Selected Title s i n Thi s Serie s

175 R . L . Dobrushin, R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Contemporar y

Mathematical Physic s (F . A . Berezi n Memoria l Volume )

174 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Netsvetaev , Editors , Mathematic s i n

St. Petersbur g

173 V . Kharlamov , A . Korchagin , G . Polotovskii , an d 0 . Viro , Editors , Topolog y o f Rea l

Algebraic Varietie s an d Relate d Topic s

172 K . Nomizu , Editor , Selecte d Paper s o n Numbe r Theor y an d Algebrai c Geometr y

171 L . A. Bunimovich, B . M . Gurevich , an d Ya. B . Pesin , Editors , Sinai' s Mosco w Semina r o n

Dynamical System s

170 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s

169 S . G . Gindiki n an d E. B. Vinberg , Editors , Li e Group s an d Li e Algebras : E . B . Dynkin' s

Seminar

168 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s

167 V . V . Lychagin , Editor , Th e Interpla y betwee n Differentia l Geometr y an d Differentia l

Equations

166 0 . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume II I

165 Yu . Ilyashenk o an d S . Yakovenko, Editors , Concernin g th e Hilber t 16t h Proble m

164 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s

163 L . A . BokuV , M . Hazewinkel , an d Yu . G . Reshetnyak , Editors , Thir d Siberia n Schoo l

"Algebra an d Analysis "

162 S . G . Gindikin , Editor , Applie d Problem s o f Rado n Transfor m

161 K . Nomizu , Editor , Selecte d Paper s o n Analysis , Probability , an d Statistic s

160 K . Nomizu , Editor , Selecte d Paper s o n Numbe r Theory , Algebrai c Geometry , an d

Differential Geometr y

159 O . A . Ladyzhenskaya , Editor , Proceeding s o f th e St . Petersbur g Mathematica l Society ,

Volume I I

158 A . K . Kelmans , Editor , Selecte d Topic s i n Discret e Mathematics : Proceeding s o f th e

Moscow Discret e Mathematic s Semina r 1972-199 0

157 M . Sh . Birman , Editor , Wav e Propagation . Scatterin g Theor y

156 V . N . Gerasimov , N . G . Nesterenko , an d A . I . Valitskas , Thre e Paper s o n Algebra s an d

Their Representation s

155 0 . A . Ladyzhenskay a an d A . M . Vershik , Editors , Proceeding s o f th e St . Petersbur g

Mathematical Society , Volum e I

154 V . A . Artamono v e t al. , Selecte d Paper s i n K-Theor y

153 S . G . Gindikin , Editor , Singularit y Theor y an d Som e Problem s o f Functiona l Analysi s

152 H . Draskovicov a e t al. , Ordere d Set s an d Lattice s I I

151 L A . Aleksandrov , L . A. Bokut 7, an d Yu. G . Reshetnyak , Editors , Secon d Siberia n Winte r

School "Algebr a an d Analysis "

150 S . G . Gindikin , Editor , Spectra l Theor y o f Operator s

149 V . S . Afraimovic h e t al. , Thirtee n Paper s i n Algebra , Functiona l Analysis , Topology , an d

Probability, Translate d fro m th e Russia n

148 A . D . Aleksandrov , 0 . V . Belegradek , L . A . Bokut 7, an d Yu . L . Ershov , Editors , First Siberia n Winte r Schoo l 'Algebr a an d Analysis "

147 I . G. Bashmakova et al. , Nin e Paper s fro m th e Internationa l Congres s o f Mathematicians ,

1986

146 L . A . Aizenber g e t al. , Fiftee n Paper s i n Comple x Analysi s

145 S . G . Dalalyan e t al. , Eigh t Paper s Translate d fro m th e Russia n (Continued in the back of this publication)

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Contemporary Mathematical Physic s F. A. Berezin Memoria l Volume

Felix Alexandrovic h Berezi n (1931-1980)

American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 17 5

Advances in the Mathematical Sciences - 31

(Formerly Advances in Soviet Mathematics)

Contemporary Mathematical Physic s F. A . Berezi n Memoria l Volum e

R. L . Dobrushi n R. A . Minlo s M. A . Shub i n A. M . Vershi k Editors

52EM4J,

American Mathematica l Societ y Providence, Rhod e Islan d

http://dx.doi.org/10.1090/trans2/175

ADVANCES I N TH E MATHEMATICA L SCIENCE S EDITORIAL COMMITTE E

V. I. ARNOLD S. G. GINDIKIN V. P. MASLOV

Transla t ion edite d b y A . B . Sossinsk y

1991 Mathematics Subject Classification. P r i m a r y 22-XX , 47-XX , 81-XX .

ABSTRACT. Th e conten t o f th e volum e i s connecte d i n variou s way s t o th e scientifi c heritag e of F . A . Berezi n (1931-1980) , a n outstandin g Mosco w mathematicia n wh o tragicall y die d i n a n accident. F . A . Berezi n discovere d ne w notion s an d idea s i n mathematica l physics , representatio n theory, analysis , geometr y an d othe r area s o f mathematics. I n particular , h e introduced th e notio n of Grassmannia n analysi s ("supermathematics" ) an d a ne w notio n o f deformatio n quantization . The topic s o f paper s i n thi s volum e includ e quantization , invarian t integration , spectra l theory , representation theory , etc .

The boo k i s useful fo r graduate student s an d researcher s working i n various area s of mathematic s and mathematica l physics .

Library o f Congres s Car d Numbe r 91-64074 1 ISBN 0-8218-0426- X

ISSN 0065-929 0

Copying an d reprinting . Materia l i n this book may be reproduced b y any mean s for educationa l and scientifi c purpose s withou t fe e o r permissio n wit h th e exceptio n o f reproductio n b y service s that collec t fee s fo r deliver y o f documents an d provide d tha t th e customar y acknowledgmen t o f th e source i s given. Thi s consen t doe s no t exten d t o othe r kind s o f copying fo r genera l distribution , fo r advertising o r promotiona l purposes , o r fo r resale . Request s fo r permissio n fo r commercia l us e o f material shoul d b e addresse d t o th e Assistan t t o th e Publisher , America n Mathematica l Society , P. O. Bo x 6248 , Providence , Rhod e Islan d 02940-6248 . Request s ca n als o b e mad e b y e-mai l t o reprint-permissionOams.org.

Excluded fro m thes e provision s i s materia l i n article s fo r whic h th e autho r hold s copyright . I n such cases , request s fo r permissio n t o us e o r reprin t shoul d b e addresse d directl y t o th e author(s) . (Copyright ownershi p i s indicate d i n th e notic e i n th e lowe r right-han d corne r o f th e firs t pag e o f each article. )

© 199 6 b y th e America n Mathematica l Society . Al l right s reserved . The America n Mathematica l Societ y retain s al l right s

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 .

'm* Printed o n recycle d paper .

10 9 8 7 6 5 4 3 2 1 0 1 0 0 9 9 9 8 9 7 9 6

Contents

Foreword i x

Felix Alexandrovic h Berezi n ( A Brie f Scientifi c Biography ) R. A . MINLO S 1

On th e Functiona l Equatio n Relate d t o th e Quantu m Three-Bod y Proble m V. M . BUCHSTABE R AN D A . M . PERELOMO V 1 5

The Differentia l Calculu s o n Quantu m Linea r Group s L. D . FADDEE V AN D P . N . PYATO V 3 5

The Penros e Transfor m o n Fla g Domain s i n ^(CP 2) SIMON GINDIKI N 4 9

Defining Relation s Associate d wit h Principa l 5l(2)-subalgebra s P. GROZMA N AN D D . LEITE S 5 7

A Generalizatio n o f the Berezin-Lie b Inequalit y A. LAPTE V AN D Y U . SAFARO V 6 9

Quantization o n Para-Hermitia n Symmetri c Space s V. F . MOLCHANO V 8 1

Integral Operator s wit h Gaussia n Kernel s an d Symmetrie s o f Canonica l Commutation Relation s Yu. A . NERETI N 9 7

Ergodic Unitaril y Invarian t Measure s o n th e Spac e o f Infinit e Hermitia n Matrices GRIGORI OLSHANSK I AN D ANATOL I VERSHI K 13 7

Cogitations ove r Berezin' s Integra l V. P . PALAMODO V 17 7

Remarks o n th e Topolog y o f the Hilber t Grassmannia n M. A . SHUBI N 19 1

Asymptotic Completenes s I. M . SiGA L 19 9

Constructive Module s an d th e Reductivit y Proble m i n th e Categor y O D. P . ZHELOBENK O 20 7

Alik Berezi n i n th e Recollection s o f Friend s 22 5

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Foreword

Over fiftee n year s hav e passe d sinc e Feli x Alexandrovic h Berezi n died . Man y things hav e change d sinc e then , an d lif e toda y i s ver y differen t fro m wha t i t wa s at tha t time . Berezi n wa s amon g thos e wh o mos t ardentl y strive d fo r changes ; unfortunately, h e lived neithe r t o se e the da y whe n hi s ideas an d wor k ha d becom e popular an d widel y accepted , no r t o work i n a period whe n scientifi c contact s wit h friends an d colleagues dispersed i n the world at larg e had become routinely possible .

The editor s o f this collection , wh o knew F . A . Berezin ver y closely , have calle d on al l thos e wh o hav e worke d wit h hi m a t on e tim e o r another , studie d unde r hi s supervision, develope d hi s ideas, were his friends o r simply wante d t o be one of th e authors o f a volume devote d t o th e memor y o f this outstandin g scientist . Th e ide a of suc h a publicatio n attracte d man y mathematician s an d physicists . Th e resul t of thei r wor k i s collecte d i n tw o volume s o f th e presen t series , whic h represen t a wide spectru m o f topic s i n mathematica l physics—F . A . Berezin' s mai n are a o f research—and relate d branche s o f mathematics .

In discussing F . A. Berezin's scientific heritage , one should begi n by mentioning his two most significan t achievements : quantizatio n theor y an d supermathematics . But hi s nam e live s o n i n representatio n theory , statistica l physics , operato r the -ory, an d th e geometr y o f homogeneous spaces . Hi s paper s ar e remarkabl e b y thei r breadth o f insigh t an d thei r conceptua l character . W e ar e no w witnesse s o f th e powerful developmen t o f Berezin' s quantizatio n an d Berezin' s Grassmannia n anal -ysis (supermathematics) . Bu t t o a n observe r fro m th e outside , th e influenc e tha t Berezin exerte d o n mathematician s an d physicist s ma y perhap s no t b e to o obvi -ous, mor e precisely , th e rol e tha t h e playe d i n makin g physic s popula r amon g mathematicians an d mathematic s amon g physicists .

In thi s volume , th e first o f two , w e hav e assemble d paper s i n grou p represen -tation theor y an d severa l mathematica l paper s i n supermathematic s an d spectra l analysis.

It wa s no t eas y t o organiz e th e interactio n o f suc h a grou p o f mathemati -cians, disperse d i n man y citie s an d countries ; al l th e organizationa l an d technica l tasks involve d i n th e preparatio n o f thi s volum e wer e successfull y carrie d ou t b y N. D . Vvedenskaya , t o who m th e editor s expres s thei r gratitude .

We hop e tha t thi s volum e wil l expres s ou r dee p respec t fo r th e memor y o f F. A . Berezi n an d wil l promote hi s work an d ideas .

R. Dobrushin, 1 R . Minlos , M. Shubin, A . Vershik 1 The editor s wan t t o announc e tha t Rolan d L'vovic h Dobrushi n die d Novembe r 12 , 1995 . H e

was a frien d o f F . A . Berezi n an d on e o f th e organizer s o f thi s collection .

ix

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Selected Title s i n Thi s Serie s (Continued from the front of this publication)

144 S . D . Berman e t al. , Thirtee n Paper s Translate d fro m th e Russia n

143 V . A . Belonogo v e t al. , Eigh t Paper s Translate d fro m th e Russia n

142 M . B . Abalovic h e t al. , Te n Paper s Translate d fro m th e Russia n

141 H . Draskovicov a e t al. , Ordere d Set s an d Lattice s

140 V . I . Bernik e t al. , Eleve n Paper s Translate d fro m th e Russia n

139 A . Ya. Aizenshta t e t al. , Ninetee n Paper s o n Algebrai c Semigroup s

138 I . V . Kovalishin a an d V. P . Potapov , Seve n Paper s Translate d fro m th e Russia n

137 V . I . Arnol' d e t al. , Fourtee n Paper s Translate d fro m th e Russia n

136 L . A. Aksent'ev e t al. , Fourtee n Paper s Translate d fro m th e Russia n

135 S . N. Artemo v e t al. , Si x Paper s i n Logi c

134 A . Ya. Aizenshta t e t al. , Fourtee n Paper s Translate d fro m th e Russia n

133 R . R . Sunchelee v e t al. , Thirtee n Paper s i n Analysi s

132 I . G . Dmitriev e t al. , Thirtee n Paper s i n Algebr a

131 V . A . Zmorovic h e t al. , Te n Paper s i n Analysi s

130 M . M . Lavrent'ev , K . G . Reznitskaya , an d V . G . Yakhno , One-dimensiona l Invers e

Problems o f Mathematica l Physic s

129 S . Ya. Khavinson , Tw o Paper s o n Extrema l Problem s i n Comple x Analysi s

128 I . K . Zhu k e t al. , Thirtee n Paper s i n Algebr a an d Numbe r Theor y

127 P . L. Shabalin et al. , Eleve n Paper s i n Analysi s

126 S . A . Akhmedo v e t al. , Eleve n Paper s o n Differentia l Equation s

125 D . V . Anoso v e t al. , Seve n Paper s i n Applie d Mathematic s

124 B . P . Allakhverdiev e t al. , Fiftee n Paper s o n Functiona l Analysi s

123 V . G . Maz'y a e t al. , Ellipti c Boundar y Valu e Problem s

122 N . U . Arakelya n e t al. , Te n Paper s o n Comple x Analysi s

121 V . D . Mazurov, Yu . I . Merzlyakov , an d V. A . Churkin , Editors , Th e Kourovk a Notebook :

Unsolved Problem s i n Grou p Theor y

120 M . G . Krei n and V. A . Jakubovic , Fou r Paper s o n Ordinar y Differentia l Equation s

119 V . A . Dem'janenk o e t al. , Twelv e Paper s i n Algebr a

118 Ju . V . Egoro v e t al. , Sixtee n Paper s o n Differentia l Equation s

117 S . V . Bockare v e t al. , Eigh t Lecture s Delivere d a t th e Internationa l Congres s o f

Mathematicians i n Helsinki , 197 8

116 A . G . Kusnirenko, A . B . Katok , an d V. M . Alekseev , Thre e Paper s o n Dynamica l System s

115 I . S . Belo v e t al. , Twelv e Paper s i n Analysi s

114 M . S . Birma n an d M . Z . Solomjak , Quantitativ e Analysi s i n Sobole v Imbeddin g

Theorems an d Application s t o Spectra l Theor y

113 A . F . Lavrik e t al. , Twelv e Paper s i n Logi c an d Algebr a

112 D . A . Gudko v an d G . A . Utkin , Nin e Paper s o n Hilbert' s 16t h Proble m

111 V . M . Adamjan e t al. , Nin e Paper s o n Analysi s

110 M . S . Budjanu e t al. , Nin e Paper s o n Analysi s

109 D . V . Anoso v e t al. , Twent y Lecture s Delivere d a t th e Internationa l Congres s o f

Mathematicians i n Vancouver , 197 4

108 Ja . L . Geronimus an d Gabo r Szego , Tw o Paper s o n Specia l Function s

107 A . P . Misina an d L . A. Skornjakov , Abelia n Group s an d Module s

106 M . Ja . Antonovskii , V . G . Boltjanskii , an d T . A . Sarymsakov , Topologica l Semifield s an d Their Application s t o Genera l Topolog y

(See th e AM S catalo g fo r earlie r titles )