Semi-simple real subal gebras of non-compact semi-simple real lie algebras V

~ars

Yd. 7, ( 1 ~

ew J~rn~munc.,c ~ r n ~ z

No. =

S]~Id[I-Sk'Id[PI.Z REAL S U I B ~ ~ A S OF NON-CO.MPACr S ] D , ~ rr~- ALGEImAS V

J.M.P.sma m~l J.F. ~ D,~m~z

or T b m m ~

Paysi~ U a i v m ~ er Sc ~t~lams, St. A a d r r ~ r~r,, Smea~

1'~ Imm~sdm er the ~ m er m ; ~ a um;.,im~ m l U= ~ .t" la s am.emq~a mmleim1~ rmt L~ alsda'a .t" Is emmdml W tlm mi~ whm .e" u¢l/w t ismm~iemL tk.riz nq~mmmi,~ t ~ ~ L k ~ ~

are & erA, (mwmmmioo (16) mal~tdi~s ~ $& (Z ~ i a tool from arF.

als~bra er rml fonm e l F . whom oomptex ~

+(9)). Itamm of ks ptestml ~ ~ q

md E. are lltma. MaW er tlmmlm osw i~mlls.

In roar l X ~ O m p s p m (¢orawzll [61. [7]. IS] sad [ t i m and Corawztl ~1]). b m s t ~ rzfzsmd to as I, lI. III and IV, aa iavadliadoa was bep= iato ~he p m b ~ of ~ a~ hal Lie aJSetn'~~.~' in anotl~ u=i-simplz real I.~ a/Zebra 2 . A aurora? sad sug,cia~ condition was daived ia I , ninety

s-.r.r,

(t38)

wbae $ is the i,,volutive automort, h;--, which Smaaas .~, 3" is the iavo;~;ve summerphism ~t~ch Ileoaatm ~ sad Y is ms automorphism of .~, such that FI"(a) ,, l~a) for tll a ia - ~ T ' ~ eonditio, wm ~ fer all ~ ia w h i ~ ~ aad ..~ a:e both ~ forms ; e a e ~ t e d by inca- ~ * e ~ considered ia I, those t,e a e m e d by the outer automorphism of A, asui Dt in 1I aad I11, nlspecfively, and the c a m where . ~ or 2 ' are semi-simple were deak ,Jrith in IV. The prcscat pap~ extends the ibvmtilgafioa to the cases where _v,, and/or ~ are zxcrptimaL la Section 2 ~pUcit forms (mLniX ~ ) an: o b t a i n ~ for th~ dzfinins represenuuioas or zn the aceptioatt Lic ~Sebras. P a t m (1171, [ISD h a cto~ ~ p, eviousty for G: sad F.. b . t his method iavoi~,m z:nbatdiag the z s c z p t i o ~ ~ zllebra ia a classical Lie a / F i n s . The method used ia this paper involves ~.ading the weililts of the ddniall represent~ion, and is easier to extend to F.,, £7 and E , . Meh~ [15] and Mehta and Sri[seT!

LM. I~GNS and L P. COPaqWSn~vastava [16] live some useful information, including the fundamentzi v.~ghts or' all cxcepdoaal Lie allebras. Sectloc 3 ¢utmin~ the inner in,,otudve zutomorphisms of all the exceptional Lie a/gebras and the real fofras generated by than. Section 4 examines the outer involudve automorphisms of F~ (the only c=cepdoaal Lie alg~ra which has outer automorpb;,mt) and tlz msl fom~ g m m u ~ ~ tham The methods and coaditiom of L 1I, 1II, and IV trc m o ~ snd eztc:sdaf to the where ~ and/or 2, are cxc~tional in Secttoa $. Section 6 couudns examples, including all the subalsebras of G# aad all tmbeddinp of $I..(2, C) in P, a~l F.~. These ~ arc ncw nm~s. Hitherto the only applicatioa of the ~ ~ l.i# alldma in dmaeat~ par~e physics has beea the use of tlm compact real form of Gz as au inte.mal symssm~ group in place of the more commoaly used 57./(3) group (cf. for example Behrends, Laudorkz . ~ Tunkaimsli [2], ~ and Sirfia [3], and aehmads [4]). The major obsutcl~ was that tim atanaans r d s t i o m i ~ ot tt~ vmiom umptimm rmi Li~ sit#m~s with

~ qutntm~~ 2. ~

rm~ Li~ a l ~ ~

~oatsin t ~ physicsU~ ~ Lomaz zmup . ~ , C) z s ,

term for © o m l ~ ~

L~ ~isc~'z of th~

Lle .Izda'm

The Dyukin diagram and Caftan matrix A of every cu:zpdonal I.~ a/pbra are ia the Appendix. The paeral method used he:e to obtain an exldicit form (nuadz r~psw szatadon) uszs the weights of the ddmial repmum~ou. The rundamemal domimm ~,~ghu ~ ) , i = 1.... .1. (rise, crank ofthestS,~rs) mustsaes~ ~ ) - 6u,j - t, ...J. where the ;b ( / - ~-, ..., 0 coastit~ a basis for the Carum subatsebnt in canonical form (c~ [14], pp. 121, 225).. This impfia that ~,(h) - (A-t)u:t(a) (tim smnmation convmst/on is ¢l:-ays implied unless othem-ise stated), ~aare the :,,. k - 1, ..., I, are the simple roots. Evz~7 irreducibb rvpnmmmtion is chasm:tm'imd by its h i g h ~ weight M - m , ~ (written as (ms . . . . . ms)) and its diaumsiou d is ~ by Weyfs dlmm~onali W formula (cf. [14]. p.2.~ !

(m,+ l) k#q - H

where the produc: is over all sequences {k~} (i =, 1 . . . . . 0 for which kjz~ is a l~si~ive1~c. and w~is the weight ofthe root ~, in the ~ diagram. ,'hen the lowest dimzmioaal repsw sentadoa may be found and its ",ve/#m calcO.ated ~ the fact that if A is a w d i ~ sad 2(,I. ecJl(z, zJ .. k (a positive imqzr), then (.4-jffiO is also # weight for j - i . . . . . k. The explicit matrix expression for the basis elements of the Carum subatZet~ arc rhea

pmastors e~ and t, sssocisted with the simple roots are found from ~qu~oas (I.4). namely

2.L As exp/Idt tern tar G# The fm~hmmml ~ ~ ~ ~: ~ ' ~ t - ~ , + ~ z , a~l ~ - z,+2~r.s. The fundam~tsl reptes~tations (I, 0) and (0, I) have dimensions 14 and 7, ~spe~vely. The welsists of the lowest d i ~ m ~ o ~ ~ p ~ t a t i o a arc z, +2a~, z ~ + ~ , z~, 0, - - z t , - z t - z # , and - z t - 2 z s. Thus the jenenttors of the C a t t u m/~18ebm m y be t s k ~ ss

n,, -d~,,Soa,t CO. X . - I . 0 . - ! . 1.0). -disSou~ (+n. - n . 2 . O. - 2 . + X . - t ) .

(I)

and the jenes~ors associated with sknplc roots amy be udam ss e, - ( I , . , + I u ) ,

andf, - - ~ ff - 1.2), wbc~ l.m is :he ~ e n ~ i o m d nmu'ix siren by (lu)~ -- 6u,6j,. The zaterators ~ _. associated with non-simi~ roots z , + ~4+ z,..., are cslaslstzd t'rom ,,, - [ e , , , ~ . T ~

=e

eu - ( + z . . . - I/i'I..., + r'~'x,...-xT..,). e.~ - 2 ( - ~ X ~ + x....,+ x,..,- I/t"x,..). euu= - 6(+x~-x.,..,). • ,..=, - 6a,., +x,~.

2.2. As erp/kit ~ r m r o r F .

~ domimmt we~Shts br P, are ~.~- 2 ~ ÷ 3 z : + 4 ~ 3 + 2 ~ . 1~~ + & z s + l z s - ~ - ~ , , . ~a - 2 . ~ + 4 ~ + 6 z a + 3 ~ . ,l, - z t + ~ z + 3 z s + 2 ~ , and tim dimansions of' the f'undsmzn~ ~ n t f i o n s ( l , 0 , 0 , 0 ) , {0, 1,0,0), ( 0 , 0 , 1 , 0 ) and (0, 0, 0, I) are 52, I274, 27"3 and 26, rwpectJveiy. The rejulm' represenutfioa /s thus (1,0, 0, 0) and tho l o w ~ ~ represam,.ion (0, 0, 0, l), a matrix mlmSmuu~n of which is si,~s bdow. The zenz~toa of the Cm~n sulmlpbrz a n ks -- disl[oul (0, O, O, + 1 , + 1 , - i . +1, - - i , - I , 0,0, 0 , 0 , 0 , 0 , 0 , 0 , + 1 , +1, - - I , + 1 , - - 1 . - - 1 . 0 , 0 , 0 ) , h# - disl[onal (0, 0, + 1 , - - 1 , 0 , 0, 0, + 1 , + 1 , --1, --1,0, 0, 0, 0, + ! , + 1 , --1, - - 1 , 0 , 0 , 0 , + 1 , - - 1 , 0 , 0 ) , bs .. diallonai (0. ÷ 1. - - ! . + 1 . --!. + I . O. - 1 . 0 . + i. +2. - ! . 0 . 0 . + i . --2. - - 1 . 0 . + ! . 0 . - I . + ! . - ! . +1. - I . 0 ) . h, - diagonal ( + 1 . - 1 . 0 . 0 . + I . 0 . - I . + i . --1. +1. - ! . + 2 . 0 . 0 . - - 2 . + ! , --1, +1, - I , + l , O , - 1 , 0 , 0 , +1, - l ) ,

(4)

170

J.M. ~r.~$ m d ~.F. C ~ P J ~ I . L

aud the m

n

e m m r , ¢ m l ~ to ~ p l ~ roots am

e~ -- I,..~+lzo.8 +It:j, +ItL],,+l,e.:v+I~.~, +Iz.~,~ + I ~ ,

roots ( ~ + ~ ,

,,z+as, as+~.., a t + c ~ + ~ ,

~+¢~+e.~, ~z+2~zs, a~+~z+~+~r~,

+2.z~+2~, ~ +2~+2~z~+~, "t +2=~+2=~+2.~, ~ + 2 = a + ~ + ~ , ~t +2~+~z~+ + 2 ~ , ~-t-2z#+4z~+2~, a z + 3 z ~ + 4 ~ + 2 ~ , 2~t+3=.a+4~+2~_.)ms be ot:~im~

~.~i]y from these..F m ~ , f, is ~ve= by - i , .

The fuadamen~t ~

weights of" ~

'rbe dime.~om of ~ fundan'~-',zl r e p ~ am 2'/, 35I, ~ , 351, 2/, and r~pec~ely. Thus fl~ relulm"rq~'esentltion is (0, O, O, O, O, I) a ~ ~ l o w ~ d i m ~ .,~pmslmml~ons are (.I, O, O, O, O, O) amd (0, O, O, O. I, 0), A ma~rb t ~ r ~ f j ~ i ~ of (I, O, O, O, O, O) is ~ 1~ow. "I'h= ~Im~smtors of tl~ Ca.,'~m subalpb~ auratl~ disso~l

hx - ~ ( - - I . h~ b~ h, hs h, -

+ ! . 0,0, O, 0 , 0 , 0 , 0 , --1,0, 0,0, --I, +1, --1, --1, 4-1, +1, +1,0, --1, + I , 0 , 0 , 0 , 0 ) , diN~oaal (0, --I, + 1 , 0 , 0 , 0 , O,O, - 1 , + 1 , 0 , ' - I , --1, +l,O, +l,O, 0,0, - 1 , + I , 0 , - I , +1,0,0,0), diagomd (0,0, - 1 , + 1 , 0 , 0 , 0 , - I , + 1 , 0 , - I , + I , 0 , 0 , 0 , --1, +1, O, - 1 , + t , 0,0, O, - 1 , +1, O, 0), diagonal (0, O, O, - I , + I , 0 , - I , + I , 0 , 0 , 0 , - I , +I, - I , 0 , + I , 0 , - I , + I , 0 , 0 , 0 , 0 , 0 , - 1 , +1,0), diagonal (0, O, O, O, - I , + I , 0 , - I , - I , - I , +1, +1,0, + I , --1,0, O, + 1 , 0 , 0 , 0 , 0 , 0 , 0 , 0 , --1, +1), diasonal (0, O, O, - I , - 1 , - 1 , ÷1, + 1 , 0 , 0 , + 1 , 0 , 0 , 0 , 0,0, - 1 , 0 , O, - I , - I , + I , + I , + I , 0 , 0 , 0 ) ,

(6)

.q'i~-SIMPL~ lt.P.,ALSUllALGZlkq,ASOF I , ~ .t.LGI~it,AS V t e d the ~

171

c e r r u p o e ~ 8 to s i m ~ roeu ~ e

t t m Itj+Xlo.lsaell6.ts+It6.t$,4.It~..~.4.II,l~. e, = I~.~+I~ a . + l , z . . + I t ~ a , + l ~ . ~ s + l n . ~ .

(7) e. - l . a +I,~, +ltz..~ + I , . . , + l . , a , + l a a ~ , . es = l,.,+ls.ss+I~.sz+Iso.s,+l,s,t,+l~.z~, • t -- I ~ + I ~ . s +14.11 +It~..~ + l l ~ . a +lzs..~. 1.4, As a p n d t fern d F,, The fundamental demimat weijlm an~ ~t = 2 ~ , + . ~ + 4 ~ + 3 e . , + 2 . ~ + f . , + 2 ~ , 2~ - 4a~ + S , ~ + 1 2 s s + 9 , ~ , + ~ + 3 z , , + t ~ , 1, - ](t~, + 12sz + l k s 4-15~, + 1 0 ~ + t ~ + ~,t), ,~ = ~2z, + 4 " , ' z + ~ . 3 + 5 = , + 4 ~ + 3z,+3.~), 7., ,,, 4 z t + S z z + 1 2 z s + 9 ~ . + ~ . , j + 3 ~ + 6 e , , . The dimemio~ of the f m .dsn~tai : e ~ e ~ t m i e m are 133, s645, 36.T7..~, 27664, 1539, 56. 912. Hence the regular ~ t a t / o n ~ ( 1 , 0 , 0 , 0 , 0 , 0 . 0 ) and the I m M a t ~ Zt,l~,u ~ t i o a is f 0 , 0 . 0 . 0 , 0, 1,0). A era:ix r s ~ s m t a t i o a , t t b , lamr t, livm. The Imerato~ of the ~ ~3alsebr~ are the diapeal ht " - - I 4 . , + I ~ . ~ + I s . s + I ~ . , - l , t . , t + I z e . ~ o + I : s a s + l x ~ , ~ - l m , m - l ~ t , ~ , + --Isl..~, +Is*aS +I~t..~+Ils.as--I, ttat + I t a . ~ - - l . t a . ~ +l*gaut --Is S.Sl -- ls~..,=+ l n j , - I ~ . ~ - I ~ s . ~ s - I N . ~ , , h: m - - I s . ~ + l ~ - - l g . ~ + I t o a o + l t s . ~ x - - | t s . ~ e + I t e . ~ e - - I ~ . , ~ + l ~ , m + I ~ , ~ - -- It-/,~-/+ lts~m - Isealo + I | t . j l - I ~ ,

hi =, - - 1 4 ~ + l s a - - I ~ + l u - l t

s.~ s + I ~ 1

+lt:~.~--lt,.~f ÷ I t , . , , - - l ~ . z t +

h, = - - I s . . ~ + I ~ - l ~ . e - l s e . s e ÷ l u a s - l t ~ , ~ + I , , a , + l s s . ~ s - - l t s . ~ a - I t , a e - -Ize..~o + Izs.zs - Izs.~s +lz,t.za,- lst.~, +Isz..~z + I ~ , ~ - I s ~ . ~ , - - I s L j 8 +

+It,a, + l~.ao-I~,a. +l:,a,-lsza.-I~

+ I~.u-l~.a.-Im.m-

+ l ~ . a . ÷l,~.J~ + I N ~ . + l~.a, + I ~ . ~ + l . t . . . + I . , ~ . , * - I , ~ - L,s.~s --

17'/ I~ -

~. 1~ l a a ~

mzt .I- 1¢- ¢ O I N W l a ~

-ls.s-I~-l.~.~,+l,.s-ltsaz-Iss.~t:t"l,,.s,-l,,~s-tt,~,+lt,~,--

+t,~,, +Z.,.,-zs,.,, +l,,.,, + I . . , , . and the I~aerumn mr:mlmad~i m flmlP~ rams sm e t m ~k~ +l;a.S +ISt.~ + X t e . m ÷ I 4 t ~ t + I u + l u . ~ + I m t . ~ + e a m Is.6 + ~ . t o

+In.st +|t&.~,+lss.~+l~t~+lss~t,+It~t+

+Iss.~,+l,~, +I,,m+Im~s, es -- l~s +I~., + l s , . s s + l , s . , ~ + l , . ~ . , s + l s s . u + I ~ + I m ~ , e , -- Is., + I , ~

+Ite.,t + l t s . , , + I s ~ . : ~ + l t ~ + I t t . s , + l ~ . a + +lu +lstm +l~.w+l~.,~, es - I~.j + I , s . , t a + l , ~ . : ~ + l t a . t , + l ~ . s ~ + l ~ e . t , + l ~ s j , + l ~ . ~ + e, -- I t j +Iu.s,+It,.~s+Is~.~t+l~t.as+lu~.m+I,~.~s+lut.a,+ "

+ I~.,~ +I,..,, +l,,.a + l u j , . e~ m ls..-,+It.~ +I~.~ + l t : . ~ + l , ~ , ~ + l , , ~ , t + I t t . w + I m . n +

?.& As ~ a d t

f,m f~ ~

The f , a ~ u a e , ~

dmteaat ~

o( t , m

~a -- 7g,+ I 4 c z + 2 0 1 t + l f ~ + 1 2 ~ + S L t + 4 ~ + 1 0 1 t o

2s -- 10~t + 2 0 = , + 3 0 ~ s + 2 4 ~ + l f ~ t l + 1 2 & t + 6 ~ + l ~ , m 8ett + 16~ts+244tt + 20~ + 15~ + |0ge + ~ + 12~° -- 6s, + 12sa+ l k a + l f ~ + 1 2 ~ + 8 ~ t + ~ + ~ . ~- -- 4st +$~s+ 12~a+ 1 0 ~ + 8 ~ + ( x t , + 3 a ~ + f m u , ~.~ m 2 a t + 4 k l s + ~ t j + ~ t , + 4 e t s + 3 g e + 2 ~ + 3 ~ , ]., m Sgt + 10=z+ lSets+ | 2 ~ + g e ~ + ( ~ + 3 " ~ T + ~ ° and the dimemima of the fmdtmoa=l ~

lowes~ ~ is sivac The ~

m

~e.preseatatioa. A m a t ~ zt,prMeata~u of (O, O, O, O, O, O, l, O) subalSebra has as ~s basis the dialoml mmrk~ Iq -- q~"~aulj.~ -T

for

i -- l, 2 , _ , , S, J -- 1,2, .... 2 4 .

(I0)

~

m.I~ALSUI~T.G3,qJLa~ OP Lm ~

V

173

the non-too ~ m S/veul~aw:.

-2 +2 -1

far far for

+I

far

~-2 +2

far for

J-S2, j-s4,

+1

far

72, 73, 74, 79, SO, S1' ~ , lOS, I06, U0, --, tL~ 121"-.. 125, 131, 1.32. 137, 151, 154, 161' 171, 174, 175, 200, _ ~ . . r ~ . 221, 231, ..., 234, / - S , 15, 24" 29, 32, 43, $O, $~l, g0,67,?$,76,Lq,_,g0,~,IgO. 101, 107, ~0~, ~7~, 127, 133, 134, 1 1 , I ~ , 140,145, 146, 147, 155, 156, 117, 162, 16, 164, I76, ._, lS1" 192, _ . IS'& ~ , 209, 214. 2~, ~ , , 237.

-2 +2 -I

let J - l O S , far ) - 1 1 0 , tot' / . S , ~ , 1 ~ , I S , ~ 9 , 2 0 , 2 4 , 2 7 , ~ . 3 £ ~ , ~ 7 , 3 1 , ~ , 4 6 , 4 7 , ~ 0 , 1 4 . 62, 6"7,70, 7S, 7~, 7d, Sl, S4" ST, . - , g~, 103, ~0~, 107, Z2& 127, 130, t~Z, 140, 146, 147, 1.~0, 1~7, 162, 163, 164, 16"7,_ , 170. 173, 191' 19S, 199, 209, 219, 222, far-./-~,~,16,21"23,2s,3s,39,~,4,s1"ss,~3,6a, TS,73,77,s2, 8.5, 91, 98, 105, 111, 121, 122, I23, 125, 131, 132, L~7, 141" 148, 151" l~t. 1.,~ 165, 172, 174, 1"75, HQ. 183, 184, l~g, 196,200,. 2O5, 210, 212, 220, 223, 224, 225, 231, _ , 234.

G I j -,,

a~ -

as.,--

+1

-2 +2 -!

~'~ " +!

./-29, J-$1, J - 8,..., 13, ~t, ..., 21, .q0, $1, .52, W, 75, _ , ?/, 84, ._, 94, 99, 100, 101, 116, 133, I~i~ 1~5, . . , 160, 172, 1711, _. _ , I17, ~ , 2O2, 214, J- 14,32,--,4~,S3,61,---,~,79,---,U2,1g~,--,106,117,-._, 1~, 135, I ~ , 137, 1#1, 173, 1~4, 171. lSS,~m, ~4"20e, 215, . . , 221' 235. L

far / - 1 6 5 , fat Y - 16"7, fro" / - 4" 9. 16, I?. 21.2~ 26. 33. 34. ~9. 44" 4.q. 48. ~ . 63, 68. 69. 72. 73. SO. ~ . S6. 91. tO0. 101. 102. 1Of,. 110, 111" 122. 123. t2L 129, 134, 135, 1:39. 145, 148, 149, 156, 164, 177, 183, 184" 188, 193, 194, 198, 20~, 218, 223, 234, 22~, 225, 229, 230, for J m$, I0, 18, ,'1~.,27, 33, 40, 46, 49, ..q},.q6, 64" 70, 74, 7.~,76, $1, $7, 92, 95, 103, 106, 107, 108, !12, 116, 117, 124. 130, 136, 140. 143, 146,!50, 152, 157. 168, 169, 170, 171, 173, 178. l&5, 157. 195. 199, 206, 207, 208, 209, 219, 226, 231, 235, 239, 24&

174

L M. l l ~ Z ~ ~ = i / . P'. ~

'-2 +2 -1

for for for

+I

for

--2 +2 --1

for for for

+1

for

--2

for for for

j an 2 4 5 )

'+2 -1 +1

for

j-2" 13" ....19, ~k _., 41, Sg, 73" 74, 75, 90, _.. 93, 95, 9~, 9U, 99, lO0, 115, '.20,127.,...,131, L~9, ....142, ~0, I~I, ~, ~'~, IT3. T73: 174, 1t17, I n , 207., 203, ~0"?, 217, 7.24, 230, 234, 7.37,

a~f i

a~u-

J-:2~ ./-228, j - 3,1o. 18. 22" 3.5. 4o. 46, .se. e~. 7o. `7~.rT, Ts. 79. rT, 92" 9.5, gt. 99, 105, 112, 117, 121, 124, 1:32, 133, 14o, 141, I42, 143,146. 152. 154. I$~. 163, 16:7, 175. 176. 171k 182" 185, 192, 19~, 196, 19'7, 204, 212, 213, ~ , 21'7, 22s, 231, 23s, 239, 240., 243. ] m 4. I 1, 19, 23, 24, ~ , 26, 36, 41, 4`7, ~ 57, 65, 71, T'Z,73" 80, U, 93, 96, 100, 104, 105, 1!3" 118, I ~ , I ~ , 17.5, I27, 128, 1~, I34, 133, 1~, 14`7, 14a, I ~ , 153, 156, 16~, 1~, 15a, 177, 179, 183" 186, IU. 193, 198, 205, 218, 229, 230, 232, 240, 244. J - 240, j - 242, j-2, 11, ~9, ~ , 4~, sT, ~ , "r3, 74, ,"& U , 93, 9~, ]00, 1~3" 11s, 122, L~7, ._, 131, 147, ..., 1St, ~,0, 161, ~ . , ] ~ , ~"r'~ 17& 174. 179. 183. 187, 18~, 193., 202, 2K3, 2207, ...,211, 214. 21& 224. 2 ~ 232" 2r7. 23~, 24~ 2,r7, 248, J - 3 " 12" 20. 21, 2~, 3`7, 4~ _ . , ~ , St. 6e. _.`70. 76, ._. 79. =9. 94, 97, lOt, 1(~, 103, 114, I19, 123, 124, 132, 133. 143, IS~. 154, 1~, ) ( ~ ~ , ~Ts, ~ , leO, 1s4. ]as, ~94. _., ]97, 2o4, ~ , ;~.~, ~ , ~9, ~ , 243" ~ . J-

2,r/,

J - l, 12" 37, SS, e6, ..., "/2, gg, 114. 119, 123..... I26. 15~., _ ..., 159, 169, 171, 180, IIM, 1~, I~WD,194, ..., 201, 205, 206, 21~, 21S, -., 223, 229, 233, ~ , 236, 238, 242, 244,

23~., 240. 243.

--2 +2 -I ~8.t am

+1

for for for

j =, lilY, -

isl,

J -- 6, 1% I.~, 33, ~L4,$1, 50, 61,68,77,85,95,96,97, 105, 106,107, 110, t16 ..., I~0, 17.8, 13"/, 138, 141, 143, I4,4, 148, 152, 153" 158, IT2., 174, ..., 181, 196, 205, 206, 207, 210, 217., 220, 231, 2 . ~ 233" ~ , 236, 23'7, 248, for j , - 7, $, 17, .... .'m., 26, 27, 34, ~ , 36, 37, ~1~,45, 46, 47, 52" $3" 54, 62, 69. "~ t , "m, ..., SZ, S~ .... 9O. lOS. 111 . . . . ,115, Z ~, "30, ~,*,., ;.. :30, 159, ..., 164, 19'7, 198, 199, 211, 213, 221,

3. xmm. impair, m m m u m ' ~ s ~ mmpu~ ~ , d m , n ~ from p = m t d ~ h m

mmpb L~ ¢ ~ . m

&x. Gmmmi h m 7 Any chief inmm"mm~moqd~m me be expmmd u ~exp h), * - l ~ h ~ to xl~ Canan s u ~ Xm~olmivams implim ~ ~exp ~ ) . ~ sinm r(~pi,) ~ be ~lpmml i

E

m.

l(i~ laL+isinha(h),

1

cmha(k)J"

whine i is the nmk olf tlw ailebra and a is • root. i n v o l u t i v a ~ impHm that fro" t'vm7

mot a. a ~ i ) - ~

wlUuma~, u i m . ~ . T h u s , mmsm~ sad s ~ k i U t coadatm for

iavetndvmsms is thsx for every simple mot z a . z . ( h ) - - n ~ ,

nj b,inl , a i a a q ~ I

-njg~ I - I, ._,~ where A b the Carton m m m ~ and hence oj - = ~ma(A'Ou. (Gamo

~ U m

sUown( [ ! ~ JIM) mat sore, m s ~

~

equivuu~ ~ ) T U e

invm,sm of the Cax'um mmximmaxs l i v ~ in dins A ~ Thus the ~ ot ~ iavolativ, i u m ' , u t ~ m a be fouad. This 1 ~ u ~ l is now al:qdled m tU tlu a . .

sa..~ I n v e t m ~ chief" ianer mnomorphisms SmznuieZ noe-immmpbic rml forms m ~ mired for 6) n . - a2 - 0, and C'U3a, - + Z n z - - l , which l~,eh - O w l s - t-.-.-.~. The If'aces of T(expPa) are 14 and - 2 . r m ~ . Thus theft aze two nml fmms of Gz. fO ~ romp,it ."~.'m Ct,m=mm by ~ = p m ) for I. - o)~, u o ~ m t ~ ~ or onboloul uimt'~ 7" "~hich leave the ~ ]mmucz in ~ d i a m m m s iuveximn C ~ - . . ' ( , x b ) - :Fax 2"s rm.

where ,II indicm m reduced to rood 7. This Izoup wiU Immdrorth be ~ to m ~ z a.ud h u , charm:m, of - 1 4 . Or) The n o a ~ form ~ by T ( ~ p It) for h n PJl~) is isomorphic to the . set of uamformatiom which Imve iavadant the above _ve~__~produ~ zm:l the indefinite ql~lr~.tic f o r m ~ + x t Y t ÷ xzYa + x s Y s ,

and J u " dcs..a for l , J - 1, ..., 7. ~ and h m a c h & m a ~ ~r 2.

w i ~ r e X - ( x , , :¢a, x s , z , T s , Ya,Ys) Stoup will l~mc~orch be mfm'md to as NO=

i.e.. X J ~

l'/6

;. M. ~ C I I ~ u d ~F. COKWm~J,

3.a. F, Iavolmive ~ iuaer ,mamocld~ms pucmXiuS uma-bmmm,p~ msl forms am ohmined for if) a, - ~ , - n , - n , - 0 , ('u') n, - +7., a, - - - l , a , - q , - 0 , sad 0ii) • s-0.~s--2, a,- +2.~,--I. which ~ves h - 0 . k - : : ~ h , , and h - m ~ l J , .

m~aivdy. The mnm or ~ a p b ) m ~

-4, md ~0, ~

.

T h i n .Isebras are isomorpl~ to me Stoups of m m s t o m s d o m in 2S ~ Si,nm in {j12 of [13] by Gantmtcber sad will be ndrerx~ to as CF. (compsct rud form. llemr* 8ted by 2"(1), chamm~ -52~ NFt. ( ~ reel form. ~ by T(exp/:drt,),

cbsr-_~_~ +4) aad ~'F.~ (

~

rud form, Sam, rod by 7Xexp ~

a.4. F4 Xavohmve ddd' hum ~ u ~ e d for (i) ~ m 0, ('d) ~ -, 0, ('j) ~ m 0,

~

~

- ~0~

u m . l s m m ~ c mul mnus m oh.

f - 1 , . . . , 6, i#4, n~ m - - 1 , I~$, as m - - 1 .

b - o, h - t ~ ( 4 ~ , + S b 2 + 1 2 k j + 1 0 k , + ~ + ~ , ) The ~

of T(e:cpb) are then 7S, - 2 a n d

~hic

re~ form of ~ t~mmxed b~ im~r ,utmmqdmm.

sad

14, rupecth~y, l'hm tbm~ are titan noa-

0) The compact form ~ by 7"(cop~ ) far h - O is isomorphic m ths set c t t m m ' o m a d m m which l u r e invariut the poshiv. ~ H m n i : i u form

x , ~ + . v , ~ , +:,,,-I, for p , q -

I,-.,6,

-

x.~

01)

:.~ m --:W sad

X - 0r~, z:,, x~, .v,,, ~ , x , , 2:t,.,. :2:,.,, 2:z.,, 2 : : . , , 2 : , j , 2:~,,, 2:s..,, 2:s,.,, 2 : , . , . 2za.,. 2:~..~, 2:~a, ?at.,, 2:s.~, 2:s..-, Ys, Yz, Y~, Y,. Ys, Y,) and tl~ cubic form in 2"/vaxiablm xpy,~

-- z ( p q ~ ) : ~ : . , : , ,

-- gm~X~A",,

(12)

wha~ #. q, :. t, u, ~. t,,/, k - 1. . . . , 6 , X, is r ~ lth elemznx ofX, aud d~squmm) is +1 for in eve,, pexmut~on of (1,2, 3 , 4 , $ , 6 ) , - ! for an-odd petmuxsxion and 0 ' 1 ~ non-am~o d e m ~ u

:,j,- +l

rot

~ 8u. mm fined below.

{~,;,k}- 0,~,21}, {],24,20}. 0 , ~ , 1 9 } , {~, 27, IS}, {2, 24, 17}. {2, = , 1~}, {7.,Z7,]0}, { 3 , ~ , ~ } (3,~,12}

{4,~},

{4,27.8}. {5,~,7}.

(7, 17, 19}, {8, 14, 20}, {9, II,21}, {m, ~ 19}, {l~, xs, ~,"}, {i3, 14, 15}:

{l,~,lS}. {2, ~S, 14}, {3,~,9},

{7,~3.2~}. {9, 16, Ig},

~..MI-SIMPL~ PEAL b'UBAL~I~BA.R OF L,~/IJ,,Gi~IP.,/~ 'V -I

¢m"

for

TI~ 811~

{~.j.,} - p . ~ 2~}. {.t.~}. {~. u. t2}. {e.u.9). {~. t2. 21}. {to, ~3, ~#},

17"/

{3, ~ 2o}. ~ ~ ~7}, {4. ~ ~s}, { ~ . u . u } . {~.22. t#}. (s.:a. t4}

{5. ~ t~}. {~. 22. ~}. {~. ~. ]0}. {e.~s}. {e.~}. F.~¢~o}. {8. tT. ~8}. P. 14. t9}. {m. t~. 2o}. {12, ts, ~6}.

I m a c h a r a c ~ of - ' R ~ d v~l be n~'cxed ~o u C ~ .

+ ; f ~ + 3 h s + ~ 6 ) is bomorphi¢ to tim ~ of u',,,~ormadom in 27 v a d a b ~ which ~ ~JB l x ~ . i ~ d d a i ~ . R c ~ d d ~ f o ~ fm'p,q

-- ! .... ,6 and

I ~ "rU*" --

for fog

/'6"1" 6
22,

and *.hecubic ferm (12), "I'nis algebnt will be refarn~ to as N£~. It Ires d m m m ~ 2. t'i')The ma
f o o l . q - 1,.-.$ aim (,r'~ -

l ~ 1"I,--,$,11.---,14,1'.--.21.~. -

1 " 6, 7, . - , 10, 15, 22, - - , 26,

and the cubic form (12). This aligebra will be r e f e r ~ to as N ~ #::d has ¢:Mrmg~ --14.

3.¢,. r.., Non- isomorphic real forms of E.: are geaerated from involutive chief iaaef

pht~

gm'~f~

(T) s4 - 0. (5') ~ " 0 , ( ~ .~ 0, ('w) "t "" 0,

-

to

t - 1, - . . 7 , 1 ~ 6 , , , " --1, i vt 7, "~ - - - I , i ~, 6 , 7 , n, "- - 1 , n, ,,, +1.

"r~.li~ (i) h - 0,

('~ b

-

~(:~. + ~ , . + a , . + ~ + ~ , + | ~ . + ~ . ) .

(iv) h - r d ( h , + 2 h , + 3 h s + 2 h , + h , + 2 h , ) .

178

I.M. ~

~/.F.

~

The tram of ~,ucp h) am 133. - 5 , 7, -25, rmpmivd~. Thus them are four non~ o r p h i c rm/forms or m, ~ , m d by i u ~ ~ : (/) The cmmtm~ form ~ by 7"(exl~h ) f m ' h -- O) is immorphic m the group of ~ in ~ ~ ~ -- - x ~ , a n d j,~ - -~,,~, p, q - 1, ._, 8, whkh leave invariaat tl~ peuidve ~ ~ f e m the bi]Zm~ fozm

for p , q - I , _ , ~ ,

and the ~ x~x.~v,.j.~, + ~ ( ~ )

form (xm.r~x~x.. + y ~ .v,,O',..v,.).

(17)

This algebra w~l be n e ~ to as ¢~--v, and h a s ~ --133. (h') The n o n . m m p ~ real foem ~ by :r(exp h) for h - .~t'n~ +2b,+3hs+~lx,+

Inv,,/nva,/a~ ths forms U@ and (17) and the ~

~

form

]~,Z,,~,~,,r~. +~,,.,.~",,,)

(~s)

for 2, - , ~ . - - 1 , 2 1 , - l , p v t 1 , 2 . This alsebra wgl be refened to as NF.~ and has c h a r a c ~ +& The u o u . m m p ~ real form Seaward by 7"(ecp h) for h - . ~ ( ' ~ + 4 h z + ~ s + lav~ns i a v a r ~ t the forms (16) and (17). This allebnt will be r e f e n ~ to u a v ~ and has a charaaa, G/" - 7 . (iv) The non-compa~ real form ~ r a t e d by T(ezp h) for h -- ~Cns +2hz+3hs+21~+ +hs+2hT) is isomorphic t o t k llr~up of tr~Mrm.mations in 56 variab~ x m and j,~ eoaue:ud by

which leave inwm,iant the forms (I6) and (IT). This al~bra wgl b~ refened to as b'~"~ and I m chamam" +25:.

3.~.~ Non-isomorphic real .Corms of E, are generau~ by involufive Lunm". c o ~ s to 0) n ~ - 0, for all L, ('6) n f - 0 , t ~ 5, ms - --I: t ' ~ n , - 0 , Z ~ 6 , n, - --1,

a

~

s t ~ a ~

oF L ~ ~

P.P.AL ~

v

179

(i) h - O , (H) h - "M(6hs+121s~+l~s~+l.qlt,+121ts+li~+4~+gl~ (~') h -- "M(41tt+l~ls+12hs+ l ~ + l s + ( J ~ * 3 b t + ( J l t t ) .

aad

Thls i ~ , ~ s that there u e t~-~ re,~ r ~ = s a £ , s , , a m ~ b~ h m ~ s . t e m ~ m . if) "rh. m m p s a real fc, rm lleaet'aed by T(=p h) for it - O. has ~ a r a c ~ r --24~ ...,4 will ~ r d a m i to ss CF~,. (TO The n o a - c o ~ real form Zmersted by ~ezph) for h - ~ ( ~ , + I ' ~ , ~ + 1 ~ + + ! ~ . + 1 2 1 ~ + ~ , , + ~ b ~ + ~ ) bss ~ +S tnd wal be n~amd to as H / ~ . ('th') The otlm' ~ msl form, Ilenerat~ by T(up b) for h - - m ' ( 4 k t + S l ~ + + l ~ , ~ + ~ a t , + t t l s , , + ~ , + ~ , + e t , ) has clmmc~ - 2 4 stud will be refmtd to ~ 2 V ~

4a. Cb~f ~

~u~q~ms

~r ~

- U ' ~ e . q ~ b)O-', wbaz b" is an inter automor#,;-,,, of .fo aad 74 exp (sd h) is a,-~'ou~ u t o n u ~ t ~ m ( z # - t,). TUmis cm 7.. (net ~ . ~ iuz to acts m m s m a t m m p m m a o f ~ srtmp ~ atstommlt~ms. ( : ; a u m a m p.3] h w s h m m tlwt £ , l m two such cotma:md cmapoaeats, o m m m a i a i a z tim ismm mstomm~ phim (Z, -/) tnd o u ~ S the ~ ~ ~ a x m p m d s m a'Im. recalkm" : (as d~b~l ~ ~ [12D ~ th, ra~ slm~. In ~b, ca~ ufZ~. ~r ~ssuch ~ -u~, !ml,...,$, ~t=t

~

m s t :,may Ze& ~1~ Zeq Zo~

" "

~.. h,, e6.~, e6.

t-

1 , . . . , 5,

i -, i, - - , $,

(20)

This rotatJou , is u m d s m d w ~ tlm tmtsformafloa z: .., z6.~, J , , l . - - , $. ze -* a4, that l e s ~ s iawuism the Dyakin diqrmm rot ~ . The omer ~ of Dr w n be expressed in thz form T(q) ( w h a t q dots am t x ~ n t to $0(21)), t , z bern t h a t is no •

f ~ Z e

,,, T(q) is comisttm with (2Dk to Z¢ mint lm mtpcmmd in a diffamm form.

4.2. As ezpndt form for Z , ia trams d £ Sincw in [Ib'] Mehtz and Srivasmvt show that the ~ ~ dimensional tatiou of' E.Q is not a;ulvtdcnt to its ~ coa~sme., tim opermion of compkm eoqj~ priori K must be an o u t ~ automorphism. It must thet~om be possible to ~ p r m s 74 as

7-, - T(d)a:.

(21)

180

J.M. ~ S

us was- done in the ~

amt J.F. C~KNWlK~

of A, in equafiou tILT), wha'e d • G, - CF.~. k" is defimd by

~ e . + ~ - e.÷Y,,, ~(..-~ - -~(..-~.

(22)

Hence d i -

di~ de,

-

~t~-

-k...,d. -k,d. ~,..~d. e~..~d,

i-

1, ..., $,

i - 1, - . , 5. i- I....,$,

(7.3)

~

One solutio~ of' (23) is the symmeu'~ mauiz (24) 4.3. b d

r u m p m m m i U~ e m t rumtue,w ~

$ i a m A -- U'Z4e~p(adl~U-t jlemtltm tim same : u l focm for all U IxdOUlli~ to the Im~ ofim~ smemm~ e ¢ 2 . , it is ealy ~ to m m i d ~ tlm cam e l ' U - L S~nce 7-.ok .,,it. h mint be d t l m from It m ~ t ( ] ~ l - i . l g S ) ÷ ~ 2 ( ] / l . 4 - 1 1 ~ - k . ~ j k j t ÷ ~ ] l O .

We require that Ze .~cp(ad ~) m u ~ be involutive and m 7-0 is iavolutive., the Zob -. h implim e~]p(ad h) mus$ be im-olufive also. Thus exp(ad It) is an involufive chief inmu" automo~hism. By reamaiag similar to that in Section 3.1. one obtains

(J.,, ~,, 2~..t,, J.~. ~

- =~(n,. a~. n3, ~ . , , . ~ A "~.

C-S)

where nt (I - l . . . . . 6) are inteSe~ ~nc~A-* is the iavene of the Casum matrix (sec A.3). Gantmach~ [13] has shown.that oedy t ~ o sets o f nj in (7.q) l a d to the p ~ - ~ i o a of uouisomorphic teal forms, namely when (~ ~ - O, i -- 1 . . . . . 6 and (u'i)., - O, i -- ! . . . . . $,

n,

- I , which ~ ~ It - 0 a n d (it') k - u r ~ , + ~ + 3 k ~ + 2 h , + ~ + 2 l O . The aummorphism Z.exp(ad h) for k - O malx 2 6 ~ into ~ e s .

-

umady

h~, h~ and ~be jenmstc,rs e. and fe ~ with the foilowb~ roots: ~j, go. ~ + :r.,,. : = + e t ~ ÷ g . . g = + g s ÷ c . + ~ , , zt + = 2 + z s ÷ e ~ + = s , gj ÷ : z + g s + g . ÷ g s + a t , , ~t=÷2g~ ÷ + ~ + ~ . , % +=z +2g~ + g . + % + f f i . . =~ ÷2ffiz+2g~ +2ffi, + g s + g . , zt +2~z ÷ 3 : ~ +2.~-I+ffi.-+%, Ca +2ffi:+3ffi~+2~,,+:ts +2ffi.. Thus 7.. has a t r a ~ of +26 aud the real form i~ntq'med by it has a char~',t~ of --2& This real form, heaa~orth r~f~'md to as NF_~. is isomorphic to the IProup of tmmi'ormatiems in 27 complex variables xp..r,,. --'0,. ( - - :,.) for p, q - 1. . . . . 6. which leave i~m'iam the cubic form x,.~;:.--

t(~lxt~):~,:..

(26)

S P . M t - ~ I , ~ REAL IK~AI,GI~R.~q OF I.~ .ALGZag~ V

I|1

the .-,,daME bein| ' m b j ~ m the madidem •r ~ a , u m a l p l ~ a Z.eqD(td~.) for ~ . - ~ ( ~ . + ~ + ~ . ~ + 2 ~ . . ~ + 2 t ~ ) e,q~ ~, lie ~ ~ 1111lsratogsee $J~ f~ ~ with za, Lga+gS4"f~, =t + g a + q + ~ t ~ q ~ a t + 2 ~ + 3 ~ a + 2 = . + a , + 2 a , into ~ aM multiplies the I t a m u ~ asmc!ated with tl~ followiql roots by ( - I): a~. za÷et,, s= +g~ ÷ z , ÷ z , . zz ÷2z~ ÷ z ~ + a , . s t ÷ a ~ + 4-~14,~÷~÷Ie, gt'k~tl+2s,14"gt+gs-ksct. =t-l-2:ta-l-2aa@2:rqt+~l.kge. a,t-I-~t|'l+3=,+2¢,+q+q. Thus the automerphim, has trace - 6 . H m the hal form It, mr-_*~ by Z.ap(adh) for h - ~0t,+~=+31t,+2k,+bs+2k,) has d.,ntr.er +6 aad will he m i m e d to as N ~ . Itisisomerphicto the Iro~p of m m s f o m s ~ m ia 27 real v a ~ x . 7,. : ~ ( - - . ~ which leave lava:ira Use form ('2,').

As 7., - T(d).g for ~ . which is exact~ the same as for..4., tl~ t~:~ery o; ette.:s~em of Ze for ~.~ ~ throush ia the same way as that descn'btd i~ th~ lagtcr pagt of Sr:,itm 2 of n The * - s t o p , , of eq,-_doas (n..9), (UAO) ~M (iLl6) are m p e a i ~

- WZT),

(26")

for get

(27)

(which f o a m ~ from (24y),

~p

. | eq,(s_)d !,) t - ~ P ( ~ t

~ - 0 C ~ for ~' - ~ ' ) , ~ ~* 0 ('~. for .t' - ,V~'*)

(whkh foaows from (23)). and

jr(w~), r(d~(c=pr(ho)-r(d-)*(cxpr(ho)- Ir(-I(~7)).

s," ~r

-

,v~. Jv~.

(2S)

The metluxts for embeddiq real ~mpte Lk algebras in n=l dmple Lie Jllgebra, Im'ea in L li, and 11I, require modification if . ~ or .W is eggt.pdomd, zlthc,ush the tw.dc prhgiplu are the same The mczssa~ and sumcizm c0ndidol ([j$) holds for an)" send-simple real Lie algc,bra. Assuming that F can be redu¢~ to the form

r - s(®

(.,,)

(tl~ FJ bang irreducible ~ t a t i o m o f .Y,, occurrialg pj dmcs in t ~ .red~.m of F) the detzgmhtadon of B may not be strailghtfot'ward, and B may not n e c m B l g ~ ' b d o q t o ,.w. If .~ is classical, then th© dgtgrminadon of B is ¢askr_ It may either be taken as I ~ithcmt

182

;.M. ~

met J.P. C O ~

lossof Zenemlky, re'isSiren in I or 111.M o n m w ~

if~ isclsssic~ tlm omuslim~ m y also

be cslcu/stafu in [~/.u tim m u m ~ of tlm ~ d o w not emm" into tlm ~ m m m L A nwmsary c o m l / t ~ for . ~ to" be a m b s / p b r a o f -~ is ths~ ..~, be a ~ ~" 3 ' , .

b ~ , ~ i s byno m s m s s S ~ m , on roman81 m ~ J p b m

h is p , m ~ , ~ b S t~s n n ~ s o / ' ~

u ~ S , ~ . ~ [19]

~ c,m m ~

Lie sJSsbau to list -II semi4im~ ~ • siren compsa Lie ,dptm,. ~ w i t h tim redaaim or d p m s i ~ e J ~ k ~ n s

of

may be found from tlm tree ia l q ~

1. E.sch bem~ut8 pmom j i m all mmdms/semi-

(cu) dim7

(1,0)÷ (0.1)~.(0,0)

(W

dim 3.4.3-0.1 j

dim7 Ate,dr dim 4 ÷ 3

,dr

,At

(2J ÷ ( I ) ÷ ( I )

(1) ÷ (1) + 3(0) dim 2 . , 2 ÷ S x l

dim 3 + 2~'2

,41 (2)+¢~÷40J dim 3 ÷ 3 + 1

rq. l.S,emisebm or ~ (0.1) G, Jm 6 ema.4quiv'~demcom~x L~em ~ (C'~, Ins t ~ t mmpm:: nml I.~emtmJSelma.) The reducdoa of the repememaskm of eazh ~ and Ss dimemtom ~ 8iveo simrde ~ of pr~imss a l j ~ r a s and the rcduaion of the r z p m ~ m d o m . The br'~nchin8 process continues unul thee: a.,': no noa-tfivi~ maxima/ s u b a l l ~ of the last subqdsalm: (Le..4.); then all ml=',~-bras have bee: inchtded.

s ~ m . s ~ . , ~ REAL S;~ALGImh~ OF Z.~ ~

V

183

t ~ m s ~ n ~ suhdSeb=s or= non-cmp=t n ~ form .~" o~.~. ~or m a q ~ u ~ ~ - ~ . ( t . o . o . o . o . o ) . ( ~ . - so(~:)~ ~ . so(st) ((t. o, o. o.o)+(o.o, o.o. o)). ( . ~ , , - B,) and ~ - ~VD==. ~ ~ is =o r=l for= of ~, ( 0 . o. o. o. 0)+(o.o,0. 0,o)) which b a mbxil~ora o~ND==(N= Seuiea 7.4 ia 1). NDto ((1,0, 0, 0, 0)+(0, 0, 0, 0, 0)÷ +(0, 0,0, 0, 0)) is hcnveva" a ~ subatlgebm o/',,v,V==, eve= tboush SO(X0) is uo¢ a uuud:~ suhdpbm d' SO(t:}.

" r ~ ~

~

P'~ =. C m ~ t t.ieatstc~as (rank ~ S) ~ P, t~ a k ~ m ~ c'r..tr.s~rsa=y o¢ t~ tbo-~t , ~ a ~ w = m t .s~" anx other t~ a4~m. thin .~" ® .s~"~ F,

toms o r ~

In order to fad out if ~ is a subalpbra of .~', the reverse or the broaching process may be used, as in .~lgure 2. Once it has been established that ~ ca,, be embedded in ~ , with a reptzsemtat/on Z' which reduces as in (29), B be/nlg unknown, there are three main

caiculatiom: the ¢~plicit form of r (usually iawoivinS lrh tits r~t~sica of S" (the automm'phtsm which ~ ~ and the ~ i o n of the cmtralM~. Then it is str~lghtforward to substitute ~imo ( L 3 ~ in at similar way to that in L I! aund HI.

~be n m ~ o~ ~ ~ ~ fo{lo,~t{u,cmq~min tbs ~ ~

~ 1 1 for e ~ [ n i e m i l, ie ~ r ~ for d,. wfm/s$ / . ~ , / ~ ~

In a n n whe~ d ~ m i s ~ y ~ ~ pare'ok m end ~ ~ expi~ x~

¢ ~ i ~

The aura ~ ~ by N ~ and Q{i{"")

of.~

for I' from ~!~ ~

~n cmbeddJns o t ' ~ iu £ , rOttD - k , , i - 1, ..., $, , ~ 1 ~ oF

ia ~ , . k m~y be

dhmmm (~-r. for

-- b,). Otberwbe a solution

is required (ba~ b; and I~ u e d ~ ~ ~ . ~ . and .~.~ In some c~scs a solution for the ,,.. may be found by ~ b,a e,p~offi, O0) ~e u e ~ l m ~ . ~ ~ L (This is

of D, and ~ (see [61 and ~ which ~ the same mdueek~ :a~l a m not qmv~ke~t (Le. ~ 1 ~ " 5 is no, an b ~ - ~c~m'le~umm). l.n such ~ s C',0) mint be ~h,ed ~ mm'~mu. I~ h o~'um simp~, ro soNc (30) by comidm.h~ ~be subse~ of the equdom for which TC,=pl"(~9 ) mmz also be am imm" ~ r(~9 tea be c h m ~ to ~:~donl to time~ TI~ eqummm

H~ ,~b a ~ ~ of ! . mtmll~.bm of.~. withou~ loss of' jpma-a,li~.

can oitas be mlved by impec~oa, by comidmin| tlm s y m m ~ p m p e m ~ of tim Carom sub~s~.~ (~.s. ca.u~ emems ~ zero, m,.isymma~ ~ -,~ ~dt,='um c l ~ ) and the i m m : h a a ~ of rows and c o l m m ~ U a l e ~ tbcm~ is morn than one oon-,conjulp~ embeddins with the s a m ~ a solmbm dmt~ ~ tory be used without loss

W{t~ r ¢gpmu~ in the form (2{) tlm ~ ~ .

i

~ m

~',(~ ( p j ) ® x ( . , ) ) ) ~ .

¢ ,all l~ of tie form (32)

where c'j" is a pj by pj .,,st,ix. By m u m p ~ n , c b e l o a p to G, amd can ~ b ~ be wrJmm c - exp 2,b, (b, t~nS d z Sene~'..~s of .~'~ wbk~ pbums some ~ c m s on ~ c~'. TheR restriaiom may ofte8 be found truingthe followimq{~bttiom{ailm:

~

X.E./G, SU1kA;,fll~k~ OF LI~ ,~A31UIIL~ V

('/) ~ s e b ® c - - )

-G~O=Ep

b~qc_

~ ' ) - IKa:pA)~'. ('m") U" ~ - - Z (',~- A - --A-'), t i m ~pAA - Ic~A+As~,L

Cw) I~A ~ - - ~ u d

d = A - - 0 C ~ A " a h B Uum (3~) b o k ~ dum a p Z A -- l + A . s i a A + A Z ( l - c m Z ) .

As shown in [5], c can m'hm be ~

('~

i ~ o ~ e form • - r(e~pf3r".

~

r b a ~ "

c¢ ~

~

,

~l

Sa m ~

o t tb~ Csrum ~

T ~ s ~s

J

m s m b m of' $C/(n~ ~

msukuic~ on dm ~ ,my m ~ u du= no • a u be T o a d

I

cue h am Su~emu~ ~r ~ , . " n u i~ is uumru~u~ kuow ~e rot= or ~esp(,qle) =

su=

in. d~mmou, tl~ r ~

r t m ~ tit ~

(u) (=p=dZ.~U.)(=~+r~) - (,~+r~)¢o~(p(~.~)+~(~-r~)~(/(Z.~). (-~ ¢ = p = 4 ~ ( , ~ - ~ ) - ~(,~-~(z.u.))-(,~.r,)~ ~(~0). c,,.,,)

- r,,,+

(vr) (apad~.Ce.+~)t(es-r~) - A/(o,-r~)+SKe..e-r.÷s)+Cr(e,..e-f....~)+ +Z~(e~,~-r,.~t) +B(e.,,..,-t'.,-.~+~(ekos-r~,~,) + (vii) (=p,Kl. Z J ( = , - ~ ) ( e ~ + r ~ ) - .~(,n+r,)+ s t ( e . . , - r , . ~ ) - a ( e . . . ~ - f . . , ~ -- D(e~u.,.e + f,~..,.n) -- E ( e . , - 4 4 + f-:.,4.~ -- F/(e~.,.p- f~**~ +Gi(e.aa4.e--f-a,4.~),

(-h+B.

-=+B. 8)

(8. a+B. tr+B.

. l-+8.8.

3a+B)

a+P. la+@)

187

- ,A(~- r ~ ) ~ ( ,

+ v~.)+

+hc~+~ . " ~ o -

(-~+p,

~)~.).

- ~+ }.~),~ + ~.)+ +,A(~- r ~ ) , ~ c ~ - v~.).

- . + p , P. -+P)

r'~+m,~,(~o- r-~.). - I r 3 , ~ o + v~.)--I ~=,~.Io- ~.). r ~)~,,~(to - FS ~.). c . ~ o r"~-s ) ~ o ~ ~.'~ ~ + +~ +

(-3,+#, -h+p,

- = + p , p)

- I, ~'~-~(lo+ F'~.)-

l

,~.o+ } " ~ : ~ -

v~.).

~& ~ ~ X' a d 5' a m Im~ I l l The mlpmzm is very simil~ w d ~ Sivm m L I r c a m I m ~ ia The f~m then the madifions A aad B in I ~ 6 bold. Mm~ lpmenlJy, the neamsry m d ~ for an embeddls~ are:

A': T(~p(r(v))c) ~ m, L~,o~ah.,.~ q ~,. r : Tnz. {T(,~pt'r(~')~)} - -,~ (,s i, , ~ _ - , . ~ ef.~. SAL ~

~

$ m l 5" m b , ~ u a ~

r,eakm 3('0) ;,, !I. Sa:fiom 3(,1), 6(c0 and 7(d) ~ ~ chnl wi~ ~ runes where .~' ~d -Gm ~ (A~ ~r D ~ T i m ~be n ~ m ~ msm am wbaz .~' or -~ is E~. Sincz ~ is s sulxdsd~ of Az, taxi im ou~r ~ m be z~lmmed i~ u~ms cd"the Ol~rstioa of complex c~.uf:ssion Jr. the 8mmsems for f., am ~ to them for A,. .S" wUlbe el" the farm ~',.~), ~ . r ) - ~"- esp(ad t')7"(¢) "~, so .5"~ win be T ( r ( . ' ) ) • • T(P(d'))- K"m - exp(ad rta'))- T ( r ( ¢ ) ) - , .-ha. Jr.., may be mk'. u TOt)- K- 7~3-'), (I" beinl upnused b the f e n (29)). The ~ fo~lows that in II $,~-tioa 3('0) up to equadoa (20). whic~ bern am8 -

e.-

Ir(_i(:. +l)).

•

~-

Q:,,÷,,.

(4o)

181

J.M. ~

~

.~.~. ~

0; i x ~ g + I , as n remit be nmllsld q z" - 1). Takinlg into tccotmt the stntctu~ o f c livlm in ( ~ ) , this coadition m to:

A: c;'O,~)'cT(a~) - xfa.a,

for ~

- sz4r+~, e).

og

A'~

=;O,j)'c';O,~)

,I0,,). for .'r' - 0t,r+,,.

-

w~=e r , ( - t ( r + t ) )

- -,z0,s}. ,rj ~

±L

(b) .~ - r.,..~" - D(. T b e ~ faiiows that liven in I l l Section 7(d) U ~' 4) ~ d Scgtio,, 8(a) (! - 4) lea,linll to t]~ amditkm: A: For ~ r# weem~g ~ tke reducem o f t U" tl~eform (29)) (1) d~er, ~ an s} tuck t/sat .;r~(IV)'s} --~ . r~(Q'b'q" -s) for z~ V b~ .~, or (2) Z ' J ~ ~ Z ' J ( Q ~ ' Q ' - ~ - Z4"~'~. ~ d~ sam~ ,a,mem. oJ'~mz. ~ tar ~ b c t / m o/ r (i.e.. p j - p y , 3 . cuension of' .5" - 7 ~ ) is tt,,,, 7 ~ , ) ~ . . w h ~

r.,, - ]'O~-x. ~-~)

(41)

. - s(® :E: O(p,)®.;)® ~co,.e.))s--.

(42)

and s is of the form J

w.~

L(m) is dzfhxgl for even m in the Appendiz of II ss (II,A.9)

and wbcn~ the 1~J satis~ns AO) are ]isted ~ and the 1~ and 1~ ' smist'yinl A@) age listed consecutively in (29). The i n v o l u t ~ condition tiara reduces to the form s c ' ( s c ~ - x('n) (n - + t , ~ c , it mu, t ~ n,,/a,,d ~ = 1). ~ s t l , e ~ ~ s l a (42) and g in (32) ~ becomm:

a: /f r , ~ z s A O ) . . ~ , c;'(p~@fp~) - I t ) . c~'@,)-~--(p,), c~"(~t)-~'(.,,.,~ - zOO, The u'sce condition is: ¢: T ~ {~(s)..g.,,. ~'(¢ap r ( ~ ) } - - a .

m,~ ¢ p ~

A, B and C am a set or necmary and suflichmt conditions for

(c) ~

-~..~-

AO). t ~

8nembeddin~

A,.

Since the out~" automorphism of E6 can be ~ in t ~ ~ way as that of Aa, the argument is the same as in II Section 3Co), with sL(r+ 1, R) replac~ by NF~ and Qi(r+i ) replaced by ? / ~ . Equation (28) is used instead of (ILl6), and condition A (for .W'

S l l ~ q t ; d P l ~ P.F.AL St~ALGlt~qAS OF ~ - N E D ~ k' (fo~ 2 " - N£~) and coad]~oa B ate a z dittos for ,.,

~

~

V

189

of n m m m 7 a d s . ~ i m z coa-

(a-) . . ~ , . E,. # - ~,. The srlpun~t siwm in 111 Section 6(d) based on 6(c) and II Section $(b~ holds far

Theam

wbm .{" - A, orb, :,r~ c k ~ wi~ in ]I ~

me a u n w h m # -

m~

~. ~

.e/~ (I) be q ~

3(c) sad I I I Sa:doa ~'o).

tlmm~ is very similm, m that Siam in 11 rmml~ 3(c) far

to r~" (i~. y~ , ~ m ~

# ~ Tty~r~- - I-% o~ ~

t~ s a m m . b ~ of.#m~'~ r ~,/~ (2~) Cu~ ~o T ~ s y is of tl~ form

y-

ZOc )®yy)e J

smunius ~ ~ rJ m i d ~ s A(1) m ~ ~ and the ~ m d r ' , ~ A(~ are F.zed c o m m n i ~ . !13,~ tl= ~ d c m : y Si..e. in { ~ ) m d (43) u d n o ~ ~ z y} a m be ~ o m , ~ ~ y]y}° - ~sl(,#. wtm~ e~ b +1 for r J msi and - 1

m/~ the im~lmi~ cce~lidoa ~ B- (1) r j ~ A(I~ ~ ~ ~msr ~m~ ( ~ ~ : ~ - - 1

(2) If I" ~,~ r'* ~

A(~ ~

#~ q'~,,)~

:~

~k~

~

m

TI~

~ , ) - ~(p,).

11m raze mzdidoa is

Ccmditiom A,'B and C form a setof ae=esm7 and z d k k m

~

w~li6om for a embed-

wire 3' outer and $ ~ m '

This case only applies if' . ~ - A t , Dr or £6. The cases not c o v ~ d by I, I I or 111

(,)

.~" = r., = w ..~ c ~ , , L This case is very similar to the case where .~" --, Ar described ;n II Section 3(d) and

190

;.M. mlUlqS sad ,LP.

5(b) sad HI Section 6(c~ The :ondi6on Siren m these secsiom a p ~ s ~ ( r ÷ ~ , ~ sad qi~r÷ . r e p ~ = i ~ ~ b~ c . ~ , ~ , ~ ~,

for s u ( r + I), ~ .

TI~ a m z i o n of K" must-~ of the fore 7~) (for y of tt~ ~ ~ in (43)) and so a mcmmry rendition i v A: r tmm bs ~]sdwdm: to r*. 7 ~ u ~ r : o c a m C g ~ s t ~ r m b a ~ o [ r ~ (~) aaa: t/t/ur (1) ~ t ~ m / ~ , m r~* t ' ~ ~ r t zztst: ~ yj ~ : ~ 7 ~ r t~ - r : ) , or (2) ~ a t r ~,~ m , , t m m , ~ o f tams ~s r~" ~s ( ' ~ fLL p~. - p~. T~ ~ ~onom thst t ~ m b U Saniom ~(d) w d SO)), tUe csmt ,taU ~V~ sad ~V~ being as for $L(/'+I, J~) and mspectivdy, but c mat ~m mzmsm'~ b~ of

~(r+.

th~ form .('~/), so the hzve/utiw, a m d a ~

~(r)_.

is:

if

~

- xJ.(r + ~, ~) or ~v.rL

wb~er' and r ~ the ~ of.~, aad.~,, ~ . exmpt when . f - £ , . " t ' ~ mdum, to ~ c o m l ~ x ~

I q ~ 7 0 0 -~.

• ;;0~)" '~[,.~,~;(p~)-..

~-

fu

~ - SL(r+ 1, ~) or N~li~o

~,

x - Ot,.. , ., hi.

Thz tram conditioa is: c: "r~-~ {ffr(¢).~ ~ p r t ~ ) } - - 6 . Coudidom A, B and C form a s ~ of mmsmry aud suMabm ~ (c)

! and q muscbemsi

far m robed-

-~" - ~ - m / . ~ a c ~ m m / . in III Section 3(c). a q must =ist such ~ ~ ' ) - r(~Q')b'), w ~ ; ~ : A. 77~ r~ ~ , ¢ s I ~ t/w ~ of r (29) mua .m,~y (1) thtr, , ~ , *5 ~ q r a u r to ~r. =oh ~ .r(~3rJc,~ - r s ( r ( q - ) b 9 / o , air b' in 3 , , a t (2) r10¢) ~ , its co./u~t., r ~ ( v ) , rl(T(Q*)b ~) must occur t ~ number of

SZ/~I.SlMPLZ ]~.AL ~ G ' l g ~ . q

Ol= I , ~ ~

V

191

"rhm the exumioa ~ ~Q') is ~q), w h m q-

u(®

0(,,.,)®,,;)+

T h e m l u m ~ t f ' o l l o w s C h a t S i v a s i n m s a : t i o n 3 ( c ) , 4 ( ' o ) u d 7 ( c ) . The involudvemndifon is (qc): - ,~l(r). wlu=e ,7" - 1. Thus '7 must be m d except whm _~ - £,. Txkins into account the su-ucturu of c and q ~ in (32) and (4S). this mduzs to the conditiom:

B: O) ,r.,,,"r~ ~ .

AO). ~ , , c;Le~ - nl(p~

:f r • ~ ,.,(2),~ T ' ~ u-,w~ =ondidoa ;. Oam

.~(~,)~"L~,) -

~r

,~(~).

c: T== {~(~=pr(~)} - -a. A. B a n d C m a s~ o / ' n ~ m a r y aad ~

~

fo~.-, ~

(.~r' and ~ arc two r ~ Lb ~Sdmm ~o¢ b o ~ c h a i c ~ t) Use ~ ~ pro=ss to r ~ the r ~ u a m or an ~ n - c n n k l p m ' ~ ~ ..% ia 2 . . Work t ~ ' o u ~ a ~ h ~ t h ~ s ~ p r ~ l y . 3) FTmd ",J~ foren ~ r,~ ~

dm~mu~ ~mcl ~

r,~ or" Z if' c ,~,,, be c q x w ~

m

~am Z,.~r.l~. 4) Cboem ,,,, S" which ~ U'.$" is xa omm, a ~

~ erom .,% ( . ~ - - ] ~ . ~ . find its ~ (Le. ~ I ~ o / ' y ,

6) Sa~ if ~be m ~ ~ c m m ' ~ x~l s u ~ k m t condhim~ (jivm ia ~ or $.~) for tbLs cBe ~-e ~ for any c in the ~ of ~ (it" i~ ~ p p e m in ~ cmdi~mn).

and ~

s o ~ q is

~..~ 5.6, ~.7 aaow~d w i n e

and ~e p i n t , ors of .~. are due same limmr combiumions ot ~ as ~he I m m x m ~ cd'.~, are oC M... If" the conditions in (6")c u u o t be simulumeously safis6ed, chcu ~ is not • sub-

aiZetn ~ .~ foc m y mp~mauukm ~

~.L ~

otr~

to r .

f a - m or ~

As w/ll be s~n from H ~ 1. there are six non~onjulWJ s ~ d m p i = mballls~'~ of G=. ro,r of these are non-conjulpUe m b ~ d i n p of A:. and t h ~ is an cmbedd;-g or Az aad of A s ~ A s . We wiU examine them in the s z w z ~ order.

(It) . ~ ,.. A t Q A s A~OAs is a subaJgebra of G2 w/th theembedding reprmenwtioa r - B{((1)@(1))O @ ( ( 0 ) ® ( 2 ) ) ) B - ' . which can be n , ~ z = l --

x. M. ~JU[NS md ~. i~. COItNW]~J. r(h| ®l) - a ( - z t ~ - z ~ + l , ~ + z ~ i J -~. r(~®h~) - a(-Zt.: + l ~ - I ~ , * ÷ I ~ - : X , : + : t , . , ~

r(e[®l) - I~Itj+I~B-*, rCl®eO - ea,~+x~,+ I / ~ . +

~.

l/~z~,)n J.

wh~e B ata be u2ms to be

ted ~e cq~k~ form of r is ebm rthi®r, - - f ~ t +h~). r(e|Ol) - |ft:m,

The' mmraliz~ ~

rG®~) - -~. r ~ ® e D - f,.

(47)

a:e et tim farm

c - ~(c,x(4)ec, z ~ ) a - , - ~ (c,. c,. cs. ca. ca. c,. c~). Thus c - exph for h - - ~ ( R - e,I). ca - I ~ d cs - d:L for both ~ otn ~ d ~ ~ ~ 1 ~ e~aS t,, ~ s (a - 0 ~ 1 ~ - 14) md ia~VG, (x - Z.umw - - Z ) . ~ , 1) • SC~itlmmmmdby T ( e K p ~ ( i ~ ® ® O) , ~ eros r . r - ( r ( ~ p ~ k , ÷ ( ~ ) ~ urue~ em ~ c~ dizim rot ~har v~h,~ or,,. s ' . r - T ( a q ~ , , + i ) i , ) r~r s ~ . ® S~(z. z) ~ uze i=vozuzi~ co=~do= canoe be mdsibd. SO,Z. z)® sz~z, 1) is pmnmd by S"

m t i l m the involuUve condiUon for m - O z a d 1. Th8 tm:e ¢oedkJoa jives an ~ l l b ~V~;z (n-- O,Z. t~ce-- --2~ S~2, C) iJ p a i n t e d bY Z', ~ thin rs n o ' m m s i o a of Z' for this nqmmntaUoa (see IV ~ 4~'~ so them is no embaddl~ ~ $o(~) ® so(z) md S~£i, 1) ® szTfz, 1) = n be a n b d d d ~- ~ forms of Gz. The pzza.zmcs ef $ ~ 2 ) ® $ ~ when embaSded in C~z ~rNOz m - K I t , +MsM ~ ( 1 . 1 ) ® ST.,'(I.I) ~ ~'G= a m - / ( 2 i s + l ~ ) , -his. t / ( e t ~ ÷ f , zz.~), Kez,,~,- t , ~ m ) . Ke,+rO. ( ~ - t r ~ Co) .~" - ~lz ~lz is a sub~LlZebraof Gz forwhich eN f ~ eftl~~ ~ is B((I, 0)~ ~(0, I)~(0,~))a -~. whi~ can be ~ ~:

r~D

- l~-lt.,+I~+I~-I~.~ll -~.

r0nD - a ( - X ~ + Zt.~+X,.~-Z,.~ll~. r(eD - n(l,a-z,~)s~. r(eD - ]t(l~-I,~)ll-*. where B can be taken to be B explicit form of X' is rhea

+I,~+l~.+t,.~+I~,+lu--I~÷l~.,.

(48)

and the

SSU~U-SSM~ZStZ~C ~ L ~ S

T~ mmM~

O~ Cm ~

V

193

r(,o - +f,. r(,~ - ~,m. ~mmm m ~ ~ fore

(~

c - B ( c , x ~ • c , x ~ • c~x(x))s-., - ~tapx.d (c8. c,, c,. cs. c~. ca. c,).

TI~ • - ~p ~ h - .z.rv~, c~ - 1,rod c, - ~ m S s n m ~ by 7 ~ p k ~ V - 0and ~ + h ; ) ,

,,,d

+Ib,0÷2,,))).

e q ~ $ ~ 9 and S C ~ l) m th~ s ' _ r k T ~ p ~

"n,,

co,,dJt

is

mJ,n,d

for,, - 0 for ~ - 57./0) andn - I for .2" - SU(2. l) and the trim comihi~ or 5"00) h CGs ( u . ~ - l~) tad S U ~ 1) in ~VG~(u'~. -- - ~ . Sr.¢~ lO it W~mu, d ~ T(a'). x. ~ ;,, da, {o. ~) is the comp~ co,,jup~, of O. o).JLro~m y - n ( L ( 6 ) + z 0 ) ) n - ' , .uS ~ a S,,:,k,, 2 r(d') - B ( l s ~ - I ~ +I,., + I , . ~ - l ~ +I,~.+I,.,)B "4. 00)

Coaclidons A and B are satkfk,d, and ]L"(d~- c is B ( c , ( l , . ~ - l ~ + l ~ ) ÷ c ~ ( I . . 6 - 1 s . s ÷ +It.j)+X,.,)B ''~, so condition C is sa;isbd for an cmbeddin| of S~r2. R) in NG. ( ~ ,,s "-2~. Thus ~ truly anbeddiap of tad forms of A:~ in ~ forms or'G: we S ~ in CG,, SO'T2, I ) i n ~G~ and 5'Z(2, A) in h'Ga. The ~ at X'(.~') m ~ J l ~ wi~ (eJ . ~ - A , which .,~ce ~o (6), (2)÷(2)+(O), ~ + 0 ) + 0 ) + (o). 'nzy ,,,~1 be m m i m d in this mSu.. ti) The mptwemadc,. (6) I m Senm'axa~ r(kD

T~ ~mion

mS ( 1 ) + 0 ) + ( 0 ) + ( 0 ) +

- -(61,.~+4f~=+~r~.O+~l,.~+41,.,+~l,.,. am b~ emb;~k~l d i n ~ y in O~ ( ' ~ B - 1% ~

~

f~

four

• r ( , D - I " ~ , + vT6-,. (fl) The c e m ~ r ~ is ~he klend~ (as r is irredua'ol~). $U(2) a d SO(l, 1) am jmemu~ by 7"(ap h~ .for h" - 0 ~ d ] ~ ; , respective. Thus 5".,~r is 7"(copO) or T(cq, ~ ( ~ + + i t ) ) , both of which s a t i ~ the invoi~ve condition. The trace ~ ~ ,,,, em~o~ SU('~ in C'G~and SU(1, 1) in NG~. (h') Whh the rcpnmmtation (2)÷(2)+ (0); As c ~ be embedded ;t, Gz in ~ e form r(h;) = 2B(-I,.~ +I~.,--I,..,+I,.~)B", r(e;,) - )/~B(It.= +I~..~+ I . j +Is.i)B -t,

(53)

194

J.M. ~

md J.F. CORJCWEI.L

wbaz B - ( I t . z + I z . , + l a . 6 - 1 , , , + I , . , +I,.~+I~.D. Thus r ~ s ) - ~ : and r(e;) - ( 3 / t " 2 ) ( - - f t z z z + * t Z z z ) - " r ~ centralizer eizmzuts as~ of the form c - B(c;'(2) ® I(3) ~ czl(1))B-:, c mast be z member of CU=, which impltm that the o f l ' - d ~ e l ~ t s t s m zero. Thus e - d i q l ( c t s , c u , czz, c ~ , c s s , c t s , c u c )

- - a p h for h m ~ l .

~

~

Ca m 1, Ctz m C ~ a z m ~ ( + ~ ) .

"r'am $'mYiS

for n - 3 in both ~ and the t n ~ ~ ~ an m a b m i d ~ of $U(2) in CGz aad SU(I, 1) in ,t'V~z. ( ' ~ The ~pmtee~fi~e ( 2 ) + ( I ) + 0 ) can be embedded in the form

rco~) - 1~2(--Is.s + I s . s ) - I ~ , + I s j - - 1 4 ~ + I , ~ . ~ ) B - t ,

r(e;) = ~I/Y(I,= +Z~.~)+I~ + I ~ , ) a " , for B~

[ oo ] 0

0 I(2)

.

Tlms rOtO - ~

and r ( e ; ) - e2, and the ~ d m m m s are of dm form c - B(c;'(2) ® I(2) S csl(3))lt-L which can t~ d ~ to the form c - . r(zzp 10r-* ,t-ith g -- ~ b2 (n -- 0, I). The involutive condition is satistq~ for ,5" - 7"(O) bat not for S" - 7"(zxp ~,~h;~ The trace coaditiou allows an embeddinll of $U~) in CUz (a - O, tram - 141 and in NGz (n ,. 1. t n ~ -- --2). * ('iv) The repreumtatioa (1)+(I)+3(0) of A s can be emlmdd~ in the f a , at

1"~s") - B ( - I I . ~ + I s . z - l s a ÷ I ~ ) a " ,

PCe;) - B(I,~ +l~)lt-%

(55)

with

B

-

It.s+Iz.i+Is.z+l~.6+Is.s÷le~+l~.T.

Thus r(hD - hs and P(eD - e,. "r~ ce~Iker eleamms are of the form B(c~'(2)® ® I(2)• c;O))B-L which cannot be put in the form (44) b ~ can be expmned in the

form

For SUf2], S'., Y - T(©). the involutive eoaditioa is o ~ ~ for ~z " J.s - 0 and ;., - 6-.ai, and then c - L The trace condition allows for aa embsddiall of $0(2) in CGs. S;,,, ~" - 7"((=q~e=hs)c) for SU(I, l), but the i ~ o h a i v , co~._i~a cannot be m t i s f ~ for ~ny value of c, so there is no embeddinl of SU(I, !). The semi-simple suballebras of real forms of Ga are summariz~ in Table I.

~

1

4

~

~

OF. I,~ X l ~ D ~ t A S

T,/~=~ k ~ n

¢ . = = ~ l.r= =I==I~

V

195

at m=l rm,m ~' G=

nm

ItmhmJaeof nqxwmtmtm

|

e

|

SO¢= ® M,'~ SCRI, 1) ® Sb'O. 1'}

O) ® (I)+(o) ® ( ~

IL

A=

nnu

1) ,Sz,,C3,~

O. o)+(e, 1)+~.~

|

(~ ¢~÷(2)+ (U} (2)~0)4. (I)

,,S'O0,I)

~% SubuJSd== or nud ror=u =r ~'o 1"he ~ oi'& m too muuemus m be ~ i= t h ~ aucir~ ==r., m m ~ u 0=~ ====b= ~ orma ~c,rms oI'B. ((o, o, o, l ) + O , o , o , o)),,=, (06)+ + ( 9 ) ) = o -!I m t = e ~ q p ot s z ~ c~ (a) ~ - ~ J~ (0,o,o, 1)+(1,0,o,o) c a b= embedded rod= rive acpl~ form for F =((o, o,o, =)e 0 , o , o , o ) • (o, o,o,o))=-' r~D - ~, r(h;) - k,, r(~;) - b=+h=, r~It,.

ro,D r(,D r(e;) r¢~

-

-h. -r, -fro. -f,.

The amu'alim~ elam~m a ~ of the fm'm

(5@

© - ]i(cix0@ • ¢;I(~+c~I(1))s "=, wl=~ch k a = e m l ~ r d CF, if ¢j - e z -

I Id

¢i.

± I . "f1=m ¢ - ap==i1~z, wb==

m == 0 , I .

.The r ~ ! forms of B, =:re SO(9-2p,:2p) fot p - O, ...,4, which am dd IWUemu~

Junto,aucemaqd=hm~ and S',,=Yis ~ m ~ h ) for (p - O}, • -- m=rhs =~(C,+,)h, + n = +6u, +-~u,) 0P = 1), ~((4+,,)b, + ~U, + su~+ . ~ ) 0 , - 2 ) , ~(¢3+,,)u, + au=+4u~ + . ~ ) 0 , - : 5 ) , • r~((2 +n)u, + ~ = + 3 u = + : ~ ) 0 P - 4). 4

The involutive condition on :/"(~,pgii.~ A,I~) is oaly = a t ~

U"an the & ate i a t e l ~ Thin

196

AM. ~

aad J.F. ~ 0 8 J ~

it is no¢ mJsged fo~p - I, 3. T l x m m condition ~

auras

of SO0) ia ¢¥, (n - 0.

u = ~ - 5=) and av~ (,, - 1, ~ - -4), ~ ( S , ~ ) iu ~v~, (, - 0 , k u m : u d $O(1,S) in ~V,'2 ( n - I, u s = - - 4 ) asd ~ ' ~ ( n - 0 , u-~m - :~). (I=) _ca' . At Th~ l~N~a~tion (I6")4-(8) o f A I c~t be ~mbedded in E. with the ~

-4),

form for

r - n((z6)+(s))a- ,.~ m a r(h;) -- ~h, +4~=+ 12~,+#~.

The o m m d i ~ " demenu am o£ the form c - B(c, K I T ) ~ c , I ( 9 ) ) B - ' , but both c, and c: mum: be 1 (so that © - 13. The iavo~mive condition is ~ for ~ . , Y beiaS both T (0) aad T ( e : p ( ~ , + ~ : + l~b,+~l~)) Ct.a for $ r . , ~ aad SU(I, I)), and the ~ condi~m allows the embekUnp $ZlO in ~¢ ¥ , and SU(I, 1) in ~ ' ~ .

(c) .e- - sz.~, c) Any mp~mmmk~ which pro~lm for u e m b e d d ~ of $£.(2, ~ musz rednce to arormw~o~x comim~z~® Z~ , ~ (Xv ® p ) ® CA"® r 0 ( m ~ ' ~ 4.~). Thus many ofth~ nqa~m~Sem of A, ~ A , may be ~

md "&e only ~bre,

r , - 4((o) ® (z)+(z)® (o))+(x) ® (~)+~(O) ® (0)~ r=((o)®(2)+(,-)® (o))+~((~)® ( x ) ) ~ ( ( o ) ® (o)), r . - ({z) ® O)+O) ® 0 ) ) + ~ ) ® ~)+(o) ® (o).

(m)

6) ~ expli~ form for F: is r(b; ® 1) - b=, r(e; ® 1') - e=, F(I ® hD - hz~h:, r ~ ® ,D - l e ~ . . ~ . I r - mr:B-:. The ~

cannot be n~:lu~d m dlaiP:mai form but ~,n be e=pNss~i aa

• = ~p~(h= + ~ , + ~

• I'[ (exp ~ ( , . + r ~ . exp~(e,-r.)).

(el)

wlmm Um )~' am zm'o ampc for axbcinj ~-e-:~,+2~, ~,+cz+c:. = , + 2 = z + ~ + 2 : , .

~,+z,,+4=,+,=.. ~ . ~ be ~ : u rCn(~t6) • ,,k.(4)$ x(e'))s-'~ w'.4 is defined in iV Set,on 4. The ~ cendi~on is however only mhdied for ~ - 0 for an a ~ d ,:~ - 0 for all • crzapt ~,+2,x,+2=, and ~ , + 3 : : + 4 , , . s + 2 ~ . when it is a multiple of" .-./. The tz'ace condition allows an ~ of" SLY,, C) in

x~

(z - o, trace -.20) and j V ~ ( ~ , ~. ~ ,

-4).

~') r z can be embedded in the form r - ]~'zB - j , where Z'(h; @ I) - h~, r(e; ® 1) -- e , : + e , ~ , I'CI ® h;) "- ~ ' z + k s , l'('I ® e;) -- e t ~ ÷ e z ~ u ~ . Again ttw ccntr-dtizer cannot be dia~nalized bu: can be e q ~ s m d as © -

['[ (up ~(e,+f~- u p ~I(e,-f,}),

whe~ ~ are zm'o ex~pt for - -

,,=+~+~

and 2 = , + ~ = , + 4 c : + 2 ~ .

(~)

~ t X Z~ m

~'~e ~

REAL $trl~,al~~mt~$ oF ~

~

V

197

b~ talma m

coad~n ( ~ ' . ~(c))• - x ~ n p ~

~ - oror~

c,~ ~a,md~

For ~ - o r o r beth ..ew ~ ~ allow. ~ m b e d d ~ ot ~ . C) 0 . . ~ ( t n ~ 21~ aad for ~ .'- ~ for both ~ the re:c, ~ allo~ ~ m b e d e ~ efSI.¢2, C) in ~VP**(trace --4). Thus $/..(2, C) can beembedded in both mn-compa~ m d farms afF6. The :qmmmuttion 1": can be embedded in the form F -'BF~B " , wba~ B is

such t ~ the e x p ~ f o ~ o~ r b r,~| ® 1) - ~:-k,. r ( l ® k~) - - ~ , + ~ + k ~ r(el ® 1) - e ~ + f , = - f = ~ . cam-~ ~s d~soz~ ~ 1 " d ~ m t ~ c ~ ~ e . ~ a n be u:bm ~s.

u • -- a p 2 r ~ , + l ~ l .

(64) ~_~

w ( . ~ ~ is d c t h ~ in IV SecUre 4. Far ~ - 0 asd 1 ,h. iavolmive ~ is m.:Isned u d th= u , ~
6J. m

bmm d ~ . e m m ~

SZ~C}

Wt ~ o ~ comlder enbaddinl~ o f $£(2, C) in mti forms of ~ As for F~, anIxddia| r ~ may be ~ if~ufe isno ~muion m Z" {s~ IV S m k m ~ "n.~ one, ~ r.pma~B m I'I - 4((~) ® (D)+fD ® ~)+(]) ®{D+~((O) ® tO)).

r , - ~ ® (2)+(2) ® (o)+4((D ® o))+~((o) ® ~ ) . T, - (l) ® (3)+0) ® (I)+(2) ® f~+2~))® (o),

(¢~

1", - (2~ ® (o)+(o) ® ~ + ( 2 ) ® (1)+(D ® (2)+(I) ® (o)+(~ ® (I)+ -,-(I) ® (I)+(o) ® (o). r , - (4)® ( o ) + ( 0 ) ® ( 4 ) + 0 ) ® (3)+(o)® (o). ~) X', c,a be embedded in the form r - BrsB" wi., F~a| ® I)- h,. r(e; ® I) - e,, r~l ® bO - h:, r(~ ® .D - e~. The a ~ e h m e ~ s c,an~ be d i a ~ but they am of the from

l'letp(,q(..+~). ~p(~t(..-~), It

(6rj

where ~ is zero except for z being ~ s + ~ , + ~ ,

cs+e4+~+e~,

~,+2az+3cs+2~+

Z'.~ is Siam by

2-_ - rCnC~xe) • ,,'.c4) • x~)n-~).

(e~)

For $ i m p , the iavolufive condition implies that ~ -- 0 for all c and ,~ is a o f ~ for all =. sad t h s a , -- ~ - m d aml az " a. - 2ma/. The tram m a d i t i e a aflom for t , a a b e / d i q ia N ~ (~ - 0 ~ u = ~ - Z4) aad ~ (~ - t, u a m - - 2 M ~ S ~ou~, Yis o u ~ r m d ~ u s 8 y is m q u i ~ ! such that 7Y~r)r ~ r . . ~ t ~ beias s i n = by y - B (J-~"ty~ @ X(7))B-: for Yt - M(2) when ! - I, .... $ and y, - M(A), with M(n)

C.~n)),,- a,.,..,. The hnvolud~ coadidon puts no ~ on r,he cmtxtliJr e l e m m ~ but the trace coadi~oa is o ~ ~-~,d fez Jv£~ (u'aa, -6"). ~ ..~ CA, ~ ~ ~ ~ ,~ ~V~, NE~ , s d ~ we~ ~ i t r , lj -~. (iJ) r~ ma be e m b e d ~ in ~ e form r - Br~a ~ wi~hr~a; ® I) - k , + k , , r ( e l ® I) - ,~.,~+f~.. r ( I ® I~) - b , - ~ . r f l • e~) - e ~ , - t ~ m . ~ ~ a t r ~ _ l ~ e l e m e ~ are of the from

where th~ ~' are zero e ~ e p t for • - =, or ~ + 2 s , + 2 s s + 2 ~ + ~ j + e ~ , be d i s l o m / k s d to the form 0 7 ~ ~ m csa be talam as

z'. - r(n(z~ 0-.(4)

@~(~

. , ' _ . ( 4 ) ~ w ~ Q z(.5))n-~),

For $ Janet ~ e inveluu~ coadh/oa implles that ~. - O f a r u~e-

which cannot

i'm)

all c a m t ~ b a m~ldple o r =

- 2 ) and is NF.~ (n - I, rowe -- + ! 4 ~ I f $ is ouu~, dam y caa be ~ m

B ( , ~ yi÷|(b'))B "~, where y~ is ~

as

far l =. 1,2 aad M(4) for i - . 3, ...,6.

The iavolutive condition p u a no rmt..ie,iee on e but the tra~. m m l i t m is only u u / s ~ for N ~ (all v a t m of • ~ ~ -e~ ('m") r -- BZ'~B - I ~

be a n b ~ i d e d i n such a m a a m r that r(h~ ® I) m 2hz+hs+hz,

d;agonalized and can be c x ~ Z',,~ cau be l~.en as

as c " r ( ~ p l o t - ' , where g is 2(h, + ' ~ z - ~ .

r . , - ~aCw(s, s) • ,...(9) • x(2))n-').

-- h~.

SEKI-SIMPt~ ]t~.AL SUILq,,GBZ~ OF LI~ AL,GimBAS V Fc~ $ im~r ~ ~$ h ~ -M~ em~I

invoh~,~ ~

is ~

y csn be ~ e n ss B ~

199

f o r ~ -- a , : u d the tram moditim Slvm

y~ O I ~ )

B -t, ~ e m y :

-~

~ M ~ y s

The ~ ~ ~ o a b m ~ f o r ~ n 2 ~ ~ - ~ ~ ~ m m h ~ (u~ce - - 6 for all vai~s of ©~ ~ $~ ~ am ~ ~

(iv) The ~

S

~

~

r - ~ , B -~ am ~ m ~

,,bin

r(~; @ I) - :~,. r ~ ; ® 1) - . , ~ u ~ - t . . . r ( l • bD - lib. e~-tuum~. ~ a m ~ m r m ~ tm t s b a ss

~ ~

~m

r(Z • eO - -

sad the c m n a l k ~ elemsts can be 4 i a S o m E ~ to the form c - ~ ~, ~zm S , ~ ' ~ , - I ~ ) + ;~'h,. For $ i n ~ , the iavoluuve coaditim ~s s a : i s ~ f o t ~ - 0,I, ~ - an~ ..rod tl~ ~ c~a¢:~i~ is s m i s ~ for e m b e d ~ s s ia NE'~(a - m - O, m e : e - 14) and ~

(~ - I orm - 1 . ~

- - ~ $ ~ . y m s y ~ a m m B ( ~ y ~ B

where y~ - M('$) for i s I , . . . , 6 n d y , - M(9). "l'he i n v o / m ~ ~ all © sad ~ ~ condition is s a ~ for m emt~ldial; in ~ vahm of e]~

-~,

is ~ fo~ (trace -- --6 let tU

(,) r - a t , n - , m b, ~ ~ r(~;®;)-~,+~,~ r(.;®z)et~+f~ts--fta~u. 1"(,I~ h~) m ht+h)+Iss, r ( [ ® e;) m e t + q + * * "r~ = m n s i ~ d e = e r a are dia~oeal of the form c - ~q~2(~L+21~--~t,--b,), and 2"_ be takm as

z ; . - ~(sL~(6~ • w ( t @ • L(~) • ,~.(~) • m ) ) n - ~

(7)

For $ i a ~ r tl:s ia~oludv~ co~:li6oa impU~ tlu~ Z = ~= (~ = O, 1) sad d~tra~c~mr~i~a T

is sstisled for,-, ~.beddinS ~ 37.(2, C) i n N ~ . For $ oum'. y cm be,,#-, ss W ~ , :r~ •

N~

~ (row-

condition h ~ for .n c ~ - 6 for ~n ~ u m d ~

the I ~ O n

for an e m ~ ~

(vi) r . c:aa be embedded in tl~ form of r - z r . i - L wha.e .'(h; ® 1) - 4 , + + 4 : + ~ , a + k , ~ - I , , . r(e; ® .') - 2 , * u ~ , - ~ . , - I ~ r , . , . r ( l ® hD - ~ + ~ 3 + + l ~ + h . ~ I*(I ® e;) - 2 e : ~ , - y ~ r , - I/~t.. T/~ ce~l~==r comim ~ d m ~ u c~ :J~ form ~ W l . so tha: ~c) is the identity ope~u~on. ~

extension of" Z" is TL~J~L(IO)•

200

,7. M. ZXD,IS md J.F. COR.qW'EIL

• w'.,,,(ze') e z(,z))n-") cad for S w tt~ iavoiutiv, u d ~ u.~litiom an uzistmd for an embeddZuz ia X ~ (u'a~ 14). For S ogre', y ma~ be takea a('M(S) • M ( ~ • • M(16) ~ I(I))B "~, and the iavolutive tad trace ~ are ~ for an anbe,:Uthn;i n ~ (ma:e - --6). $Z.(2. C') am ~ ~ / a 2VF..~m~'/V',l~ ~d~ ~ 1"..

% C,mdmSmm mmmm~ , m s m m s d s~ C2, ¢) I,, t'v Seatm 9,,it wM sbowa ~ t h ~ Ziz or rml L~ ~ mammn"S $Z..'2. C') 8iven by lhsmt u d P.,atm~,.[1] w m ~ The rmults of tltis m,p~ iadkam that i b m are mzw ex,:epemd Lie t l p b r ~ w l d ~ m m i a ~ C'),, wtzi~, wu,,. d m not meaeomd ;,,, (1]. Zt has also been show. ,I.-. SZ.t2,, C') may be ~ us ~ mn-majW~s, subaljebru o f a real form of a

a

~

~

mmtx., ~ .m~majasm ~

~

T'm followias list ~

i. ~ i .

all eswp.

bactm, stta' . . ~ s ~ . , ) :

,,v.~('z).

,tmmm. ~ mmmm, C m m m m s m ,ql em~ ~ , m m oc -n ~ m ~ ~ L b ~ ,t.l=

m s . ~. D ~ m

Cmma

m m r i x A --

d i q m m for ~

3

-:] ,,.[: '] 2

"

"at P'~ 4. l:bakfa dlasram for ¥.

Cu'um mstrix .~ -

2 --1 0 0

-1 2 -2 0

0

0 0

-I 2 --I

-1 2

,

ak"-~ m

642 g 63 432

"

SEMI-SflVfP~ RP.AL S [ , ~ X L G r n . ~ OP ~

~

V

q

iq~. s. D.~dm d ~ . a m ~m. ,V.,. m

C.~

m~ix A -

A ~|

Im

2 -1 0 -1 2 -1 0 -I 2 0 0 -1 0 0 0 0 0 --1

0 o 6 0 0 0 -1 0 --I 2 -1 0~ -1 2 0; 0 2 0

s 6 $ 10 12 8 4 6 612 18126 9 4 8 12 10 2; 6 " 2 4 6 $43 3 6 9 636

~4:

F'~. 6. D , ~ m armwam fro' &

C,m.tn mau'ix A -

o o 2 -1 0 0 0 -1 2 -I 0 --1 2 --1 0 0 0 --1 2 -1 0 0 0 --1 2 0 0 0 0 -1 0 0 -I 0 0

o 0 0 0 --I 0 0 -1 0 2 0 0 2

201

202

,L M. ~

aed J.F. ~

A "4 m

"2 3 4 3 36 8 6 48129 36 9 ~½ 24 6 5 i 2 3 2½ 2 4 6 4,1,

21 42 63 5 2i 42 2 s½ 3 1½

24 6 ~ 3 ~½ 3~

A.S:

q

Fq. 7. D ~ a , , d i m ~

2 -1 -I Can,-, ~ A -

0

0 0 0 0 0

0

2 -I

-1 0 0 0 0 0

for y.,

2

0

0

0

0

0

0

0

2 --1

0

0

2 -I --1+ 2 0 -1 0 0

0 --1 2 0

-1

-1 0 0 0 -I

0

-I 0 0 0

0

--4 7 ZO 8 6 4 2 7 14 20 16 12 g 4 ~0 20 ."q) 24 18 12 6

A~ "

81624201510512 6 12 18 15 I2 4 8 12 10 8 2 4 6 5 4 $ I0 I$ 12 9

8 ' 3 6

4 3 2 3

0

0

O0"I

J" 10 15

9i ~! 8! :.

tEIqUt.lmClS (II Carat, A. 0., m t p.. ~,=v.t, emc. I t ~ . s~.. A ~ OW,Sk $SS. [21 S,dsr~a, L E., L. F. t.andoriu, aad a. T,aketmk P h i . W'~. t a OW~, 1082. ~ ~ S.. ted A. Stdi,.. tU#. 142 0see,, ]e~.

[41 ~ It..F,, /b/d. 142 (tiM/k, HOI. |~ corm,raft J. F.. g.tlmm Math. Ph).t. 2 (19"71),229.

0i l

- I

"

21 -,.,I

['71 --. ~

Z (lYe. 2n.

•.,5 JJ~. 3 (lssr'~ 5q.

rJl D*~nth4E. B,, Trod3, MmL ~ J ~

Olin. 1 ( t g ~

(

~

~ , ~ s ~ MU~. Sin. Trmui.Ses4m2,

24.5).

[:~ G, nanwm,, 1~.,~ r , ~ht~. ( ~ z . S ~ k )

N.S. S (,r;) (rag). ~Ot.

[tM --, J ~ . n ( t s m ) . 3 8 ~ P,sdnrehm Y-~bem~Jqu~ Unimsh6 de ~

( p m p ~ C]U~.t~. Feb. tg"~).