nhóm con chuẩn tắc của nhóm tuyến tính tổng quát trên vành chính quy von neumann

trình bày về nhóm con chuẩn tắc của nhóm tuyến tính tổng quát trên vành chính quy von neumann

, ~ 4 ? ,

TUYEN TINH TONG QUAT TREN VANH

\j x E A , 3 YEA: xyx = x

thuong cua A cling la vanh chinh qui von Neumann

Ta khao sat cac nh6m con chuffn dc cua GL(M) trong tnfong h<;1p

M c6 co scihuu h?n Khi d6 ,GL(M) ~ GL~(A)

Xet n 2:3 va A la vanh chinh qui von Neumann hay t6ng quat hall,

A la vanh kC'th<;1p voi don vi 1 ma A/Rad(A) la vanh chinh qui von Neumann, ta se khao sat cac nh6m con cua GLn(A) chuffn boa bCiiEn(A) Chung la nhung nh6m con H dtNc xac dinh bCiidiSu ki~n: t6n t?i duy nhit ideal B cua A thoa En(A,B) c H c Gn(A,B).

2.1 MQt s61{hai ui~m va Huh cha't cd sd :

NQi dung cua m\lc nay neu ten mQts6 Hnh chit co ban nhit cua cac tr~nsvection so cip Cac nh6m con En(A), Gn(A,B), En(A,B) clla GLn(A)

se du<;1cdinh nghIa , mo ta cac ph§n ti1'va nhung Hnh chit co ban clla

chung nham ph\lc V\lcho ph§n tiC'ptheo.

Xet x,y E A , cac transvecsion so cip thoa nhling Hnh chit sau :

2) (Xij rl = (-X)ij

3) [x~i , ykI] = In nC'u j =1=k va i =1=l

Sur ra: xl] y =Y .xlj

xij ykl (- X)ij = ykl

4) Xik yjk =yjk.xik

xki.ykj=ykj.Xki

5) [Xij ,yjk] =(xy)ik

j =1=k va i=1=l

i , j , k khac nhau d6i mQt

i , j , k khac nhau d6i mQt

i , j , k khac nhau doi mQt

voi gk la cQt k cua g, g'tla dong l cua g-l.

15

vdi gk la cQt k cua g , g'l la dong l cua g-1

2.1.b Nhom con En(A) :

Trang I.l.b , ta da dinh nghla :

En(A) = < xij , I ~ i ofj ~ n va x E A >

Vdi B la ideal cua A, En(B) chI nh6rn can cua En(A) sinh bai

n

+ gik Yg steilest

i,s,t=ln

M~nh d~ 5 : ( xem [1] , 1.2.26 trang 36 )

.Vdi n C::3 , i :;z!:j va x E A, nhom can ehuctn tde sinh b(JixUtrang E,lA) La E,l <x» , trang do < x > La ideaL eua A sinh b(n x.

Dge bi~t, hhom can ehuctn tde sinh b(Ji lij trang E,lA) LaE,lA).

2.1.c Nhom con Gn(A,B) :

Ki hi~u Gn(A,B) chi anh ngu'<;1c cua Him GLn(A/B) qua d6ng ca'u chinh t~c :

V01:,. g eij = eij.g \-I'v 1*J, 1,J. ., = 1, ;2" ,n va n -"> 2 , ta co :

= a In (ma tr~nvo huang)

Ta chung minh a E Cen(A) :

'v'~ E A, (a In) (~ In) = (~.ln) (a In)

V~y a E Cen(A)

Cu6i cling, ta chung minh a kha nghich.

g EGLn(A) ~:3 g-I E GLn(A) : g g-I =In

Ta chung minh g-l ding thuQc Cen( GLn(A))

Th~t v~y, 'v' h EGLn(A), g-I.h = g-I (h.g).g-l

-I

V~y, g-l E Cen(A) => g-l = f3 In

(a.1n ) ( f3.1n) = In => (a.f3).In =>

Do do : a E Cen(A)*

M~nh df 6 da chang minh xong

En(A,B) =< g xij g -1I g E En(A) va xij E Gn(A,B) , i oFj >

En(A,B) =< yjixij(_y) ji I XE B, YE A,l:::; i oFj:::; n >

Chung minh :

f)~t: E = < yii xij (_y)ji I x E B, YEA, 1 :::;i oFj :::;n >.

HiSn nhien: E c En(A,B), ta chI din chang mhlh: En(A,B c E.

Xet phftn ta sinh ba't kl cua En(A,B) co dC;lng:

= [yk l , xij ] xij

h =yjl xij (_y)jl =xij (-X)ijyjl xij (_y)jl

=xij [(-x)ij, yjl] =xij.(-xy)il E E. k =j va I = i:

h =y.ii Xij (- Y)ji E E

V~y h E E trong mQi tntC1ngh<Jp

V~y : En(A,B) = E

Mt%nhd~ 9 da: chung minh Kong

Phdn tie'p theo ta xet eae ma triln boan vi : ]a ma tr~n ma m6i

dong va m6i cQt chI co duy nha't mQt phftn ta khac 0 1a 1 M6i ma tr~nhoan vi p tlidng ung 1 phep the" (j E Sn co dC;lng: p = (8j,cr (j))

Suy fa: p.q =In ~ p =( <>i ,0 1(j) )

Xet : p Xkt.P -1 = In + (8 i,O"U»)k ,X.( i,CT -I (j) )t

(CQt k) (dong t)

= X er(k), er- I (I)

GQi p = (8 k ,er(I)) la ma tr~n hoan vi , ta co :

2.2 Nh6rn con chufin tdc cua nh6m tuye'n tfnh tang quat tren vanh chfnh qui yon Neumann:

Xet A ]a vanh ehinh qui van Neumann, ta se di de'n 2 dinh Ii m6

ta cae nh6m can H eua GLn(A) ehu§'n boa bdi En(A).

GQi B la ideal eua A Nhom can H eua GLn(A) duQe gQi la nh6m can mde B ne'u: En(A,B) c H c Gn(A,B)

Dinh Ii 1 neu cae tinh eha"teua nhom EnCA,B)va eho ke't qua mQi

nh6m can mtie B eua GLn(A) d€u ehu§,riboa bdi EnCA).Dinh Ii 2 eho

ta chi€u ngtiQe l~i : mQi nhom can eua GLnCA) ehu§'n boa bdi En(A) d€u la nhom can mde B Tli do, ta co th€ mo ta tfit ea cae nhom can eua GLnCA)ehu§'n boa bdi EnCA)thong qua cae idea] eua vanh A.

Tit d6, fa co' EnCA,B) ehudn fde lrong GLnCA).

(b) En(A,B)::J [ EnCA), GnCA,B)].

Suy ra mQi nh6m can mue B eua GLnCA) d~u ehudn hod biJi En(A).

(c) Khi n 2 3 , fa e6

EnCA,B) =[EnCA),EnCB)] =[ GLn(A),EnCA,B)] = [Eil(A~,H]

wJi H la nh6m can mue B.

(d) V&i A la vanh ehinh qui van Neumann, la co

EnCB) = EnCA,B)

H(m naa, v&i n 23, fa e6 EnCB)= [EnCB), En(B)].

CluIng minh

= 0

=> u'v' + UnVn = 0

=> d' =1-u'v'Taco:

( 1 - Und-1vn ) d'= ( 1 - Und-I Vn) ( 1 + UnVn)

d' ( 1 - Und-1vn ) = (1 + UnVn) ( 1 - Und-l Vn)

= 1 - Un d-I Vn+ UnVn - UII(Vn Un) d-I Vn

= 1 + UnVn- und-1vn - UII(d -1 ) d-I Vn

V nun

v'u

J

n d

- 0

(

0 ')d'-l d) E En(A,B) , ( Ke't qua tU [5], ~2)

Truong hqp 2: ::3i E { 1, 2 , , n} : ViE Rad(A).

Ntu i = n :

Soy ra : 1 + VnUnE GL1(B) => In + V U EEn(A,B) (tnf<Jng hQp 1)

Ntu i < n: xet phep the- (j = ( i

( ma tr~n co du'<;1cb~ng cach hoan vi cac cOt i & n cua u)

vnO = Vi E Rad(A) => In + va UOE En(A,B)

In + v u = p.p-l + p.pv.up p-I =P On + va DO)p-l E En(A,B)

Truong hqp tang quat:

vn ( 1 - x Vn) UnE Rad (A)

1 + VnO - X Vn)Un kha nghich trong A

1 + Vn(1 - X Vn)Un E GL1(B)

g := In + v(1 - X VB)U E En(A,B) (tnrong hQp 1)

Ph§n tie'p thea, ta ch(tng minh : h := 1n + V X Vn U E En(A,B)Xet:

V \1-1 =V n-l ( 1 - X V n ) X V n

t =In + Ul VI E En(A,B) (TnfOnghQp 2 voi i = n-I)

Ph§n tie'p theo , ta chllng minh En(A,B) chu!Inuk trong GLn(A) :

Ta thl/c hi~n cac bu'oc san :

1 Tn(oc bet xet cac phgn tU'bgt kI g E GIn (A) , xij E Gn(A,B),

2 Tie'p then , ta xet cac ph§.n ti't g E GIn(A), h =yii Xi j (-yy i, i *-j ,

X E B,YEA, ta CO :

g h g-1 = g yji Xij(_y)ji gol = (g yji) xij(g yii rl E En(A,B)

( Ap d\lllg bttoc 1 cho g y-ii)

= TI g hi g -I E En(A,B)

i=1

Chung minh l.b: [En(A), Gn(A,B) ] c En(A,B)

Ta thl,fchi~n qua cae buoc :

a) Trudc h€t, ta chung minh : [1 ij, g ] E En(A,B) ,\/ g E Gn(A,B).

[11,.I,g]

gia siT (i, j ) = ( 1 , n ), khi do :

VIlEB

(Xvn) I.il E En(B)[ (xvn )1.0 , g ] =(xvn )1.n (1n - VXVnW) E En(A,B)

modulo En (A,B) Ta co :

[(1 - XVn)1,n , g ] = (1 - XVn)1,n, (1n - v(1 - XVn)w )

[(1 - XVn)1,n, g ] = (1 - XVn)I,ll ( In + 'lU )

vdi VllUn= -vn(1 - XVn) Wn=(Vn X Vn -Vn)W E Rad(A)

Suy fa: d := 1 + VnUn E GLI(B)

, D6ng thai, xet trong GLI1(A/B) va GLj(A/B), ta co :

vlund -1+ 1 - XVn E B

1 n

(

In-I(1-XVn) , 0 V'~"d-') E E,,(B)

= (xvn )1,n[(1- XVn)I,n ,g] (- XVII)I,n [(xvn )I,n , g] E En(A,B)

b) X6t phtln til' thuQc [En(A) ,Gn(A,B)] cI~ng I ylJ, g J , YEA, i :;i:j,

gE Gn(A,B) :

En(A) la nhom con chu§'n t~c sinh boi ] l,n trong En(A) nen :

3ZEA: yi,.i = Zkl 11,11(-Z)kl,k;t:.l.

D~ ktt thuc chung minh phfin lb, ta gQi H la nh6m con mo.c II

va chung minh H chuan boa bdi En(A) :

Ta co : En(A,B) c H c Gn(A,B)

V xij E :Bn(A), V h E H, xij h (-x )ij .h-1 E tEn(A) , Gn(A,B) ]

Ma: [En(A), Gn(A,B) ] C En(A,B) C H

Sur fa xij h (-x )ij.h-1 E H => xij h (-x )ij E H

Chung minh loc :

X6t ph~n tl'i sinh ba't ki cua En(A,B) co d~ng :

yj i xij (-y )j i , X E B, YEA ~ I ~ i -:(::.i ~ n

Taco:

yjixij(_y)ji = [yji,xij]xij

= [yj i ,Xi j ] ( I.X )ij

= [yji,xij].[1ik,xkj] (ChQnk-:(::i,j)

Sur fa yji xij (_y)ji E [En(A), En(B)]

Hi~n nhi~l1, ta co : [ En(A) , En(ll) ]c [ GLn(A), En(A,B) ]

Chung minh [GLn(A), En(A,B) ] c En(A,H) :

X6t ph~n tU' [g, h] E [ GLII(A), En(A,B)], g E GLn(A), h E En(A,B)

g h g-l E En(A,B) ~ [ g , h ] = g h g-l.h-1 E En(A,B)

Do do : [ GLII(A), En(A,B) ] c' En(A,B)

.

V~y ta CO : En(A,B) =[ En(A),En(B)]= [GLn(A), En(A,B)]

D~ ke't thue ph~n Ie, ta ehu'ng minh En(A,H) =[ En(A), H) :

V?y En(A,B) = [En(A),H ]

Chung minh l.d :

En(A,B) c En(B)X6t ph~n tti sinh bat ki cua En(A,B) q<;lng (- yi i) Zij yi i, ( ZE B ,

YE A, 1 ~ i =I:j ~ n ) Vi En(B) dl«;1Cchu~n hoa hdi t?P cac ma tr?n

hoan vi, ta co th€ giil sU' (i, j ) = (1 , 2) Ta chang minh :

Vi A chinh qui van Neumann nen t6n t~i phfin tU' x E A thoa :

z = z x z Khi do :

h := (- y ) 12 Z2 1Y12

= (-xzy)1 2 (Xzy)1 2 (-y )12 Z2 1yl 2 (-xzy)1 2 (Xzy)1 2

=(-xzy)1 2 ( xzy - Y)12 Z2 I( Y - xzy ) 12 (xzy) 12

vdi (xzy)12 E En(B)

X6t g :=( xzy_y)12.Z21 ( Y- xzy )12

Ta l~p cong thlic tinh aI2b21(-a)12, a, bE A

Ap dl,lng cong thlic tren cho a =(xz-I)y E B , b = z

0 \

0 I I

(1+(xz-l)yz (xz-l)yzx 01 (1 (l-xi)yzx 0 01

z(1 - xz )yzx = zyzx - zxz yzx = zyzx - zyzx = 0

(1 + (xz-1)yz)(1 - xz )yzx + (xz-1)yzx =(1 + (xz-l)yz - ] ) (xz-l)yzx

=(xz-] )yz(xz-] )yzx

=(X7,-])(yzxz - yz) yzx

=(xz-l )(yz - yz) yzx

=0

Do do : g =t = «xz -1)yzx) z « I - xz )yzx) E En(B)

Suy fa h = (- xzy)12 g (XZy)12 E En(B)

Chung minh En(B) = [En(B), En(JJ)] vOi n 2 3 :

Hi~n nhien [En(B),En(B)] c En(B)

Ta cling co :

V ZlJ E En(B), zij=(zxZ)ii , x E A

zij= (ZXZ)ij=[(ZX)ik, zkj]E [En(B),En(B)] (chQn k:f=i,j)

V~y : En(B) = [En(B),En(B)].

Dinh Ii 1 da chung minh xong.

D~ di dSn dinh 11 2, ta c§n mQts6 b6 d~ Trang ph§n sau , ta ki

hi~u A ia v~nh kSt hqp voi ddn vi 1 va thai man di~u ki~n A/Rad(A)

la vanh chinh qui van Neumann,

2.2.b B6 d~ 1 :

Cho n 2 3, a E A, a :;r 0 va a kh6ng La LtcJC dla 0 H La nhom con

cua GLn(A) chwin hod biJi tqp cac aU, i :;rj thod tlnh chat H chaa

g =(gij) thod gn] = 0 va t6n tc;zii, j sao cho g kh6ng giao hoan vcJi

[akj, g]= akj bl,n ( bj-2)j-2,n (-a)kj,(-bi-2)j-2,FI (_b])I,n, (abl,n

= akj bI,n (-a)kj (-bI)I,n (abj )k,n=akj (-a)ki (abj /,n =(abj )k,n

V~y H chlia (abj )k,n-:f=In

Truong hqp 2 : gnl = gn2= = gn,n-I= 1 - gnn = 0

g aLn= aLng => g ( In + a.ej n) = ( In + a.ej n) g

= TI (g i,n )in i=1 :;t In, (g khong giao hmln voi aij)

g =

.gn-I.n

Nen t6n t~i i < n sao cho gi n:;to

Suy ra g thoa cae dieu ki~n cua trtiong h<;1p1

Ntll tUn t(li ml)t i nho hdll n thod g khong giao hoall vai ain :

[g,a' in] =g. ( a'.in g.-a-1( )i n, ) E H

[g, a.i,n] = g a g( i,n -1) ( ) -a i,n = (1 +n gi.a.g'n) (. -a )i,n

( gi, g'n l~n ltTQtla cQt i, dong n cua g va gol)

Th~t v~y :

-1

1g.g = n =>

=>

=> g' n = (0 0 0 1)Tli d6:

0 1

'" g I i a g2ia

0 0

0 0 , - a I (dong i)0

0

n gi a g n -a ::;:: gli a g2 i a -a gi i a gn-I,i' aV~y [g, ai,n] E H thml diSu ki~n tn1ong h<;1p1.

Suy ra H chua illQttransvection so ca'p khac 1n.

gl.a.g'i co dong cu6i cling la : (0 0 0)

In + gla.g'i COdong cu6i cling la : (0 0 0

Do do : (1n+ gl.a.g'i ).(-a )1i co dong cu6j cling la : (0 0

E>(;t~t h:=g,a[ Ii] =g.a.g.-a[ Ii -I ( )Ii ]E.H

1 )

0 1)

Ta co : hn 1=hn 2 = = hn,n-l = 1 - hnn = 0

transvection so ca'p khac 1n

Ngli(fC [{Ii, lltll g giao hoall vui llU;Ji a I,i, 2 .s i s ll, ta co :

l,i l,i 'o.r-I'

(In + ae Ii ).g =g (In + a e Ii) , \j i = 2,3, , n.

Ntu t611t{li sf/ i = 2, 3 , , n-1' thoa g khong giao hoan vai ai ,I:

h =a,g=a.g.-a[ i ,I ] i,l [ ( )i,1.g.E-I] H

Llong t 01: g -a '.g = n-gi a.g 'I

Voi gi 1a CQt i cua g, g'l 1a dong 1 cua gol

Tli (1) ta co : gni=0

h thml cae di~u ki~n cua tru'ong h<Jp 2.

H chua mQt transvection so cffp khac In

0)

I )

'Ngll{fCltJi nill g giao hoan vf1iffl(Ji ai,l, 2 ::; i ::; n-l, fa co :

g.ai,l = ai,l.g

g(In + a.e i,l ) = (In + a.el,i)g

g.a,e i,l = a.ei,l g

Suy ra ': g12.a =g13 a = = gl,n.a = 0

B5 dS 1 da:chung minh xong.

2.2.c B6 d~ 2 :

Cho H la nh6m con cua GLiA) va chulln hod !xii EiA) , n ;: 3 .

Ne'u H khong la tam cua GLiA) the H chlia milt transvection 'Ie!c{{p khdc In

Chung minh :

Truong hqp 1: H chua g =(gjj) thO<l gn, I:;: 0 va t6n t~j k-::/= 1 saGcho g khong giao hoan vai 1k,1E En(A) Khi do H va g thoa cac di€uki~n cua b6 dS 1 V~y H chua illQt tran~vection so ca"pkhac In

Truong hQp2 : H chua h =(hij ) thoa hn2 -::/=0 va t6n t~i YEA, thoa

V~y: g khong giao hmln vai 121.

Suy ra H va g thoa cac diSu kit%nct1a trtiang h<;1p1

H chua mQt transvection sd ca'p khac In

Truong hqp 3: H chua g = (gij) ~ CeDeGLn(A)) va gn,l= 0

Ne"u t6n t~i cac sC; k,l khac nhau sao cho g khong giao hoan vai

chua mQt transvection sd ca'p khac In

Ngti<;1c l~i, ne"u g giao hoan vai

V~Y H chua [g, y12 ] la mQt transvection sd ca'p khac In '

Truong hqp 4 : H chua h = (hij) ~ Cen( GLn(A) ) va h22E GL](A)

Ntu (h-l),rl = 0 :

Ta co h-1 E H va h-1 ~ Cen( GLn(A) ), (h-I )n,1= 0 nen H

thml cac diSu ki~n Clla tnfdng h<;1p3,

V~y H ch((a mQt transvection sd ca'p khac In'

gn,1+ gn,2'Y =(h-I )n 1.h21 -(h-I )n 1 h22 + (h-l )n t h22 h22-1 (h22 - h21 )

= (h-l)n 1.h21 - (h-I)n 1 h22 + (h-I)11 1 h22 - (h-I )111.h21

Nen H va g thoa cae diSu ki~n cua truong h<;1p2.

Ne'u g E CeDe GLn(A) ) thl g.f' = f '.g

0 - - 1

0 1 0

-1 0

(0 -1 0I

H *-Cen(GLn(A)) ~:3 h = (hjj)E H: h ~ Cen(GLn(A))

A/Rad(A) la vanh chlnh qui von Neumann Den :

nay mall thu~n vdi p "*0

Vdi Z E Rad(A) => (h-1h 1.Z E Rad(A) => g22 E GL1(A).

H va g thoa cac di~u ki~n cua traCinghqp 4.

B6 d~ 2 da chung minh xong.

2.2.d B6 d~ 3:

Cho H la nhom can cila GL,lA) va chwin hod biJi E,lA) , n 2 3

Neu H chaa xU, trong do x E A, 1s i ;r:j s n va B la ideal 2 phiacila A sinh biJi x thi: H:::J E,lA,B).

Chung luinh :

[x ii, t kl] =xii. t kl (- x)ii (- t kl) E H

[tkl,xij]=tkIXii(-t)kl (-X)iiE H

Ta co cac traCing h<jp sail :

V~y, trong mQi tru'ong h<;1p,ta CO XkI E H .

Xet phfin tusinh bfft ki cua En(A,B) co dCJ.ng:

B6 de 3 da chung minh xong

2.2.e Bjnh Ii 2: Giil sa n;:::3 va AIRad(A) la vanh chinh qui van

la nh6m con nlllc B, nghfa la EiA,B) c H c G,lA,B), V(ji B lel ideal cuaA.

Chung minh :

GQi H la nhom can cua GLn(A) chugn boa bdi Eo(A)

Voi phfin tIt X bfft ki thuQC A, gQi Bx la ideal cua A sinh bdi x,

ta co Bx= xA = Ax

B~t B = { X E A / Eo(A,Bx) c H } , ta se thvc hi~n cac bu'ocsan :

a) Chung millh B faideal cua A:

V~y: 0 E B => B;;t 0

GQi x,y la 2 phfintu bfft ki thuQc B, ta chungminh : X - Y E B:

Eo(A,Bx-y) c H:

Phfin tu sinh bfft kl ctla Eo(A,Bx_y)co dCJ.ng:

h =tji (( x-y) Z )ij (-t )ji ,voi t, z E' A, i;;t j

=tji (xz-yz )ij (-t)ji

=~i (XZ)ij.(_YZ)ij(-t)ji

=tji (XZ)ij (-t )ji .tji (-yzij (_t)ji

xz E ax =>tji (XZ)ij (-t)ji E Eo(A,Bx) c H

53

-yz E By ~ ~i (_yz)ij (-ti E En(A,By) C: H

Sui fa: h E H

V~y : En(A,Bx-y) C H ~ x - Y E B

Cu6i cling ta chung minh \/ x E B, \/y E A, xy E B va yx E B

En(A,Byx) c H

Phfin tasinh ba'"tkl cua En(A,Bxy) co d?ng:

h = tji «xY)Z)ij (-t)ji, voi t,ZE A,i-:f:.j

= tji (X(YZ»ij (-t )ji

= tj i [ Xik, (YZ)kj](-t )j i , (ChQn k ,-:f:.i, j )

=tjixik (YZ)kj(-X)ik (_YZ)kj(-t)ji

(YZ)kj(_X)i\ _YZ)kjE En(A,Bx)c H~ tji (YZ)kj(-x)ik(_YZ)kj.(-t ).iiE HSuy fa: h E H

En(A,Bxy) C HChung minh En(A,Byx)c H hoan toaD tu'dng tl!.

V~y, B la ideal 2 phia cua A.

b) Chu1lg 11li1lhEn (A,B) C H :

Phfin tu sinh ba'"tkl cua En(A,B) co d?ng :

c.l Tntoc hSt, ta co A 1a vanh chlnh qui Vall Neumann nen AIB Ia

vanh chfIlh qui von Neumann Do do (AIB)/ Rad(A/B) cling Ia vanh

chinh qui van Neumann.

c.2 GQi H'= q>(H), q>(En(A) ) = En(A/B) Ta chung minh H' du'Qc chuffn boa bdi En(AIB):

La"y ba"t kl h' E H' , (x)ij E En( AIB ) , ( i:f=j), ta chung minh :

c.3 Chung minh H' c Cen GLn(AIB) :

chua mQt transvection sd ca"p (X,)ij khac' ma tr~n ddn vi cua GLn(AIB)

0 I

bn)

(i,i)

0) I

=>[xi.ig,ljk]E H

Ta cling co :

[Xi.ig, Jjk] =Xi.ig.1.ikg-1(-x)i.i(_1)jk.

=xij Ijk (-l).ik g l.ik g -1 (-X)i.i (-l).ik

=xij Ijk [(_1).ik,g] (-X)ij (_l).ik

57

=Xij Ijk (-X)ij (-l)jk tjk xU [(-l)jk,g] (-X)ij (-l)jk

=[Xij ,ljk] Ijk xij [(_l)jk,g] (-X)ij (-l)jk

g E Gn(A,B) => [(-l)jk ,g ] E [En(A) , Gn(A,B)] c En(A,B) ,(dinh Ii l.b)

Ta suy ra :

]jk xij [(_1)jk, g] (-X)ij (_1)jk E En(A,B)

[Xijg, ljk] =Xik.t , t E H

Xik = [ xij g , Ij k] t -1 E H

Do b6 d~ 3, ta co :

En(A,Bx) c H => X E B , di~u nay mall thua"n voi X ~ B

Suy ra H c Gn(A,B)

Dinh Ii 2 da dlf<;Jcchung minh

D~c bi<;t, ne"u A Ia vanh chlnh qui van Neumann va n;?: 3 , til dinh

11 I.d, En(B) = En(A,B) nen co th~ suy ra :