Lie Groups an d Invariant Theor y Ernest Vinber g Editor

The leaders of the Seminar on Lie Groups, A. L. Onishchik and E. B. Vinberg, with guests of the seminar A. G. Elashvili, B. Kostant, V. Kac, A. A. Kirillov , A. M. Vershik, and A. Dzhumadildaev (Jun e of 1990) .

American Mathematica l Societ y

TRANSLATIONS Series 2 • Volum e 21 3

Advances in the Mathematical Sciences—56 (Formerly Advances in Soviet Mathematics)

Lie Groups an d Invariant Theor y

Ernest Vinber g Editor

American Mathematica l Societ y f | Providence , Rhod e Islan d

http://dx.doi.org/10.1090/trans2/213

A D V A N C E S I N T H E M A T H E M A T I C A L S C I E N C E S

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

V . I . A R N O L D

S. G . G I N D I K I N

V . P . M A S L O V

2000 Mathematics Subject Classification. P r imar y 22-06 ; Secondar y 14L24 .

Library o f Congres s Car d Numbe r 91-64074 1 ISBN 0-8218-3733- 8

ISSN 0065-929 0

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Contents

Preface vii

Real Form s o f Comple x Reductiv e Group s Actin g o n Quasiaffin e Varietie s D. N . AKHIEZE R 1

Component Group s o f the Centrahzer s o f Unipotent Element s i n Semisimpl e Algebraic Group s A. ALEXEEVSK I 1 5

Classification o f Pseudo-Riemannia n Symmetri c Space s o f Quaternioni c Kahler Typ e DMITRI V . ALEKSEEVSK Y AN D VICENT E CORTE S 3 3

On th e Canonica l Embedding s o f Certai n Homogeneou s Space s IVAN V . ARZHANTSE V AN D DMITR I A . TIMASHE V 6 3

Classification o f Goo d Grading s o f Simpl e Li e Algebra s A. G . ELASHVIL I AN D V . G . KA C 8 5

The Separatio n o f the Continuou s Spectru m o n Symmetri c Space s o f Cayle y Type SIMON GINDIKI N 10 5

Connected Sum s o f Compac t Manifold s an d Homogeneit y V. GORBATSEVIC H 11 3

On Curvature s o f Section s o f Tenso r Bundle s P. I . KATSYL O 12 9

Affine Structure s o n Filifor m Li e Algebra s YUSUPDJAN KHAKIMDJANO V 14 1

Lie Superalgebr a Structure s i n C* (n; n) an d H' (n ; n) ALEXEI LEBEDEV , DIMITR Y LEITES , AN D ILY A SHERESHEVSKI I 15 7

On Isotropi c Super-Grassmannian s o f Maxima l Typ e Associate d wit h a n Odd Bilinea r For m A. L . ONISHCHI K AN D A . A . SERO V 17 3

Ideals o f Heisenberg Typ e an d Minima x Element s o f Affine Wey l Group s DMITRI I . PANYUSHE V 19 1

vi CONTENT S

Projective Dualit y an d Principa l Nilpoten t Element s o f Symmetri c Pair s VLADIMIR L . P O P O V 21 5

On Representation s o f Carta n Typ e Li e Superalgebra s VERA SERGANOV A 22 3

Construction o f the Exceptiona l Simpl e Li e Algebra s E. B . VINBER G 24 1

Short S03-Structure s o n Simpl e Li e Algebra s an d Associate d Quasiellipti c Planes E. B . VINBER G 24 3

Preface

This i s a collectio n o f papers writ te n b y participant s o f o r peopl e closel y relate d to th e Mosco w semina r o n Li e group s an d invarian t theor y heade d b y E . B . Vinber g and A . L . Onishchik . Starte d abou t 196 0 a s a semina r o n Li e groups , i t wa s conceived a s a continuatio n o f E . B . Dynkin' s semina r o n Li e groups . (Abou t the latter , se e th e remarkabl e prefac e b y S . G . Gindiki n t o "Li e Group s an d Li e Algebras: E . B . Dynkin' s Seminar" , AM S Translations , Serie s 2 , vol . 169 , 1995. ) I t inherited som e tradition s o f Dynkin' s seminar , i n particular , producin g result s an d conjectures o n a soli d experimenta l basis , fo r whic h th e classificatio n o f semisimpl e Lie group s an d thei r representation s provide s a n excellen t material .

In th e 1967/196 8 academi c year , th e semina r wa s devote d t o studyin g th e the -ory o f algebrai c group s an d t o developin g th e theor y o f semisimpl e Li e group s o n this basi s (a n ide a goin g bac k t o C . Chevalley) . Thi s resulte d i n a boo k "Semi -nar o n Algebrai c Group s an d Li e Groups " b y E . B . Vinber g an d A . L . Onishchik , which first appeare d i n a mimeographe d for m i n 1969 . Th e author s use d pre -liminary draft s o f talk s prepare d b y th e participant s o f th e semina r E . M . An -dreev, V . G . Kac , B . N . Kimelfeld , an d A . K . Tolpygo . (Th e table s o f tenso r decompositions wer e compile d wit h th e assistanc e o f A . G . Elashvili , V . G . Kac , B. N . Kimelfeld , B . O . Makarevich , an d V . L . Popov. ) A considerabl y revise d an d expanded versio n o f th e boo k wa s publishe d i n Russia n i n 1988 . It s Englis h trans -lation wa s publishe d b y Springer-Verla g unde r th e titl e "Li e Group s an d Algebrai c Groups" i n 1990 .

The invariant-theoreti c componen t o f th e semina r wa s base d o n th e previou s work o f E . B . Vinberg , A . G . Elashvili , E . M . Andreev , V . L . Popov , V . G . Kac , D. I . Panyushev , an d P . I . Katsylo . Fo r severa l year s a separate semina r o n invarian t theory wa s heade d b y E . B . Vinber g an d V . L . Popov . I n 1986 , i t wa s combine d with th e semina r o n Li e groups , an d i n 1986-200 0 V . L . Popo v wa s a codirecto r o f this join t seminar .

A collectio n o f paper s b y participant s o f th e semina r wa s firs t publishe d b y th e American Mathematica l Societ y i n th e serie s "Advance s i n Sovie t Mathematics " under th e titl e "Li e Groups , Thei r Discret e Subgroups , an d Invarian t Theory " i n 1992. Th e presen t volum e i s the secon d suc h collection . I t consist s mainl y o f origina l papers wit h ne w result s i n geometr y an d topolog y o f homogeneou s space s o f Li e groups, structur e o f Li e algebras , algebrai c transformatio n groups , supermanifold s and Li e supe r algebras.

We hav e als o include d translation s o f tw o ol d paper s o f topica l interest , b y A. Alexeevsk i an d b y E . B . Vinberg , whic h ar e almos t unknow n i n th e Wes t becaus e they wer e publishe d onl y i n Russian .

vii

viii P R E F A C E

Many forme r participant s o f th e semina r no w liv e abroa d bu t retai n clos e contacts wit h us . I n particular , whe n visitin g Russia , the y ofte n giv e talk s a t our seminar . Stil l considerin g the m t o b e participant s o f th e seminar , w e in -vited som e o f the m t o contribut e t o thi s volume . W e ar e gratefu l t o D . V . Alek -seevsky, A . G . Elashvili , S . G . Gindikin , Yu . Khakimjanov , V . Kac , D . Leites , an d V. Serganova , wh o kindl y agree d t o participate .

Many paper s o f thi s volum e ar e devote d t o A . L . Onishchik , wh o reache d hi s 70th birthda y o n Novembe r 14 , 2003. Al l the participant s o f the semina r wis h hi m good healt h an d lon g year s o f furthe r activit y a s a scientis t an d teacher .

E. B . Vinber g

Titles i n Thi s Subserie s

56 Ernes t Vinberg , Editor , Li e group s an d invarian t theor y (TRANS2/213 )

55 V . M . Buchstabe r an d I . M . Krichever , Editors , Geometry , topology , an d

mathematical physic s (S . P . Novikov' s seminar : 2002-2003 ) (TRANS2/212 )

54 S . G . Gindikin , Editor , Li e group s an d symmetri c spaces . I n memor y o f F . I .

Karpelevich (TRANS2/210 )

53 M . V . Karasev , Editor , Asymptoti c method s fo r wav e an d quantu m problem s

(TRANS2/208)

52 Yu . M . Suhov , Editor , Analyti c method s i n applie d probabilit y (TRANS2/207 )

51 M . A . Shubi n an d M . S . Agranovich , Editors , Partia l differentia l equation s (Mar k

Vishik's seminar ) (TRANS2/206 )

50 V . Turae v an d A . Vershik , Editors , Topology , ergodi c theory , rea l algebrai c geometry .

Rokhlin's memoria l (TRANS2/202 )

49 Michae l Semenov-Tian-Shansky , Editor , L . D . Faddeev' s semina r o n mathematica l

physics (TRANS2/201 )

48 L . Lerman , G . Polotovskii , an d L . Shilnikov , Editors , Method s o f qualitativ e theor y

of differentia l equation s an d relate d topic s (TRANS2/200 )

47 R . A . Minlos , Seny a Shlosman , an d Yu . M . Suhov , Editors , O n Dobrushin' s way .

From probabilit y theor y t o statistica l physic s (TRANS2/198 )

46 Vladimi r Arnold , M a x i m Kontsevich , an d Anto n Zorich , Editors , Pseudoperiodi c

topology (TRANS2/197 )

45 Ya . Eliashberg , D . Fuchs , T . Rat iu , an d A . Weinstein , Editors , Norther n

California symplecti c geometr y semina r (TRANS2/196 )

44 Alexande r Astashkevic h an d Serg e Tabachnikov , Editors , Differentia l Topology , Infinite-Dimensional Li e Algebras , an d Application s (D . B . Fuchs ' 60t h Anniversar y

Collection) (TRANS2/194 )

43 A . Yu . Morozo v an d M . A . Olshanetsky , Editors , Mosco w Semina r i n Mathematica l

Physics (TRANS2/191 )

42 S . Tabachnikov , Editor , Differentia l an d Symplecti c Topolog y o f Knot s an d Curve s

(TRANS2/190)

41 V . Buslaev , M . Solomyak , an d D . Yafaev , Editors , Differentia l Operator s an d

Spectral Theor y (M . Sh . Birman' s 70t h anniversar y collection ) (TRANS2/189 )

40 M . V . Karasev , Editor , Coheren t Transform , Quantization , an d Poisso n Geometr y

(TRANS2/187)

39 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Geometr y o f Differentia l

Equations (TRANS2/186 )

38 B . Feigi n an d V . Vassiliev , Editors , Topic s i n Quantu m Group s an d Finite-Typ e

Invariants (Mathematic s a t th e Independen t Universit y o f Moscow ) (TRANS2/185 )

37 Pete r Kuchmen t an d Vladimi r Lin , Editors , Voronez h Winte r Mathematica l School s

(Dedicated t o Seli m Krein ) (TRANS2/184 )

36 V . E . Zakharov , Editor , Nonlinea r Wave s an d Wea k Turbulenc e (TRANS2/182 )

35 G . I . Olshanski , Editor , Kirillov' s Semina r o n Representatio n Theor y (TRANS2/181 )

34 A . Khovanskii , A . Varchenko , an d V . Vassiliev , Editors , Topic s i n Singularit y

Theory (TRANS2/180 )

33 V . M . Buchstabe r an d S . P . Novikov , Editors , Solitons , Geometry , an d Topology : O n

the Crossroa d (TRANS2/179 )

32 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Topics i n Statistica l an d Theoretica l Physic s (F . A . Berezi n Memoria l Volume )

(TRANS2/177)

31 R . L . Dobrushin , R . A . Minlos , M . A . Shubin , an d A . M . Vershik , Editors , Contemporary Mathematica l Physic s (F . A . Berezi n Memoria l Volume ) (TRANS2/175 )

TITLES I N THI S SUBSERIE S

30 A . A . Bolibruch , A . S . Merkur'ev , an d N . Yu . Netsve taev , Editors , Mathematic s

in St . Petersbur g (TRANS2/174 )

29 V . Kharlamov , A . Korchagin , G . Polotovskii , an d O . Viro , Editors , Topolog y o f

Real Algebrai c Varietie s an d Relate d Topic s (TRANS2/173 )

28 L . A . Bunimovich , B . M . Gurevich , an d Ya . B . Pesin , Editors , Sinai' s Mosco w

Seminar o n Dynamica l System s (TRANS2/171 )

27 S . P . Novikov , Editor , Topic s i n Topolog y an d Mathematica l Physic s (TRANS2/170 )

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

Dynkin's Semina r (TRANS2/169 )

25 V . V . Kozlov , Editor , Dynamica l System s i n Classica l Mechanic s (TRANS2/168 )

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

Equations (TRANS2/167 )

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

(TRANS2/165)

22 N . N . Uraltseva , Editor , Nonlinea r Evolutio n Equation s (TRANS2/164 )

Published Earlie r a s Advances i n Sovie t Mathematic s

21 V . I . Arnold , Editor , Singularitie s an d bifurcations , 199 4

20 R . L . Dobrushin , Editor , Probabilit y contribution s t o statistica l mechanics , 199 4

19 V . A . Marchenko , Editor , Spectra l operato r theor y an d relate d topics , 199 4

18 Ole g Viro , Editor , Topolog y o f manifold s an d varieties , 199 4

17 D m i t r y Fuchs , Editor , Unconventiona l Li e algebras , 199 3

16 Serge i Gelfan d an d Simo n Gindikin , Editors , I . M . Gelfan d seminar , Part s 1 and 2 ,

1993

15 A . T . Fomenko , Editor , Minima l surfaces , 199 3

14 Yu . S . Il'yashenko , Editor , Nonlinea r Stoke s phenomena , 199 2

13 V . P . Maslo v an d S . N . Samborskii , Editors , Idempoten t analysis , 199 2

12 R . Z . Khasminskii , Editor , Topic s i n nonparametri c estimation , 199 2

11 B . Ya . Levin , Editor , Entir e an d subharmoni c functions , 199 2 10 A . V . Babi n an d M . I . Vishik , Editors , Propertie s o f globa l attractor s o f partia l

differential equations , 199 2

9 A . M . Vershik , Editor , Representatio n theor y an d dynamica l systems , 199 2

8 E . B . Vinberg , Editor , Li e groups , thei r discret e subgroups , an d invarian t theory , 199 2

7 M . Sh . Birman , Editor , Estimate s an d asymptotic s fo r discret e spectr a o f integra l an d

differential equations , 199 1

6 A . T . Fomenko , Editor , Topologica l classificatio n o f integrabl e systems , 199 1

5 R . A . Minlos , Editor , Many-particl e Hamiltonians : spectr a an d scattering , 199 1

4 A . A . Suslin , Editor , Algebrai c K-theory , 199 1

3 Ya . G . SinaT , Editor , Dynamica l system s an d statistica l mechanics , 199 1

2 A . A . Kirillov , Editor , Topic s i n representatio n theory , 199 1

1 V . I . Arnold , Editor , Theor y o f singularitie s an d it s applications , 199 0