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trình bày về nhóm con chuẩn tắc của nhóm tuyến tính tổng quát trên vành chính quy von neumann
CHUaNG , ~ ? , 2: NHOM CON CHUAN TAC CUA NHOM K , ,.? , " , TUYEN TINH TONG QUAT TREN VANH CHINH QUI VON NEUMANN Vanh A dU la d6ng ca'u nhom Nh~n xet : Cell (GLn(A/B)) la nhom chu~n t~c cua GLn(A/B) nen Gn(A,B) la nhom chu~n t~c cua GLn(A) M~nh d~ 6: g = (au) E Cen(GLn(A) = a ,In, g a E Cen(A) (g Lama tr~n va hudng) Chung minh : all"""", g.eij a1n 1,0 00000000 = [0 ::: } = anI t (i,j) a II omo eij.g = lo.k , ! (i,j) °} [ ~ n1 ( C(jt j ) ~ aln \ I a J nn ao =( a .10.ln a (dong j ) ) , V 01: \-I' g eij = eij.g v , 1*J, 1,J=1,2 ; " " - "> ,n va n - , ta co : " [ O:::O} = anI ( a~j"J dong i t cQtj a 11 => = a J J f ak i = lajt =0 'v'k;i: i 'v't;i:j La"y i , j l~n htQt b~ng 1,2, 3, , n , i * j , ta co : all = a22= = ann= a akl = , V k ::;c l all g = [ 0 ann ] rao °a) = a In (ma tr~nvo huang) Ta chung minh a E Cen(A) : 'v'~ E A, (a In) (~ In) = (~.ln) (a In) ~ ~ a a ~ In = ~a In ~ = ~a 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 -I -1 =g ( gh) g = h g V~y, g-l E Cen(A) => g-l = f3 In , (f3 E Cen(A) (a.1n ) ( f3.1n) = In => (a.f3).In => a.f3 Do : a E Cen(A)* M~nh df da chang minh xong Ap dl,mg m~nh de cho tam ) = => a kha nghich A (GLn(AIB)), ta co : 1VI~nhd~ : gECenGL,lA/B) g=a.I Trang d6 : E (Cen(A/B)) * - , 11 ') I I=lod Til ta co thS mo ta Gn(A,B) bc3im~nh de saD : M~nh d~ : r (aij) E Gn(A, B)i l a - =0 ~J- - aji - a JJ- au = Oij.a , i oF j, a E (Cen(AIB))* ex (mod B), a E (Cen(A/B))* 2.t.d Nhom COllEn(A,B) : Ki hi~u En(A,B) chi nhom chu§'n t~c cua En(A) sinh bdi cac transvecsion sd dip xi.j Gn(A,B) En(A,B) Nh~n xet: = < g xij g -1I g E En(A) va xij E Gn(A,B) , i oF > j En(B) c En(A,B) c G,lA,B) M~nh d~ 9: Voi n ~ , ta co : 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, Ta chung minh : En(A,B) YEA, oF :::;n > :::;i j = E 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( _y)k l , i * j , k *- I, = [yk l , xij ] xij h B va YEA X E Ta co cae tn1C1ng h u'v' => d' + UnVn =0 = - u'v' Taco: ( - Und-1vn ) d'= ( - Und-I Vn) ( + UnVn) = + Un Vn - Un d-I Vn - Und-I (Vn Un ) Vn = + UnVn - Und-l Vn - UIId-I ( d -1 )Vn -I -I = + Un Vn - Un d Vn - Un VII + Un d VII = d' ( - Und-1vn ) = (1 + UnVn) ( - Und-l Vn) = - Un d-I Vn+ UnVn - UII(Vn Un) d-I Vn = + UnVn- und-1vn - UII(d -1 ) d-I Vn = 1+ -I -1 UnVn - Und Vn - UnVn + Und VII =1 V~y ( - und-I vn)d' Do d6 d' Khi d6 : E ~ d'( - und-I Vn)= , Suy fa d' kha nghich GI1(B) va (d' rl = - Und-I Vn VI = VU (U' [VnJ In-ol +V'U' In + VU = ( VIUn (V nU ' V 111-1+ v' U' = l+vnun ) VnU' v v'U' = Un) ( I 1.1 n J nun v'u n d VnU' J E)~t : ", -1 , ,a ,- 1n - + v u - V un d VnU = 111- + v'(1 =111-1 - Un d-1 Vn) U' + v'(d'ylu' V,~"d') Ta ch1?ng minh : In + vu ~ (l"~ XetO v€ phai VIUO d) l l ) d v u' v ( 10-1 ld-Ivo d -I v u v'u d +,V'U' 01 I OJ v>J U V~y : In +vu 1) u' ( ln I 10-1 =( o V';"d-') (~ d ) Voi : ' n-I (1 o V Un d ld-Ivn H' 1) 01 1) (lool (a +V'U d-Iv u' = (10-1 V.~"d-') (~ = (a d) : 10-1 g:= ( o (~ (10-1 -I (1n-I ) ' l d-Iv u' a In + vu E En(A,B) 01 (o ) E E' B n( ) d) En(A,B) l d -I V n U I 01 d 01 Suy ': g12.a = g13 a = = gl,n.a = g12 = g13 = = gl,n = Ti'i (1) va (2) cho ta tr~n cheo Khi : [g, aij] ij g a g = g aij g-l -1 , (2) g la ma tr~n cheo, suy g-l cling la ma (-a )ij E H ' = 1n + gi.a g j (gi , g'j la cQt i, dong j cua g, g-l ) oI r ~ I g OO \I g aij g-1 I I la = In + " (0 , 0) g jj I lo) (0 0'1 I I I° ° =In+IO "',, I I o ""'" l~ - gjj ag'jj o "" 0J , ij - ( gii.a.gjj ) ij , , ij ii ij - ( gii.a.g jj ) ( -a ) - ( gij.a.g,jj - a ) [g, a ] - [g, aij ] ]a transvection sd cffp khac In thuQc H B5 dS da:chung minh xong 2.2.c B6 d~ : Cho H la nh6m cua GLiA) va chulln hod !xii EiA) , n ;: Ne'u H khong la tam cua GLiA) khdc In the H chlia milt transvection 'Ie! {{p c Chung minh : H chua g = (gjj) Truong hqp 1: va t6n t~j k saG cho g khong giao hoan vai 1k,1E En(A) Khi H va g thoa cac di€u ki~n cua b6 dS V~y H chua illQt tran~vection so ca"pkhac In thO :3h E GLo(A): g.h *-h.g => gll.h *- h.gll => :3YEA: Khi : [ g, /2] = g y12 g-\_y)12 EH gll.Y *-y.gll 12 12 [ g, Y ] = ( gll 1n ) Y ( gll 12 = gll Y gll -1 -1 ( -y ) 1n) (-y ) 12 12 l = gl\ g~IIIY 0 = , , ~ ' gll -y+gl1ygl~1 '" -I , gll l l~ j -1 glJ ,'0 = ( -y + gll Ta co: (g~11 -g~:y -1 12 Y gll ) y.gll *- gll,Y => Y *- gll.Y gll -1 => - y + gll Y gl1-1*-0 12 -1 12 [ g, Y ] = ( -y + gll Y gl1 ) *- In V~Y H chua [g, y12 ] la mQt transvection sd ca'p khac In ' Truong hqp : H chua h = (hij) ~ Cen( GLn(A) ) va h22E GL](A) Ntu (h-l),rl = : Ta co h-1 E H va h-1 ~ Cen( GLn(A) ), (h-I )n,1= nen thml cac diSu ki~n Clla tnfdng h~t : g :;z!: sd ca'p khac In' 0: = h-1 11,2 h (-1)1.2 Ta co : 11,2.h (_1)1.2 E H => g E H g =( (1n + (h-1)1.1.h2 ).(-1 )1,2 Voi (h-1)1 la cQt cua h-I h2 la dong cua h H va h-1 (h-l)lI (h~1)21 g = ( In + (h21 h22 h2n) ) (-1)2,1 (h -I) n I go t = (h-t)o gn =(h-t h2t - (h-1)n t h22 )n h22 X6t hai truong Truong h (h-I)n Truong Voi =0 (h22)"1 = (tnli gia thie't (h-I )n,1 =f::- ) h t g ( gij.- - h-I p.I,n h ) C;l Ne'u g E Cell GLn(A) thl g.h In D dO:g h =p' ~ = h.g h ,suyra:g=p' Khi do: pi,n E Cell GLn(A) mall thu~n vdi p "* V~y , g ~ Cen GLn(A) Ta co: g = In + (h-1)IP hn in nen pI,n la ma tr~n vo hadng, di~u Vdi (h-t)I la cQt cua h-I , hn la dong g22 = + (h-1h1.P.hn2 = + (h-1 h 1( - hn2 x) hn2 = - (h-1h1 n cua h (hn2.x.hn2-hn2) = - (h-1 1.z h 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 B6 d~ 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 Neu H chaa xU, x E A, s i ;r: j s n va B la ideal phia cila A sinh biJi x thi: Chung luinh : Xet tEA, ~ k ;r:I sn, = xii [x ii, t kl] H:::J E,lA,B) t kl ta co : (- x) ii (- t kl) E H [tkl,xij]=tkIXii(-t)kl (-X)iiE H Ta chung minh XkI E H , vdi k,l ba't kl thai! l~k;r:t~n Ta co cac traCing h Xk I -l=i: Xk I Xk i (x Ok i = [ X k.i, 1j i ] k' X -Z=j: * k = i: k' f kl X = = =(1.x) = [I I, K I]] =X =(1.x ) = [I k'" E ] k' k' ] f H ::::> k'" I, X I]] E ta co I ;r:i -l;r:j: -l=j: Xkl Xk I = Xii = (x.I)it = [x = xii E H 52 ii, Iii] E H x H kl E H EH * k = i" ta co l;r i .i l;r Xk l = = (1.x ) j l = [ 11i, X it ] Xj l X it = (x.1 ) it = [ x i.i, 111] E H ~ l = i Xk I = X pi Xj i = (1.x) Ji = [ 11P,X pi] Xk I E H ,( chQn p;r i ,j ) = (X.1)pi = [xpJ, 1Ji] X pj = (1.x)p1 = [ 1pi, X i1] E H ~ V~y, X piE H ~ XkI E H mQi tru'ong h Z = xt , tEA Zkl = (xt)kl = [Xkp, Soy fa : y I k II tPI] = XkP tPI (-X)kp(-t (-y) I k E H )pl E H, (chQn p ;;tk,l) V~y : En(A,B) c H B6 de da chung minh xong 2.2.e Bjnh Ii 2: Giil sa n;:::3 va AIRad(A) la vanh chinh qui van Neumann.Khi d6, mQinh6m can H cua GLiA) chwin hod bcJiEiA) d~u la nh6m 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 fa ideal cua A: 0,,2 =1 '" H => Eo(A,Bo) c H (bB de 3) E V~y: E B => B;;t GQi x,y la phfin tu bfft ki thuQc B, ta chung minh : X - Y E B: Ta co Eo(A,Bx) c H va Eo(A,By) c H , cfin chung minh 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 = tji (xz-yz )ij (-t)ji = ~i (XZ)ij (_YZ)ij(-t)ji = tji (XZ)ij (-t )ji tji (-yzij t, z E' A, i;;t (_t)ji xz E ax => tji (XZ)ij (-t)ji E Eo(A,Bx) 53 c H j -yz E By ~ ~i (_yz)ij (-ti Sui fa: E En(A,By) C: H h EH 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 Ta co En(A,Bx) c H, din chung minh En(A,Bxy) c H va En(A,Byx) c H Phfin ta sinh 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 )ji , (ChQn k ,-:f:.j ) i, =tjixik (YZ)kj(-X)ik = tiixik (-ti Voi : ~iXik(-t (_YZ)kj(-t)ji tji (YZ)kj (-X)ik (_YZ)kj(-t)ji i E En(A,Bx) c H (YZ)kj(_X)i\ _YZ)kjE En(A,Bx)c Suy fa: H~ tji (YZ)kj(-x)ik(_YZ)kj.(t ).iiE H - h EH En(A,Bxy) C H Chung minh En(A,Byx)c H hoan toaD tu'dng tl! V~y, B la ideal phia cua A b) Chu1lg 11li1lhEn (A,B) C H : Phfin tu sinh ba'"tkl cua En(A,B) co d?ng : h =yjiXi.i(_y)ji,voi XE B,YE A,i-:f:.j Ta co : x E B Den En(A,Bx) C H D6ng thai: x E Bx ~ yji xij (_y)ji E En(A,Bx) c H ~hEH V~y: En(A,B) c H c) Chu1lg mi1lh H c GII(A,B): 54 c.l Tntoc hSt, ta co A 1a vanh chlnh qui Vall Neumann nen AIB Ia vanh chfIlh qui von Neumann 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), (x)ij h'(-x)q ta chung minh : E H' Ta co : h' = q>(h), h E H = q> (Xij (X)ij h' (-X)ij = h EH ~ q> ).q> (h ).q> «-X)ij) ( (X)ij h (-X)ij) (X)ij h (-X)ij E H Do (H) (X)ij h' (-X)ij E H' V~y H' du'Qcchuffnboa bdl En(AIB) c.3 Chung minh H' c Cen GLn(AIB) : Ta chang minh bang philo chung: gia stYH' r:r.Cell GLn(AIB) Khi , AIB va H' thoa man cac diSu ki~n cuab6 dS Den H' chua mQt transvection sd ca"p (X,)ij khac' ma tr~n ddn vi cua GLn(AIB) Do t6n t(;liph§n ttY x ~ B d~: - 0\ 0 (X,)ij = r l> X ~ "' ' 1J ( i, j ) Do : (x,)ij = q>(h)voi: 55 bl h= °1 bi X I (Dong i ) bJ t CQtj Cae ph~n bi eua h thml: bi -1 E B , V i = 1, , n f)~t g = (-x)ij.h, ta eo : 01 r~1 g = (-xij bi I X l~ -, l~ I bn) (1 I - I I 01 I I -x O t t J °1 (i,i) °, 0) (11 I l a I q 56 x t I (i,i) (i,j) ~ I bJ f (i,j) 01 rbl bi : I = I ' t I l~ x-xb J ~ t I' I bJ r (i,i ) (i,j ) (bl °l I I = I I I b; on x(1- b j) I I bJ l~ x( - bj ) E B => g E Gn(A,B) D6ng thai: xij.g = h E H X6t [xi.ig,J.ik]=xi.ig.1jkg-1(-x)i.i(-1).ik, Xii g E H => g -1(-x)ij =>[xi.ig,ljk]E (chQn E H:::::>l.ik g -1(-X)ij (_l)jk E H H Ta cling co : [Xi.ig, Jjk] k :;t:i,j) =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 57 (_l).ik = Xij Ijk = [Xij = Xik (-X)ij (-l)jk ,ljk] tjk xU [(-l)jk,g] Ijk xij [(_l)jk,g] , Ijk xij [(_1)jk,g ] (-X)ij (-X)ij (-l)jk (-X)ij (-l)jk (-1)j k g E Gn(A,B) => [(-l)jk ,g ] E [En(A) , Gn(A,B)] c En(A,B) ,(dinh Ii l.b) Ta suy : xij[(-l)jk,g](-x)ijE ]jk xij [(_1)jk, En(A,B) g] (-X)ij (_1)jk E En(A,B) [Xijg, Ijk] E XikEn(A,B)c [Xijg, ljk] =Xik.t XikH , t EH 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 mall thua"n voi X ~ B V~y H' c Cen GI.iA/B) Suy H c Gn(A,B) Do E.iA,B) c H c Gn(A,B) Dinh Ii da dlf(En(A) ) = En(A/B)... xzy )12 Ta l~p cong thlic tinh aI2b21(-a)12, a, bE A ( a 1 = I I a12b21 (-a)12 I I b (1 -a In + 0) l ) I (CQt cua a12) (ab = In + I I I l b -aba -ba (D()ng 0 01 0\ \ Ap dl,lng cong thlic tren... Truong hQp t6ng quat: H *-Cen(GLn(A)) ~:3 h = (hjj)E H: h ~ Cen(GLn(A)) A/Rad(A) la vanh chlnh qui von Neumann Den : :3 X E A/Rad(A): hn2 = hn2'' ~ hn2 Suy fa : z:= hn 2.x.hn - hn E Rad(A) ~ Bi;}t p