math . (2) 85 (1967), 383-429. MR 37 #1526. A. M. Perelomov and V. S. Popov 140. Casimir operators for U(n ) and SU(JI), Jadernaj a Fiz . 3 (1966) , 924-93 1 = Sovie t J . Nuclear Phys. 3 (1966), 676-680. MR 34 #5446. 141. Casimir operators for the orthogonal and symplectic groups, Jadernaja Fiz . 3 (1966) , 1127-1134 = Sovie t J. Nuclear Phys. 3 (1966), 819-824. MR 34 #5447. F. Peter and H. Weyl 142. Die Vollstdndigkeit der primitiven Darstellungen einer geschlossenen kontinuirelichen Gruppe, Math . Ann. 97 (1927), 737-755; Russian transl. , Uspehi Mat . Nauk 2 (1936), 144-160. P. K. Rasevskii 143. On some fundamental theorems of the theory of Lie groups, Uspehi Mat. Nauk 8 (1953), no. 1 (53) , 3-20. (Russian) MR 15,9. 144. The theory ofspinors, Uspeh i Mat . Nauk 1 0 (1955), no. 2 (64), 3-110; English transl. , Amer. Math. Soc. Transl. (2) 6 (1957), 1-110. MR 17,124. 145. Associative hyper-envelopes of Lie algebras, their regular representations and ideals, Trudy Moskov. Mat. Obsc. 15 (1966), 3-54 = Trans. Moscow Math. Soc. 1966 , 1-60 . M R 35 #4269. 0. Schreie r 146. Abstrakte kontinuierliche Gruppen, Abh. Math. Sem. Univ. Hamburg 4 (1925), 15-32. 1. E. Segal *147. An extension of Plancherel's formula to separable unimodular groups, Ann. of Math . (2) 52(1950), 272-292. MR 12,157. Seminaire "Sophus Lie" 148. Theorie des algebres de Lie. Topologie des groupes de Lie. Secretaria t mathematique , Paris, 1955; Russian transl., II, Moscow, 1962 . MR 17,384. A. I. Sirota and A. S. Solodovnikov 149. Non-compact semi-simple Lie groups, Uspehi Mat. Nauk 1 8 (1963), no. 3 (111), 87-144 = Russian Math. Surveys 18 (1963), no. 3,85-140. MR 27 #5862. R. Steinberg 150. A general Clebsch-Gordan theorem, Bull, Amer. Math. Soc. 67 (1961X 406-407; Russian transl., Matematika 6 (1962), no. 5,142-143. MR 23 #A971. M. H. Stone 151. Linear transformations in Hilbert space. III. Operational methods and group theory, Proc. Nat. Acad. Sci. U.S.A. 16 (1930), 172-175. 152. On one-parameter unitary groups in Hilbert space, Ann. of Math. (2) 33 (1932), 643-648. T. Tannaka 153. Uber den Dualitdtssatz der nichtkommutativen topologischen Gruppen, Tohok u Math . J. 45 (1938), 1-12 . N. Tatsuuma 154. A duality theorem for locally compact groups, J. Math. Kyoto Univ . 6 (1967), 187-293 . MR 36 #313. J. Tits 155. Classification of algebraic semisimple groups, Proc . Sympos. Pure Math., vol. 9, Amer. Math. Soc., Providence, R. I., 1966, pp. 33-62; Russian transl. , Matematik a 1 2 (1968), no. 2, 110-143. MR 37 #309. 442 BIBLIOGRAPHY

B. L. van der Waerdcn 156. Die Klassifikation der einfachen lieschen Gruppen, Math. Z. 37 (1933), 446-462; Russian transL, Uspehi Mat. Nauk 4 (1937), 258-274. H. Weyl 157. Theorie der Darstellungen kontinuierlicher halbeinfacher Gruppen durch lineare Trans- formationen. 1, Math. Z. 2 3 (1924), 271-304; II , 2 4 (1925), 328-395; Russian transl . of selected parts, Uspehi Mat. Nauk 4 (1937), 201-246. A. Young 158. On quantitative substitutional analysis, Proc. London Math . Soc. 33(1900), 97-146; 34 (1902), 361-397. D. P. Zelobenko *159. Harmonic analysis of functions on semisimple Lie groups. I, Izv. Akad. Nauk SSS R Ser. Mat. 2 7 (1963) , 1343-1394 ; Englis h transl. , Amer . Math . Soc . Transl. (2 ) 54 (1966), 177-230. MR 31 #2353. 160. Description of all irreducible representations of an arbitrary connected Lie group, Dokl. Akad. Nau k SSS R 13 9 (1961), 1291-129 4 = Sovie t Math . Dokl. 2 (1961), 1076-1079. MR 24 #A1969. 161. Classical groups. Spectral analysis of finite-dimensional representations, Uspehi Mat . Nauk 1 7 (1962), no. 1 (103), 27-120=Russian Math. Surveys 17 (1962), no. 1,1-94. MR 25 #129. 162. The theory of linear representations of complex and real Lie groups, Trudy Moskov. Mat. Obsc. 12 (1963), 53-98=Trans. Moscow Math. Soc. 1963,57-110. MR 29 #2330. •163. A description of a certain class of Lorentz group representations, Dokl. Akad. Nauk SSSR 121 (1958), 586-589. (Russian) MR 21 #2920. *164. Linear representations of the Lorentz group, Dokl. Akad. Nauk SSS R 12 6 (1959), 935- 938. (Russian) MR 22 #906. *165. Symmetry in the class of elementary representations of semisimple complex Lie groups, Funkcional. Anal, i Prilozen. 1 (1967), no. 2,15-38 = Functional Anal. Appl. 1 (1967), 103-121. MR 36 #2743. *166. The analysis of irreducibility in the class of elementary representations of a complex semi- simple Lie group, Izv. Akad. Nauk SSS R Ser. Mat 32 (1968), 108-133 = Math . USSR Izv. 2 (1968), 105-128. MR 37 #2906. •167. Operational calculus and theorems of the Paley-Wiener type for a semisimple complex Lie groups, Dokl. Akad. Nauk SSSR 170 (1967), 1243-1246 = Soviet Math. Dokl. 8 (1967), 1348-1352. MR 34 #7728. •168. Analogue of the Cartan-Weyl theory for oo-dimensional representations of a semisimple complex Lie group, Dokl. Akad. Nauk SSSR 175 (1967), 24-27 = Soviet Math. Dokl. 8 (1967), 798-902. MR 35 #5553. D. P. Zelobenko and M. A. Nalmark •169. A characterization of completely irreducible representations of a semisimple complex Lie group, Dokl. Akad. Nauk SSSR 17 1 (1966), 25-28 = Soviet Math. Dokl. 7 (1966), 1403- 1406. MR 35 #1715. C. PAPER S ON APPLICATIONS OF GROUP THEORY IN THEORETICAL PHYSIC S V. Bargmann and M. Moshinsky 170. Group theory of harmonic oscillators. I, II, Nuclear Phys. 18 (1960), 697-712; 23 (1961), 177-199. MR22#11942;#13201. V. B. Beresteckii 171. Dynamic symmetries of strongly interacting particles, Uspehi Fiz . Nauk 85 (1965), 393- 444 = Soviet Physics Uspekhi 8 (1965), 147-176. BIBLIOGRAPHY 443

F. J. Dyson 172. Mathematics in the physical sciences, Sci. Amer. 211 (1964), no. 3,128-146. MR 29 #2146. V. A. Fock 173. Zur Theorie des Wasserstoffatoms, Z. Physik 98 (1935), 145-154. W. B. Fowler and N. P. Samios 174. The omega-minus experiment, Sci. Amer. 211 (1964), no. 4,36-45. I. A. Malkin and V. I. Man'ko 175. Symmetry of the hydrogen atom, Jadernaja Fiz . 3 (1966) , 372-382 = Soviet J. Nuclear Phys. 3 (1966), 267-274. MR 34 #3934. M. Moshinsk y 176. Bases for the irreducible representations of the unitary group and some applications, J . Mathematical Phys. 4 (1962), 1128-1139. MR 32 #7052. Ja. A. Smorodinskii 177. The unitary symmetry of elementary particles, Uspehi Fiz. Nauk 84 (1964), 3-36=Soviet Physics Uspekhi 7 (1964), 637—655. MR 31 #4467. J. J. de Swart 178. The octet model and its Clebsch-Gordan coefficients. Rev . Moder n Phys. 35 (1963), 916- 939. MR 29 1983. N. Ja. Vilenkin and Ja. N. Smorodinskii 179. Invariant expansions of relativistic amplitudes, Z . £ksper. Teoret. Fiz. 46 (1964), 1793- 1808 = Soviet Physics JETP19 (1964), 1209-1218. MR 31 #6572 . E. P. Wigner 180. Symmetry and conservation laws, Physics Today 17 (1964), no. 3,34-40.

D. COLLECTION S OF PAPERS 181. Elementary particles and compensating fields, "Mir**, Moscow, 1964. (Russian) [Collection of translations, edited by D. Ivanenko.] 182. Group theory and elementary particles, "Mir", Moscow, 1967 . (Russian) [Collection o f translations, edited by D. Ivanenko. ] 183. High-energy physics and the theory of elementary particles, "Naukova Dumka** , Kiev, 1967. (Russian) MR 36 #4872. SUBJECT INDEX algebra central series associative, 26 in a Lie algebra, 248 Clifford, 38,336 in a Lie group, 254 commutator, 54,57 centralizer, 106 group, 63 cnamber,Weyl,300 Lie, 16 character bicomplex, 115 of an abelian group, 59 classical, 277 of a representation, 81,210 commutative, 18 primitive, 212 compact, 259 class of conjugate elements, 81 complex, 39 coefficients, Clebsch-Gordan , 65,231 complex semisimple, real form of, 27 5 commutator complexification, 39 in a Lie algebra, 16 regular, 110 in a group, 16 exceptional simple, 277 in an associative algebra, 16, 27, 63,156 linear, 27,246 complexification nilpotent, 242,247 of a complex Lie algebra or group real form, 39,275 (bicompiexification), 115 reductive, 241,256 of a Lie algebra or group, 39 semisimple, 241,253 regular, 110,297,306 simple, 241 component exceptional, 277 of a representation, 47 solvable, 242,243,246,252 of the identity, 9 universal enveloping, 63,156,169,368 connected component of the identity, 9 automorphism of a group, 7 coset in a group, 37 inner, 7 covering, universal, 20,85,295 involutory, 276 criterion mirror in SOOO, 331 of 8emisimplicity, 253 averaging, Hurwicz, 71 of solvability (Cartan), 252

Cartan-Weyl,265,272 decomposition Gel'iand-Cetlin, 189 Cartan, 276 bicomplexification, 115 Cartan-Weyi, 285,348,349 block algebra, 307 Fitting, 250,262 Gauss canonical parameters, 16 generalized, 288,317 center inGL(n),29 of a compact Lie group, 185,295 in semisimple complex Lie groups, 287 of a group, 21,240 GraminGL(n),30 of a Lie algebra, 240 Iwasawa, 291 of a universa l envelopin g algebra , 158 , polar 169 in a quaternion algebra, 35 central element in a group, 21 inGL(«),28 444 INDEX 44 5

derivation group 6 in a Lie algebra, 238,254 abelian, 7,254,255 inner, 238,254 algebraic, 282 determinant, Weyl, 234 analytic, 14 diagram commutative, 7 Dynkin,272,277 compact, 69 weight, 120 complete linear GL(n), 10,26,37 Young, 140 complex orthogonal 0(n, 0,3 1 differential cyclic, 84 of a representation, 88 discrete, 9 of <*(<*), 202 finite, 7 division algebra, quaternion, 35 fundamental, 21,295 automorphisms, 35 Lie, 13 divisor bicomplex, 115 discrete, 21 compact, 69 normal, 8,239 complex, 39 connected, 13 element, regular, in a Lie algebra (group), hypercompact, 113,306 295 local, 107 exponential, 26 nilpotent, 254 mapping (exp), 15,26,291 reductive, 254,256 factor algebra, 240 semisimple, 254,256 factor group, 240 simple, 254,256 factor space, 36 solvable, 254 fiber space , 36 linear, 10 form classical, 33 bilinear, 30 locally connected, 13 Killing-Cartan, 252,253,272 locally euclidean, 12 covariant (contravarient), 42 matrix, 10 differential, 22 nilpotent, 255 right (left) invariant, 22 orthogonal 0(n), 11,31,32,69 Hennitian (sesquilinear), 30 orthogonal unitary OU(ft), 32 multilinear, 42 parametric, 12 real, of Lie algebras (groups), 39,275, Poincare, 21,295 276 proper orthogonal SO (a), 32,330 symmetric, 42 proper unitary SU(»), 32,36,37 formula pseudo-orthogonal 0(p,q), 32,6 9 FreudenthaTs, 355 pseudo-unitary U(p, q), 32,69. Kostant's, 365 rotation SO(3,J0,7,35,100 Weyl's for characters, 214,220,221, 362 simply connected, 20 function, wave, 422 solvable, 254 spinor Spin(n), 38,335 symmetric S(»), 83 generator symmetry, of an operator, 55 of a group, 13 symplectic Sp(n), 31,32,324 of an irrational toroid, 293 symplectic unitary Sp U(n), 32 of an irrational torus, 85 unimodukrSL(ii), 27,37 geodesic unitary U(n), 28,32,37,69,84,85 in a Lie group, 24 Weyl, 120,299 compact, 291,292 446 SUBJECT INDEX

harmonics, spherical operator solid, 100 Bose,153 surface, 100,390,391 Casimir, 90,91,158,159,368 homeomorphism, 36 Hamiltonian, 55,422 homomorphism, 18 innmtesimal (generating), 88 intertwining, 49 ideal Laplacian,55 analytic, antianalytic, 116 mass-energy, 154 direct sum, 115 orbit, 21 in a Lie algebra, 239 of a point (trajectory), 46,51 in an associative algebra, 60,280 oscillator, harmonic, 153 indicator system, 179,180, 228, 318 , 319, 376 parameters inversion, 7 canonical, 16 isomorphism Cayley-Klein, 35 local, 19 Parseval identity, 75 of SO(3, R) and SU(2), 35,239 polynomial, harmonic, 101 of Lie groups, 18 principle of analytic continuation, 112 kernel of a homomorphism, 18 of complete reducibility, 70 of duality (for completely reducible lemma, Schur's, 56 matrix algebras), 61 continuous analog, 421 product direct matrix, 10 of groups, 11 Cartan structure, 268 of topological spaces, 36 Clebsch-Gordan, 224,307 Kronecker, 307 Hermitian (selfadjoint), 27 locally direct, of topological groups, 115 irreducible, 59 semidirect, 415 Pauli, 33 tensor positive definite, 27 of operators, 41 reducible, 59 of representations of Lie algebras, 8 9 trace, 24,27 of representations of groups, 48, 224 , transformation, 22 383 matrix elements Young, 136,314 of a group, 52 orthogonality property, 73 SU(2),96,97 quaternion of a representation, 52 complex, 33 measure conjugate, 33 Haar, 24 norm of, 35 on a compact Lie group, 69 real, 35 of a set (volume), 24 quotient topology, 36 on a group, 24 on a right (left) invariant group, 24 radical of a Lie algebra, 256 metric, Riemannian, 23 relations, Clebsch-Gordan, 303 multiplet of Lie groups, 342 representation multivector, 135 analytic (antianalytic), 110,117 cyclic with highest weight, 350 normalizer, 106 faithful (isomorphic), 65,70,298 SUBJECT INDEX

induced, 414 discrete, 9 multiplicity of, 48 locally compact, 69 of a group, 45,46 structure constants, 17 adjoint, 239 subalgebra completely reducible, 47 Cartan, 261,285,294,298 contragredient, 48,57 automorphisms, 270 degree (dimension) of, 47 of a Lie algebra, 17 dual, 57 subgroup quasiregular, 50 analytic, 17,107 reducible, 47 Borel,398 regular, 50 Cartan, 288,294,298 semiredicuble, 104,394,408 commutator, 255 topologically irreducible, 53 invariant, 8 unitary, 49 maximal compact, 298 of a Lie algebra, 88,238 normally imbedded, 317,377 adjoint, 157 one-parameter, 15 contragredient, 89 of a generating operator, 87,88 8emireducible, 104 regularly imbedded, 377 spinor, 101,333,380 stable, 50 representations, equivalence of, 47 subspace root subspace, 250 invariant, 47 roots irreducible, 47 fundamental system, 302 nontrivial, 47 in a Lie algebra, 250 sum, direct, of ideals, 115 of a system o f semisimple Lie algebras, symmetrizer, Young, 142 262,268,271 central, 143 of a system of simple Lie algebras , 27 8 symmetry about a point, 291. simple, 266 tensor, 23,41 semigroup, 136 covariant (contra variant), 41 series metric, 23 Campbell-Hausdorff, 17 mixed, 42 Fourier skew-symmetric, 24,42 on complex Lie groups, 71,75 on homogeneous spaces with compact symmetric, 42 groups of motions, 81 theorem with characters, 81 Birkhoff-Witt,158 set Bolzano-Weierstrass, 68 abstract, 6 Bumside, 60,82 compact in a metric space, 68 Cartan, 298 of reducibl e (irreducible ) matrices , 5 9 Chevalley,366 signature, 134,140,315 Engel, 246 space global, 70 homogeneous, 44 Hahn-Banach, 77 universal, 50 Heine-Borei, 68 linear, 10 Kirillov, 418 conjugate (dual), 40 Levi-Mal'cev, 256 topological, 53 Lie, 244,245 topological, 9 global, 255 compact, 68 Naimark,410 connected, 9 Paley-Wiener,406 448 8UBJECT INDEX

Peter-Weyl (fundamenta l approxima - dominant, 225 tion), 70 field o n a Lie group, 22 Stone-von Neumann, 411,412 covariant (contravariant), 22 Stone-Weierstrass, 74 highest (lowest ) weight , 93 , 121 , 127 , Wedderburn, 61 312,350 theory integral, 359 of duality, 6^, 307 invariant, 47 of representation of finite groups, 82 strictly dominant, 360 torus (toroid) vacuum, 153 maximal, 293 two-dimensional, 36 weight trajectory of a point, 46,51 highest (lowest), 93, 121, 127 , 312 , 35 0 translations, 7 inductive, 132,314 infinitesimal, 127,184,31 5 universal of a Lie algebra, 249 covering, 20 Weyl chamber, 300 of a compact Lie group, 85,295 Weyl reflection, 95,99 enveloping algebra, 63,156 winding of a torus , irrational , 45 , 46 , 85 . center, 169,368 105 linear group, 341 £-invariants, metho d of , 130 , 137 , 138 , vector 227,375 analytic, 107 Z-multiplier, 183 differentiable, 88 Other Titles i n This Series , jr L , (Continued from the front of this publication) 106 G.- C Wen , Conformal mapping s and boundary valu e problems, 199 2 105 D . R. Yafaev, Mathematica l scatterin g theory: Genera l theory , 199 2 104 R . L . Dobrushin, R . Kotecky, an d S. Shlosman , Wulf f construction: A globa l shap e from loca l interaction, 199 2 103 A . K . Tsikh, Multidimensional residue s and thei r applications, 199 2 102 A . M. Il'in , Matching of asymptotic expansion s of solutions of boundar y valu e problems, 199 2 101 Zhan g Zhi-fen , Din g Tong-ren , Huan g Wen-zao , an d Don g Zben-xi , Qualitativ e theory of differentia l equations , 199 2 100 V . L. Popov, Groups, generators, syzygies, and orbits in invarian t theory, 199 2 99 Nori o Shimakura, Partial differentia l operator s of elliptic type , 199 2 98 V . A. Vassiliev , Complements of discriminants of smooth maps : Topolog y and applications, 199 2 (revised edition, 1994 ) 97 Itir o Tamura, Topology of foliations: A n introduction , 199 2 96 A . I. Markushevich, Introduction to th e classical theor y of Abelia n functions , 199 2 95 Gnangchan g Dong, Nonlinea r partial differentia l equation s of second order , 199 1 94 Yu . S. Il'yashenko , Finiteness theorems for limit cycles, 199 1 93 A . T . Fomenko an d A. A. Tuzhilin, Element s of the geometr y an d topology of minimal surface s i n three-dimensional space, 199 1 92 E . M. Nikishi n an d V . N. Sorokin, Rationa l approximation s and orthogonality, 199 1 91 Mamor u Mimur a an d Hirosi Toda, Topology of Lie groups, I and II , 199 1 90 S . L. Sobolev, Some applications of functional analysi s in mathematical physics, third edition, 199 1 89 Valeri i V . Kozlov and Dmitril V . Treshchev, Billiards : A geneti c introduction t o th e dynamics of systems with impacts, 199 1 88 A . G . Khovansku, Fewnomials, 199 1 87 Aleksand r Robertovic h Kemer , Ideals of identities of associativ e algebras, 199 1 86 V . M. Kadets an d M. I. Kadets, Rearrangement s of serie s in Banac h spaces, 199 1 85 Miki o Ise an d Masani Takeuchi , Li e groups I, II, 199 1 Z4 Da o Tron g Th i and A. T. Fomenko, Minimal surfaces , stratifie d multivarifolds , and the Platea u problem, 199 1 83 N . I. Portenko, Generalized diffusion processes , 199 0 82 Yasutak a Sibuya , Linea r differentia l equation s in the complex domain : Problem s of analyti c continuation, 199 0 81 I . M. Gelfan d an d S. G . Gindikin , Editors, Mathematical problem s of tomography, 199 0 80 Junjir o Nognchi an d Taknshiro Ochiai , Geometric functio n theor y i n severa l comple x variables, 1990 79 N . I. Akhiezer, Elements of the theory of elliptic functions, 199 0 78 A . V . Skorokbod, Asymptotic methods of the theory of stochastic differential equations , 198 9 77 V . M. Filippov , Variational principle s for nonpotential operators , 198 9 76 Philli p A. Griffiths , Introductio n to algebraic curves, 198 9 75 B.S . Kashi n and A. A. Saakyan, Orthogonal series, 198 9 74 V . I. Yodovich, The linearization method in hydrodynamical stabilit y theory , 198 9 73 Yu . G . Reshetnyak, Space mappings with bounded distortion, 198 9 72 A . V. Pogorelev, Bendings of surface s an d stability of shells, 198 8 71 A . S. Markus , Introduction to th e spectra l theor y o f polynomial operato r pencils, 198 8 70 N . I. Akhiezer, Lectures on integra l transforms , 198 8 69 V . N. Salff, Lattice s with uniqu e complements, 198 8 68 A . G . Postnikov , Introductio n to analyti c number theory, 198 8 67 A . G . Dragalin, Mathematica l intuitionism : Introductio n to proof theory, 198 8 (See the AMS catalo g fo r earlie r titles)