Luận án Nhóm con của nhóm tuyến tính tổng quát trên vành chính qui Von Neumann

, . ~ 4 ? , CHUaNG 2: NHOM CON CHUAN TAC CUA NHOM K , ,.? , " , TUYEN TINH TONG QUAT TREN VANH CHINH QUI VON NEUMANN Vanh A dU<;1cgqi la chinhqui vonNeumannnC'u: \j x E A , 3 YEA: xyx=x NC'uA la vanhchinhqui vonNeumannthl mqiideal va mqivanh thuongcua A clingla vanhchinhquivonNeumann. Ta khaosat cacnh6mconchuffndc cua GL(M) trongtnfongh<;1p M c6coscihuuh?n.Khi d6,GL(M) ~ GL~(A). Xet n2:3 vaA lavanhchinhquivonNeumannhayt6ngquathall, A la vanhkC'th<;1pvoi donvi 1 ma A/Rad(A)la vanhchinhqui von Neumann,tasekhaosatcacnh6mconcuaGLn(A)chuffnboabCiiEn(A). Chungla nhungnh6mconH dtNcxacdinhbCiidiSuki~n:t6nt?i duy nhit idealB cuaA thoaEn(A,B)c H c Gn(A,B). 2.1MQts61{haiui~mvaHuhcha'tcdsd : NQidungcuam\lcnayneutenmQts6Hnhchit cobannhit cuacac tr~nsvectionsocip. Cacnh6mconEn(A),Gn(A,B),En(A,B)cllaGLn(A) sedu<;1cdinhnghIa, motacacph§nti1'va nhungHnhchit co banclla chungnhamph\lcV\lchoph§ntiC'ptheo. 2.1.a MQt s6Hohcha'tcua traosvectioosdc;1p: Xet x,y E A , cactransvecsionsocip thoanhling Hnhchit sau: l)xij .yij~(X+y)ij, l~i=l=j~n 2) (Xijrl =(-X)ij 3) [x~i, ykI] = In nC'uj =1=k vai =1=l .. k I k I .. 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=kva i =1=l j =1=k va i =1=l i , j , k khacnhaud6imQt. i , j , k khacnhaud6imQt. i ,j , k khacnhaudoi mQt. 6) VgE GLn(A) , Vy E A, V k, l E {I, 2, ...,n } kl -I 1 'g.y .g = n+gk.y.g I voi gk la cQtk cua g, g'tla dong l cuag-l. 15 Chung minh : C~ctinhcha'tta I dSn5 dedangsuyra ta dinhnghla,ta trlnhbay chungminhtinhcha't6 : Vdi g=(gij), g-I=(g'ij) EGLn(A),YEA,tac6: - ~ g e -1- ~ g' e kl - I +g - L , g - L t t ' y - n y.ekli,j=1 1J 1J s,t=1 S S k l -1 ( I ) -1 g.y g =g n +Y ekl g -1=g.g =In =In = In = In = In = In = In n . n + c.~ gijeij)'yek[. I g'stesl I,J=1 s,I=1 n n + ( .:?:gijyeijekl) ( I g'stest) 1,J=1 s,I=1 n n + (?:gikyeikekl) ( I g'steSl) 1=1 s,I=1 + (t gikyeil) (~ g'stest)1=1 s,t-l , + gk.y. g l vdi gk la cQtk cuag , g'l la dong l cuag-1 2.1.bNhomconEn(A) : Trang I.l.b , ta dadinhnghla: En(A)= Vdi B la ideal cua A, En(B) chInh6rncan cua En(A) sinhbai cactransvectionsoca'p bij, b E B . n L I+ gikYgsteilest i,s,t=l n L '+ gikyglteileZt i,t=l n L '+ gikyglteit . i,t=l M~nhd~5 : ( xem[1] , 1.2.26 trang36) .Vdi n C::3 , i :;z!:j va x E A, nhomcanehuctntdesinhb(JixU trangE,lA) La E,l La ideaL eua A sinhb(n x. Dgebi~t,hhomcanehuctntdesinhb(Ji lij trangE,lA) LaE,lA). 2.1.cNhomconGn(A,B) : Ki hi~uGn(A,B) chianhngu'<;1ccuaHimGLn(A/B)quad6ngca'u chinht~c: q>: GLn(A) (aij) ) GLn(A/B) ~ (ai) Ta co : q> la d6ngca'unhom. Nh~nxet : Cell (GLn(A/B)) la nhomconchu~nt~ccuaGLn(A/B) nen Gn(A,B) lanhomconchu~nt~ccuaGLn(A) M~nhd~6: g =(au) E Cen(GLn(A) g = a ,In, a E Cen(A) (g Lamatr~nvahudng) Chungminh: (i,j) = [0:::0} anI t ( C(jt j ) all"""", a1n 1,0 00000000 g.eij = eij.g = l omo } [ a \ ( ~ II aln I ao a. a. o.k. ° ~ J = .10.ln ) (dong j ) , n1 a ! nn (i,j) V ,. \-I' . ., 1"2" "> 2 .-01: g.eij = eij.g v 1*J, 1,J=, ; ,n va n - , taco : ." [ O:::O } . = anI ( a~j"J dongi t cQtj f a.. =a.. 11 J J => ak i =0 'v'k;i: i lajt =0 'v't;i:j La"yi , j l~nhtQtb~ng1,2, 3, , n , i *j , taco: all =a22=. . . =ann=a akl =0 ,. V k ::;cl g = [ all 0 ]0 ann rao °a) =a.In (matr~nvohuang) Ta chung minh a E Cen(A) : 'v'~E A, (a In) . (~.In)=(~.ln).(a.In) ~ a ~.In = ~a.In ~ a ~ =~a V~y a E Cen(A). Cu6icling,tachungminha khanghich. g EGLn(A)~:3g-IE GLn(A):g.g-I=In Ta chungminh g-lding thuQcCen(GLn(A)). Th~tv~y,'v'h EGLn(A),g-I.h =g-I(h.g).g-l -I ( h) -I h -1=g . g g = .g 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~nhdf 6 dachangminhxong. Ap dl,mgm~nhde 6 chotam (GLn(AIB)), taco : 1VI~nhd~7 : , . (f3 E Cen(A) ) a.f3=1=>a khanghichtrong A gECenGL,lA/B)g=a.I Trangd6 : a E (Cen(A/B))* , r- 0 ')- 11 I I=lod Til dotacothSmota Gn(A,B)bc3im~nhdesaD: M~nhd~8 : r a.. =0 (aij) E Gn(A, B)i - ~J-..- . . l aji - a JJ- ex , i oFj, a E (Cen(AIB))* au = Oij.a (mod B), a E (Cen(A/B))* 2.t.d.Nhom COllEn(A,B) : Ki hi~uEn(A,B) chi nhomcon chu§'nt~ccua En(A) sinhbdi cac transvecsionsddip xi.j trongGn(A,B) . En(A,B) = Nh~nxet: En(B) c En(A,B) c G,lA,B) M~nhd~ 9: Voi n ~3 , taco : En(A,B) = Chung minh : f)~t: E =. Ta chungminh: En(A,B)=E. HiSn nhien:E c En(A,B),tachIdin changmhlh:En(A,Bc E. Xet phftntasinhba'tkl cuaEn(A,B)codC;lng: . h =yk l xij( _y)k l , i * j , k *- I, X E B va YEA. =[yk l , xij ]. xij Ta co caetn1C1ngh<Jpsau: . k *-j va I *-i : [ ykl, xij ] =In~ h=Xij E En(B)c E . k *-j va I =i: [yki, xij ] =(YX)kj~ h=(YX)kj :Xij x E B ~ yx E B ~ (yx)kj.xij E E ~ h E E. . k =j va I *- i: 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.iiXij (- Y)ji E E V~yh E E trongmQitntC1ngh<Jp. Do do: En(A,B)c E V~y: En(A,B)=E Mt%nhd~ 9 da:chungminhKong Phdn tie'ptheo ta xet eaema triln boan vi : ]a matr~nma m6i dongva m6icQtchIco duynha'tmQtphftnta khac 0 1a 1. M6i matr~n hoanvi p tlidngung1 phepthe"(j E Sn codC;lng: p = (8j,cr(j)) Ta co :p -1=«>i,o--I(j))' Th~tv~y, d~tq=( i,o--I(j)), ajj =1;1 i ,0-(k) . k ,CT -I (j) . i ,0-(0--}(j)' 0--1U) ,0--1(j) , ( so'hC;lngkhac 0 duynha'tungvoi k =(j -IU) ). p.q =(aij) V~y : ajj =bij , '\! i,j -1 ' 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 (CQtk) (dong t) 81,0-(k) = In+ - I -I X (8 t,o- (1),...,8 t,o- (n)) 80 ,0-(k) 0 = In + x ( 0 ... 1 ... 0) !rI dong cr(k) CQt cr-I (t) 0 0 = 10 + I I 0...... I .'rIL~£!!!L(j(i<:) -fO cQ'tcr -let) = X er(k),er- I (I) M~nhd~10: vdi n ~3 : En(A,B)dli(/Cchu6nhodbiJi tqpcdclnatrqnhodnvi. Chungminh. X6tph~ntitsinhcuaEn(A,B)dc;wg: kt ij ( ) kt B A .. ky.x.-y ,XE ,YE ,1*-J, *-t GQi p = (8 k,er(I)) la ma tr~nhoanvi , ta co : kt ij C ) kt -1- kt -I ij -I ( ) kt -I P y.X -y P - P . Y .p .p.X P .p. -y p =yo- (k) 0--let) .xo- (i)o- -I (j) ,(-y)o- (k)o- -let) EEnCA,B). 2.2.Nh6rn.conchufintdccuanh6mtuye'ntfnhtangquattren vanh chfnhqui yonNeumann: Xet A ]avanhehinhquivanNeumann,tasedide'n2 dinhIi m6 tacaenh6mcan H euaGLn(A) ehu§'nboabdi En(A). GQi B la idealeua A. Nhomcan H eua GLn(A) duQegQila nh6mcanmdeB ne'u: En(A,B)c H c Gn(A,B). DinhIi 1neucaetinheha"teuanhomEnCA,B)vaehoke'tquamQi nh6mcanmtieB euaGLn(A)d€u ehu§,riboabdi EnCA).DinhIi 2 eho tachi€u ngtiQel~i: mQinhomcaneuaGLnCA)ehu§'nboabdi En(A) d€u la nhomcanmde B. Tli do,taco th€ motatfiteacaenhomcan euaGLnCA)ehu§'nboabdi EnCA)thongquacaeidea]euavanhA. 2.2.aDinhIf 1 : Gidsa A/Rad(A)la vanhehinhquivanNeumann,n 22 va B la idealeuaA, fae6.. (a) EnCA,B) ehuaedemalr(ind(lng In +vu, lrongd6 v la n- d)1 frong A, u la n - dongtrongB va u v =0 . Titd6,faco'EnCA,B) ehudnfdelrongGLnCA). (b) En(A,B)::J [ EnCA), GnCA,B)]. SuyramQinh6mcanmueB eua GLnCA)d~uehudnhodbiJi En(A). (c) Khi n 2 3, fae6.. EnCA,B)=[EnCA),EnCB)]=[ GLn(A),EnCA,B)]=[Eil(A~,H] wJi H la nh6mcanmueB. (d) V&iA la vanhehinhqui vanNeumann,laco .. EnCB) = EnCA,B) H(mnaa, v&i n 23, fae6.. EnCB)=[EnCB), En(B)]. CluIngminh. Chung minh 1. a : (v ) v=(Vj)= lvJ ,VjEA,i=l,...,n U=(Uj)=(U', un) , ujEB,j=l,...,n Truong h(jp 1: 1 + VnUn E GL1(B) f)~t : d =1+Vn Un d' =1+UnVn => d'- 1 E B UV =0 => II "UovoL.., 1 1 I =0 => u'v' + UnVn=0 =>d' =1- u'v' Taco: ( 1 - Und-1vn) d'= ( 1 - Und-IVn) ( 1+ UnVn) = 1 + UnVn- Und-I Vn - Und-I (VnUn) Vn = 1 + UnVn - Und-l Vn- UIId-I ( d -1 )Vn 1 d-I d-I=. +Un Vn - Un Vn - Un VII + Un VII = 1 d' ( 1 - Und-1vn) = (1 +UnVn)( 1 - Und-l Vn) = 1 - Und-I Vn+UnVn- UII(VnUn) d-I Vn = 1+UnVn- und-1vn- UII(d -1 ) d-IVn 1 d-I d-1= + UnVn- Un Vn - UnVn+ Un VII =1 V~y ( 1- und-Ivn)d' ~d'( 1- und-IVn)= 1, Suyfa d' khanghich. Do d6 d' E GI1(B) va (d' rl = 1- Und-IVn Khi d6 : VU [ VI J (U' Vn = ( v'U' V n U ' Un)= In +VU = ( In-ol+V'U' VIUn ) = VnU' l+vnun ( 1 +v'U'11-1 VnU' E)~t : ,- 1 ", d-1 ,a ,- n- 1 + v u - V un VnU = 111-1 + v'(1 - Und-1Vn) U' =111-1 + v'(d'ylu' Ta ch1?ngminh : In + vu ~ (l"~ XetOv€ phai : V,~"d') (~ ( 10-1 g:= o V.~"d-') (~ 0 ) (10-1 d ld-Ivo u' =(a0 VIUO ) (lool d ld -I v uI0 01 1) (a +V'U d-Iv u' =l v0 u' 0 0 v'u 1 d OJ (10-1 +,V'U'. v U0 v>J ( 10-1 V~y : In +vu = o Voi : V';"d-') (~ 0 ) ( l ln--I d d -I V U In (1 ' d-I ) (1 01n-I V Un l n-I ) E E' Bo 1 ' d-Iv0 u' 1 n( ) ( a 0 )In + vu E En(A,B) o d En(A,B). v I 1.1 J n V nun v'u J n d 0 ) (10-1 01 d ld-IvnH' 1) 01 1) 01 d Ta c~nchungminh: ( a . 0 ~) E En(A ,B) Tnloc he't,taphantich: (~UI ~)=(~" 0 ) ( 111_1 1 0 - yldl-I ) ( 111-1 1 - u' :') (~"' 0 , d,-J j = ( 1+V1dl-IUI - U I ~J = (-:' ~) Tht vy , xetvSphcH: h :=("' - V'd") C ) (n-J ') ("-' 0 'n-I d'--J)1 -- u' = Co-' - v'r'l C n (h y I d ,..1, ) n-J ). 0 1 :- UI d 1- I = ('" - vor') (1 y' d 1-\ , In-I - u'yldl-J+d,-1)1 - u I = ("-' - Y'd'-I, (1 v'd 1-\ , ) In-I (-llIV'+l)dl')1 - u' = ("-' -V'd'.') (1 y I d 1- \ , l n - I d '. d' J)1 - U I =('" -Vt'd"'J e,,-: v ' dJ ,- 'J-u V~y : C:I OJ ~ (~"' -Vldl-I ) ( 1n-I 1 - u' ~) (~n-I n (~"' ~I'-'J Ta cung co: (1 +u 'y 'd' -1). d' =d' +U 'y , = 1- u 'y , + U 'y , =1 Suy fa : (~"' Nen: ( a -u' Vdi: (~"' d,-1 - 1 + U'y' d,-l -V'dl-I J ( 1- n-I 1 0 = (~"-' - ( 1n-I - 0 - ( 111_1 - 0 - v' (1 + "; y'd,-I )J - V'-V't:'V'd'-'] - VIUIVldl_l ] ( 1 11-1 1 0 - VIU . 'Vld'-I) ( 1 1 j 0"-1 J=(~~' -V'U~V'd") (~" - VIUIV'd'-I J ( 1n-I 1 ' -H' ( In-1 Suy fa: 0 VI ) -I ( In-I .1 -H' - v;] : ')' VI ) --! (111-1, 0) ( 111_1 1 - U I 0 0 ) EE (B) ( In-I 1 n, 0 ~) (~n-I ~')E En(A,B) VI )1 E En(A) V . I ) ( In-1 0 J1 0 d,-I V~y : (-:' 0 ) ( 10-1 1 E En(A,B) 0 0 ') d 1-.1) Nen: (~ ~)=C: J(~I Ta clingco : (~ ~) =(~ Nen: (~ 0 ) ( 10-1 d . E En(A,B). 0 O J ( 10-1 1 E En(A,B) 0 0 J ( 10-1 d'-I u' ~J ~) (~" (:) 0 J ( 1n--1 d,-I u' OJ (~"' ~J (d" 0 ') ( 1 1 d'-I) ; ~)=(~"-' Suyfa: (~ ~) E En(A,B). Tlido : ]0+ vu E En(A,B) . ( In-1 (1'--1 l1' ~) E En(B) . 0 ') d'-ld) E En(A,B) , ( Ke'tqua tU [5], ~2) ( 0) (1,_, O)( OJ (O, 0 ) (I", 0 ') (1, Jd EEn(A,B) 0 d,-I d d' 0 d'-I) on-- Ta co: ("-I 0) (1"_- OJ ("-' 0 ) = (1"-J ,,) (:t ad')d,-I u' d' 0 Truong hqp2: ::3i E {1,2 , ... , n} : ViE Rad(A). Ntu i =n : Vn E Rad(A) =>VnUnE Rad(A) =>1+VnUn khanghichtrongA Soy ra : 1+VnUnE GL1(B) =>In +VU EEn(A,B) (tnf<JnghQp 1). Ntu i < n: xet phep the- (j =( i p =( 0j ,0 (k) ). £)~t: n) Vamatr~nhO<lnvi tlfdngltng ° v =pv= (OI.a(l)Vl +OI.a(2)V, + ...+OI,a(,,)V" I I O'2,u(1)V 1 +52,crC2)V 2 + ... +O'2,crCn)V n I I . I l~","(l)Vl +O","CZ)VZ+ ...+o"."c,,)V,,) (VI) I I I . I I I = I V II I <Em~?I1~mi I. I l~J ( matr~nco du'Qcb~ngcachhoanvi cacdong'i & n cuav) UO=U P =(UIOlp(l)+:..+ullb;l,a(l)'Ul~,a(2)+"'+Ullb;1,O"(2)'...,U1O1,acn)+...+Unb;],acn)) =(U1 ,. . . , Un ,...Ui ) ( matr~nco du'<;1cb~ngcachhoan vi caccOt i & n cua u) Ta co: uOvO=up.pv =u(p.p)v=0 vnO =Vi E Rad(A) => In + va UOE En(A,B) In+ v u =p.p-l+ p.pv.up.p-I=P On +vaDO)p-l E En(A,B) Truonghqptangquat: - - Voi A / Rad(A)la vanhchinhqui vorlNeumann,taco : ::3xEA:vn.x'Vn=Vn vn - Vn. X . Vn E Rad (A) (vn - Vn . X . Vn). UnE Rad (A) vn( 1- x Vn).UnE Rad(A) 1 + VnO - X Vn)Un khanghichtrong A. 1 + Vn(1 - X Vn)UnE GL1(B). g := In + v(1 - X VB)U E En(A,B) (tnrong hQp 1) Ph§n tie'pthea,tach(tngminh: h := 1n+ V X VnU E En(A,B) Xet: t = (- Vn-l.xt-1,n( In + VX VnU) ( Vn-l:Xt-1,n =In ( . )n-l,n ( )( ) n-l,n+ - Vn-l.X . V X VnU . Vn-l.X =.In + ( ) n-I n )- Vn-l.X '.V X Vn (U(Vn-l.xt-1,n) D~t U 1=U .( Vn-l.xt-I,n (la n- dong trongB ) 1 ( )n-l,nV = - Vn-l.X .V X Vn (]a n- cQt trongA ) 1 1- ( ) n-l,n ( )n-l,n -. - 0U V - U . Vn-I.X - Vn-l.X .V X Vn - U v X Vn - 1 0 '\ (VI '\ . II V21 . I I . I X.Vn ;. ~. -V"l'.xJ U (1 \0 VI =I' l~ 0 ... ( VI lI V2 VI =I . I XVn l v - V XV Jn-I V nn-I n V \1-1 =Vn-l ( 1 - X. Vn ) X Vn =Vn-l ( XVn - X . Vn X Vn) =Vn-I X ( Vn - Vn X Vn) VI Vn - Vn X Vn E Rad(A) lien vln-I E Rad(A), ta SHYra : t =In +Ul VI E En(A,B) (TnfOnghQp2voi i =n-I) h - ( )n-l,n t ( )n-l,n E (A B)- Vn-1.X - Vn-l.X En, g.h =(1n+ v(1 - X Vn) U ) .( In + V X VnU ) = In + v(1 - X Vn) U + V X VnU + v(l - X Vn) U . V X VnU =In +v(1 - X Vn) U +V X Vn U =In + vu - V X VnU + V X VnU =.In + vu g , h E En(A,B) => In + VU E En(A,B) Ph§n tie'ptheo, tachllngminhEn(A,B)chu!Inuk trongGLn(A): Ta thl/chi~ncacbu'ocsan: 1. Tn(oc bet xet cac phgn tU'bgtkI g E GIn(A) , xij E Gn(A,B), i 7=j , taco :X E B. En(A,B) du'Qchu§'nboabdi t~pcacmatr~nhoanvi Henco th~ghi sU'(i, j ) =(1,n).Taco: g xl,ng-l = In +v X w (VI' I v21 voi V =I . I lvJ Ia cQt 1 Clla g . ( ) 1 , d' ?-1W = WI W 2 ...W n a ong n cua g . WVlaph~nti'ta dong n , cQt1 clla -1 1 " 0g g = n Den WV = £>~tu=XW , taco UiE B ,'\1i = 1,2,...,n. UV=(xw)v=X (wv) =x. 0 =0 Suyfa : g.x1,n.g-l= In+vu E En(A,B) 2. Tie'pthen, taxetcacph§.nti't g E GIn(A), h=yi i Xi j (-yyi , i *-j , X E B,YEA, taCO: 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-i) 3. Cu6i cling, xet gE Gln(A) , h E En(A,B) ta co : n-l h =Il hi , i=l vdi hi la ph§.ntit sinh d<;1ngy-iixij (-yi cua En(A,B). n-l g h g-1=g ( Il hi) g-l i=1 n-l =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~nquacaebuoc: a) Trudc h€t, ta chung minh : [1ij, g ]E En(A,B) ,\/ g E Gn(A,B). . . .. .. I [11,.I,g] =11,.I.g(-I)l,.I.g- En(A,B) c1t(Qcchu~nboa bdi t~pcac malr~nhoanvi Denco the gia siT (i, j ) =( 1, n ), khi do : [11,Il,g]= 11,Il.g.(-1)I,n.g-I =11, n ( 1n - VW ) ydi y = (:J H\CQt 1cuag, W =( W' W,,) Iii di, ng n cuag~ I w.v laph§.ntud()ngn , cQt1 cua g-l:g= In ~ wv=O. A/Rad(A) la vanh chinh qui van Neumann lien : .3X E A : VnX Vn- Vn E Rad(A) Ta chung minh (xvn)I,n va g giao hoan theomodulo En(A,B). Ta co: h:= In - vxvnWE En(A,B) (TntonghQpt6ngqUCttdiu (a),apdvngchou =- w ) gE Gn(A,B) va Vn la ph§.ntiTthuQcd()ngn , cOt 1 cua g nen Do do: X'VnE B VIlEB (Xvn) I.il E En(B) [ (xvn)1.0, g ] =(xvn)1.n(1n- VXVnW) E En(A,B) Tiep theotachungminh (1 - x Vn)I,n cling giao hoan voi g theo modulo En(A,B) . Ta co : [(1 - XVn)1,n, g ] = (1 - XVn)1,n, (1n - v(1 - XVn)w ) £)~t u:= -(1 - XVn) W =( u' un). [(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 +VnUnE GLI(B) £)~t: ,- 1 ", d-I ,a .- n-1+v U - V Un VnU Chungminhtu'dngtv Callla , taco : (1 ) 1,n (1 ) (1 ) 1,n ( In-I -XVn n +vu = -XVn 0 V'l . lnd-Ij ( a 1 ) 0 0 ) (1,,1 0) d ld--'vnu' d g E Gn(A,B) => V2, v3, .', , Vn - 1 E B d-I d-I . dol B=>V2Ull ,v3un ,...,vn-IUn E, , D6ng thai, xet trong GLI1(A/B) va GLj(A/B), taco : g = a - , I ; g-l - = f3 - , I vdi a - , f3 E Cen(A/B)* (l-xv,,) I,n (1" (1 . . . I-xv,,) (1 V'U"do') 1°' - - ° I I - -. v,u"d' II = -- II OJ.. vur,1- - - - I ' ,,( I' , . , . ,. lo ) l-- v d,1. , . . 1 n-Illn )0 . . . . 1 (1,.. vlud-'+I--xv) I "n I 10 1.. v2u"d-I .1= I l' : : : ; <, l:"d', jO. . . . . 1 g -I. g - =I => a - . B - = 1 => VI Wn - = 1 . Vn E B => xvn - = 0 u =-(1 - xvn)W => un=-(1 - XVn) WII => U" =- W n d E GL1(B) =>d-l E GL1(B) => ~ - == 1 VI . U" . d-I =VI (- w. " ). I - =- 1 Nen : vlund-1+ 1 E B vlund-1+ 1- XVn E B 1n ( In-I (1-XVn) , 0 V'~"d-') E E,,(B) In (a (1-XVn) , (1n+Vu) E En(B) \0 0 )d En(B) Ta co: (~ 0 ) . d E En(A,B) (Chang minh 1a ) [( 1- xvn)l,n, g] =(1-xvn)l,n (1n+vu) E En(A,B) Tom l~i, taco [(xvn)1,n,g], [(1-xvn)1,n,g] E En(A,B) , tU(10: [11,n,g]=[(xvn+l-xvn)l,n,g] = [(xvn)I,n(J -XVn)1,n, g] = (xvn)1,n(1-xvn)1,ng (-1 +XVn)1,n(- XVn)1,11g-I = (xvn)l,n. (1 - xvn)l,ng (-1 + xvn)l,ng-1. g (-x.vlI)l.ng-1 = (Xvn)I,n.[(1-xvn)l,n, g ](-xvll)I,II(xvn)I,n.g.(- xvn)l,ng-l = (xvn)1,n[(1- XVn)I,n,g] (- XVII)I,n . [(xvn)I,n, g] E En(A,B) b) X6t phtlntil' thuQc[En(A),Gn(A,B)] cI~ngI ylJ, g J , YEA, i :;i:j, gE Gn(A,B): En(A) la nhomconchu§'nt~csinhboi ] l,n trongEn(A)nen : 3ZEA: yi,.i = Zkl 11,11(-Z)kl,k;t:.l. [yi.j,g]=Zkll1.n(_Z)kl g Zkl (_I)I,n (-Z)kl g-1 . =Zkl. 11,n. (- Z)kl g Zkl . (-1 )l.n . (- Z)kl g-1Zkl .(- Z)kl £)~th=(- Z)kl .g.Zkl g E Gn(A,B)=>h E Gn(A,B) Tli chungminhph~n(a),taco : [11,n,h] E En(A,B). [yi,j,g]= Zkl 11,nh(-1)1,n h-1 (-Z)kl =Zkl [11,n,h] (-Z)kl EEn(A,B) D~ktt thucchungminhphfin lb, tagQiH la nh6mconmo.cII vachungminh H chuanboabdi En(A): Taco : En(A,B)c H c Gn(A,B) V xij E :Bn(A),V h E H, xijh (-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 =>xijh (-x )ij E H V~y H chu~nboabdi En(A). Chungminhloc: Chung minh En(A,ll) c [En(A), En(ll) ] : X6t ph~ntl'isinhba'tki cuaEn(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)] En(A,B) c [En(A), En(B) ] Hi~nnhi~l1,taco : [ 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) En(A,B) la nhom con chu§'nt~ccua GLII(A),( chang minh la) nen : g h g-l E En(A,B) ~ [ g , h ] =g h g-l.h-1E En(A,B) Do do : [ GLII(A), En(A,B) ] c' En(A,B) . Suy fa : En(A,B) c [ En(A),EII(B) ]c [GLn(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'tthueph~nIe, taehu'ngminhEn(A,H) =[ En(A), H) : Taco : En(A,B) c H c Gn(A,B) En(B)c En(A,B)c H Dodo: En(A,B)= [En(A),En(B)]c [En(A),H ] D6ngthoi : [En(A),H ] c [En(A),Gn(A,B) ] c En(A,B) V?y En(A,B)=[En(A),H ] Chung minh l.d : Chung minh En(B) =En(A,B) : Ta daco: En(B) c En(A,B) , chI c~nchungminh : En(A,B) c En(B) X6t ph~ntti 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~nhoahdi t?P cac matr?n . hoan vi, taco th€ giil sU' (i, j ) = (1 , 2). Ta chang minh : h:=(_y)12z21y12EEn(A,B), (zEB,YEA) Vi A chinhqui vanNeumannnen t6nt~iphfintU' x E A thoa: z =z x z . Khi do: h := (- y )12Z21Y12 = (-xzy)1 2(Xzy)12 (-y )12Z21yl 2(-xzy)1 2(Xzy)12 =(-xzy)12. ( xzy - Y )12 Z2I( Y - xzy )12. (xzy)12 vdi (xzy)12E En(B). X6t g :=( xzy_y)12. Z21. ( Y - xzy)12 Ta l~pcong thlic tinh aI2b21(-a)12,a, bE A. Ap dl,lngcongthlictrencho a=(xz-I)y E B , b =z ab =(xz-l)y.z ba =z (xz- 1)y =(zxz)y- zy=zy - zy =0 -aha=-a (ba)=0 Ta co: g=a12b21(-a)12 ( a 1 I 1 I a12b21(-a)12 =In + I I b (1 -a 0 ... 0) l ) I (CQt 2 cua a12) (D()ng 1 cua (-a)12 ) (ab -aba 0 ... 01 I b -ba 0 ... 0\ =In+ I 0 0 .\I l 0 ) (xz - l)yz I z g =In + I 0 l 0 0 ... 0\ I 0 0 ... 0 I 0 . . '1 ~j00 0 \ 0 I I Ij Ta co th€ pMintich g thanhd<;lng: rl . (xz-\)yzx 0 ". 01 (I 0 01(I (I-xz)yzx 0 ... 01g = l ~ ~ :..::::::::..O J I~ I :::::::.o J II~ ~ '.'~~~.'..'.O J ' 0 : I ~O 1 ~O..." 1 = «xz -1)yzx )12. Z21.« 1 - xz )yzx )12E En(B) Th~tv~y, xet : (1 (xz-l)yzx 0 ... 0\ (1 0 0\ (1 (l-xz)yzx0 ... 0 \ t := I I ~ ~ ::::::::..o J I!~ 1 ::::::::.() J l l ~~""""'~"""":"'::::::~:"~" J I 0 1 ~O I 0 1 (1+(xz-l)yzxz (xz-l)yzx ... 0\ (1 (I-xz)yzx 0 ... 0\ = I I ~ 1. ...::::::...O J I ~ l"."".""::::::::"~ J I 1 ~o' I. (1+(xz- 1)yz 0 0 ... I z 1 0 ... = I 0 l 0 0 . . (1+(xz-l)yz (xz-l)yzx ... 01 (1 (l-xi)yzx 0 ... 01 =I z . 1... 0 110 1 0 I l j \ )0 1 ~O 1 (1+(xz- l)yz (l +(xz-l)yz)(l- xz)yzx+(xz-l)yzx 01 =I z z(l- xz)yzx+1 0 I l , j0 1 Taco: 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 /. 12 21 J 2 Do do : g =t=«xz -1)yzx) . z . « I - xz )yzx) E En(B) Suy fa h = (- xzy)12g (XZy)12E En(B) V~yEn(A,B) c En(B) . ChungminhEn(B)=[En(B), En(JJ)] vOin 2 3 : Hi~nnhien[En(B),En(B)]c En(B) . Taclingco : V ZlJ E En(B), zij =(zxZ)ii, x E A zij= (ZXZ)ij=[(ZX)ik,zkj]E [En(B),En(B)] (chQnk:f=i,j) Nen En(B) c [En(B),En(B)] V~y :En(B)=[En(B),En(B)]. DinhIi 1 dachungminhxong. D~didSndinh112,tac§nmQts6b6d~. Trangph§nsau, ta ki hi~uA ia v~nhkSthqpvoiddnvi 1 vathaimandi~uki~nA/Rad(A) lavanhchinhquivanNeumann, 2.2.b.B6d~1: Chon 2 3,aE A , a :;r0 va a kh6ngLaLtcJCdla 0 . H Lanhomcon cua GLn(A) chwinhodbiJi tqpcac aU, i :;rj thodtlnhchatH chaa g =(gij) thod gn]=0 vat6ntc;zi, j sao cho g kh6nggiaohoanvcJi ij 1<. '<a, - l :;rJ - n, Khido, H chaa mQtransvectionsacapkhac ill' Chung minh : n-I Tru'onghqp1: g=Il b;,n voi bi E A va t6n t~i j <n thoa bj -:f=().i=1 ChQns6nguyendu'dngk thai: k <n va k -:f=j , taco: kj ( kj ) H [ kj ] - kj ( kj ) -I Ha .g. - a E. => a, g - a .g. - a .g E. [ kj ] - kj b I,n b j,n b n-I,n ( kj )(b )n-I,n (b) j.n (b) I,na , g - a 1 ... j ... n-I -a - n-I ... - j ... - I NSuj =n - 1 , ta co : [akj,g] =akj bIl,n ...bjj,n (-a)kj(-bj)j,n... (- bI) I,n NSuj <n-1 : bn-In-I,n(-akj).(-bn-dn-I,n=(-a)kj[aki,bll_tll-I,n]=(-a )ki Dodo: [ kj ] = kj b I,n b.j,n b n-2,n(- ) kj (-b ) n-2,n (-b.)j,n (-b ) I,na , g a I "'.1 ... n-2 a n-2 "'.1'" t =akjblI,n ...bjj,n (-a)kj(-bj)j,n... (- bt) I,ll V~y: [akj,g] =akjblI,n ...bjj,n(-a)kj(-bj)j,n...(- bI) I,n,V j S;n - 1 =akjbI,n b. j-I,n (-a)kj [ kj b.j,n ] (-b. )j-J,n (_b) I,n. "'.1-1 . .a,.1.' .1-1 ... I =kj bI,n b. j-I,n (- )kj (ab .)k,n(-b. )j-I,n (-b )I,na ... .I-I a .1' .I-I ... 1 = kj bI,n b. j-I,n (-a)kj(-b. )j-I,n (-b )I,n (ab.)k,na ... .1-1 . .1-1 ... I . .1 [akj, g]= akjbl,n ...( bj-2)j-2,n(-a)kj,(-bi-2)j-2,FI... (_b])I,n,(abl,n = akjbI,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 Ntu g giao hoanvui ai,n, vUi i batki nhi}ho.nn : g aLn=aLng =>g ( In +a.ejn) =( In +a.ejn) g => g.aein=a.eing 0 0 . . 0 . . 0 g l.ia g2,ia 0 0 = agn,1 ag",2 agn,n (dong i) 0 g n.ia 0 0 => J gj; a = 0, V j,ti 19.. a = ag = aII nn gji= O,\ij:;ti => (gii - 1)a=0 gji = 0, \i j:;ti => gii =1 L§y i tu 1 den n -1 taco : 1 . . 0 0 .1 . 0 gl,n g2,n n-I = TI (g i,n ) in i=1 :;t In, (g khong giao hmln voi aij)g = .gn-I.n 0 . . . 1 Nen t6n t~i i < n saocho gin:;to. Suy ra g thoacaedieu ki~ncua trtiongh<;1p1 V~y H ehua IDQttransvectionsd dip khac In gtt :::I 0) (0 0) I gll .. gl,n a I = I ' a I I . I (dong i) g n.l 0) \,,0 0) "gn,t .. gn.n Ntll tUnt(liml)t i nho hdll n thod g khonggiaohoall vai ain : [ in ] ( in -1 ( ) i n ) Hg,a' =g. a'. g.-a , E [ .i,n] ( i,n -1 ) ( ) i,n (1 + ' ) ( ) i,n g,a = g. a . g . -a = n gi.a.gn . -a ( gi, g'n l~nltTQtla cQt i, dong n cuag va gol) Ta chungminh: g'n=(0 0 ... 0 1) Th~tv~y: -1 1g.g = n => r ~ ( -I ) j ~gok g k j = l~gnk(g-')kn= an j ,jsn-l ann { <g-I)nj=0 ,j:<:n-l (g-I)nn=1 => => g'n=(0 0 ...0 1) Tli d6: r gl' 1 g2:'i I (0 0 ... gJja 0 0 ... g2ja , gi a g n= a(O...Ol)= gn-I.i .gn-I,ia 0 () 0 ... () 1 0 0 1 '" g I i a ... g2ia ] 0 ... 0 1 ... 0 0 On+gia g'n)(-a)i,n= ... ... ... ... ... ... 0 0 ,.. - a I (dong i) 0 0 0 ... gn-I,ia ... 0 ... 0 0 ... g lia g 2; a I 0 0 - :.+,~i~aj Suyra : (1 + ' )( ) i,n ( . ) I,n ( ) 2,n ( + ) i,n ( ) n-l,n n gi a g n -a ::;::gli a . g2i a -a gi i .a . ... gn-I,i'a V~y [g, ai,n]E H thmldiSu ki~ntn1ongh<;1p1. SuyraH chuaillQttransvectionsoca'pkhac 1n . TruonghQpt6ngquat: Ntu tallt{Iii thod:g khollggiaohoallvo'iali, (2::;i::;ll), taco: [ Ii ] ( Ii -1 )( ) li (1 ' )( ) Ii g, a = g. a . g -a = n+gl.a.gi -a ( gl, g\ l§n h1Qtla cQt1,dong i cuag va g-I) , gl.a.g i = r gJ" g2,1 I' l~~-" a(g'il...g'in) gl.a.g'icodongcu6iclingla : (0 0 ... 0) In+ gla.g'i COdongcu6iclingla : (0 0 ... 0 Dodo: (1n+gl.a.g'i).(-a)1i codongcu6jclingla : (0 0 ... E> ~ t h [ Ii ] [ Ii -I ( ) Ii ] H(;t :=g,a =g.a.g.-a E. 1) 0 1) Ta co : hn1=hn2 =...= hn,n-l=1- hnn=0 Suy ra h thoacacdiSuki~ncuatrtionghQp2. V~y H chua illQt transvectionsoca'pkhac 1n Ngli(fC [{Ii, lltll g giao hoall vui llU;Ji aI,i , 2 .s i .s ll, taco : l,i l,i 'o.r-I' 2 3a .'g=g.a ,vl=, ,...,n. 1 0 0 1 I... .. = 0 0 (In +aeIi ).g=g. (In +a e Ii) , \j i =2,3,... , n. ae Ii .g=g .ae Ii , \j i =2,3,... , n. (C{Jti) , , 0 a 0 gll g',n 0 . . . . 0 gll gl,n = gn,l gn,n g n,l g n,n (agi1 agi2 ... ag.. ... I 11 l:. . . . . . . 0 { agii = gIla ag.. = 01] agin l ( 0 10 I _I . -I J l~ , \j i =2,3,... , n , \j j *-i { ag.. = g a1I 11 . gij = 0 ' \j i =2,3,... , n , \j j *-i Ta l(;lixettiSp 2 khaDang: (qt i) . . 0 a 0 0 . . . . 0 g J I a g 21 a g 3 I a g n 1a (CQt i) (1) 0 ) 0 I I 0 I oJ . Ntu t611t{lisf/ i =2,3, ..., n-1'thoag khonggiaohoanvai ai,I: h [ i ,I ] i,l [ ( ) i,1 -I ] H=a,g=a.g.-a .g.E I::'\~ h" ( ) il -I 1 '.Llong t 01: g . -a '.g = n- gi .a.g 'I Voi gi 1aCQti cua g,g'l 1adong1cuagol Tli (1) taco : gni=0 gt,i g2,i , gi .a.g 1= a.(g'lJ gn-I.; ° , g 12 g'ln) Suyra : gi .a.g'l COdong n la ( 0 0 In+gi .a.g'l COdong n la (0 0 h co dong n la (0 0 ... I ) V~y: h thmlcaedi~uki~ncuatru'ongh<Jp2. H chua mQttransvectionsocffpkhac In 0) I ) . 'Ngll{fCltJinill g giaohoanvf1iffl(Ji ai,l,2 ::;i ::;n-l, faco : g.ai,l =ai,l.g g(In+a.ei,l ) =(In + a.el,i)g g.a,ei,l =a.ei,l. g ° ...,..,..,...° g. :l. .. , 0 dongi gli.a o l g2i.a ..., . . . . go,.a '. '. '. '. oj = ~ ~ 0 1 al . .g ;) I .' . oj I I! dong i = Suy ra ': g12.a=g13.a=...=gl,n.a=0 g12=g13=... =gl,n=0 , (2) Ti'i (1)va(2) chota g la matr~ncheo,suyra g-l clingla ma tr~ncheo. Khi do: [g,aij]=gaij. g-l (-a )ij E H ij -1 1 'g a . g = n + gi.a.g j (gi, g'j la cQt i, dong j cua g,g-l ) g aij. g-1 = In + r oI . I ~ gOO I \I I . la (0 " I lo) , 0 ... g jj ... 0) (0 0'1 I. . . . . . . . . .. . I I° ° ... gjjag'jj ... o I =In+IO ""'" o J I l~ "',, "" 0 - ( , )ij- gii.a.gjj [ ij ] - ( , )ij ( )ii - ( , )i jg, a - gii.a.gjj -a - gij.a.g,jj - a [g,aij ] ]atransvectionsd cffp khac In thuQcH B5dS 1 da:chungminhxong. 2.2.cB6 d~2 : Cho H la nh6mconcuaGLiA) vachullnhod!xiiEiA) , n ;: 3 . Ne'u H khongla tamcua GLiA) the H chliamilt transvection.'Ie!c{{p khdcIn Chungminh: Truonghqp1: H chuag=(gjj) thO<l gn,I:;: 0 va t6nt~jk -::/= 1saG cho g khonggiaohoanvai 1k,1E En(A) . Khi do H va g thoacacdi€u ki~ncuab6dS 1. V~y H chuaillQttran~vectionsoca"pkhac In. TruonghQp2 : H chuah =(hij) thoa hn2 -::/=0 va t6nt~i YEA thoa, hnl +hn2'Y =0 £>~tg:=(gjj):=(_y)2,I.h.y2.1,taco gEH. ( I ... 0 (h" h" ... h...J r1 ... 01g = l ~.~ .~. ::: .~. l ~~.t ~~~ ::: ~~~ J l '~' ,~, ::: .~, j0 ... ... 1 hnt hn2' ... hnn 0... ... 1 Suyfa : gn1=hnl +hn2 Y=0 gn2 =hn2-::/=0 Ta chungminh gkhonggiaohoanvai 12,1: S 121 /" h;:o t ? t ' d' " 1 l'uy fa : g. cop an U ~1ong n, cQt a: gnJ+gn2 hIt h'2 '" hl.n 1 (1 ... 01 -yh'l +h21 -yhl2 + h22 I l: 1 01... -Yh,..+h,"j ,.. J = I ... ... ... ... ... hill hn2" " hnn ... ... glt gIn I e ,0] 121 I' . I l 1 J g. = gill goo) .l r:;. 0 121.g= 0 . . 1 gll . . ginI . I gnl . : gJ N" 121 " h;: t? t . d' " 1 1,en. .gcop an ti' c;t!ong n, cQt a: gn!. D6ngthai: gn2*-0 =>gol+go2*- gol Suy ra : g 121*- 121g . V~y: g khong giao hmln vai 121. Suy ra H va g thoacac diSu kit%nct1atrtiangh<;1p1 H chuamQttransvectionsd ca'pkhac In. Truonghqp3: H chuag=(gij)~CeDeGLn(A)) vagn,l=0 Ne"ut6nt~i cacsC;k,l khacnhausaocho g khonggiaohoanvai 1k,tE En(A) thl H. va g thoacac diSu kit%ncua trtiangh<;1p1 Den H chuamQttransvectionsdca'pkhac In. Ngti<;1cl~i,ne"ug giaohoanvai l k,lg. = g.( 1+ekI) = l kl /. . k I /, , VOl mQI *-, ta CO: l k,l .g ( 1+ekl ).g g.ek1 = ekl'.g Thlfchit%nttidngtlfph~nchungminhcl'Iamt%nhdS 6 , taco g la matr~nvo htiang, dodo : g = gll.ln , gll E A* g ~CeDeGLn(A)) =>:3h E GLo(A): g.h*-h.g =>gll.h *- h.gll =>:3YEA: gll.Y*-y.gll Khi do: [ g, /2] =g. y12g-\_y)12E H [ 12 ] ( 1 ) 12 ( -1 1 . 12g,Y = gll. n Y gll. n) (-y ) 12 -1 ( 12 = gll Y gll -y ) = gl\ g~IIIY0 .., ~ j l 0 ,.. ..' gll l (g~11-g~:y 0 '" -I , gll l~ 0 -1 glJ 1 -y+gl1ygl~1 0..,'0 0 1= 0 1 ( -1 ) 12= -y +gll Y gll . -1 Ta co: y.gll *-gll,Y => Y *- gll.Y gll =>- y +gll Y gl1-1*-0 [ 12 ] ( -1 12 g, Y = -y +gll Y gl1 ) *- In V~Y H chua [g, y12] la mQttransvectionsd ca'pkhac In ' Truong hqp 4 :H chua h =(hij)~Cen(GLn(A)) va h22E GL](A) Ntu (h-l),rl =0: Ta co h-1E H va h-1~ Cen( GLn(A) ), (h-I )n,1= 0 nen H thml cac diSu ki~n Cllatnfdngh<;1p3, V~y H ch((a mQttransvectionsd ca'pkhac In' va h-1 Ntu ( h-l ) III :;z!:0 : E>~t: g=h-1. 11,2.h (-1)1.2 Ta co : 11,2.h (_1)1.2E H =>g E H g =( (1n+(h-1)1.1.h2).(-1 )1,2 Voi (h-1)1la cQt1 cuah-I h2 ladong2 cuah (h-l)lI (h~1)21 g=( In + (h21 h22 ... h2n) ). (-1)2,1 (h-I) nI go t=(h-t)o 1 .h2t - (h-1)nt h22 gn2=(h-t )n1h22 X6t hai truong h<;1p: Truong h<;1pgn2=0 : (h-I)n 1h22=0 => (h-t)n I. h22 .(h22)"1 =0 => (h-I)n 1=0 (tnli gia thie't (h-I )n,1=f::-0 ) Truong h<;1pgn2 =f::-0: Voi YEA, ta co ,: go 1+ gn 2'Y =(h-I )n 1.h2 1 - (h-I)n I .h2 2 +(h-I )n I h2 2Y ChQny =h22-1 ( h22 - h21 ), tasuyfa : gn,1+gn,2'Y =(h-I )n 1.h21 - (h-I )n 1h22+ (h-l )n t h22 . h22-1(h22- h21) =(h-l)n 1.h21- (h-I)n 1h22 + (h-I)11 1h22 - (h-I )111. h21 =0 Nen H va g thoacaediSuki~ncuatruongh<;1p2. Truonghqp5: H chuah=(hij) ~Cen( GLn(A)) va hn2~0 . D~tf: = (~1 10) Ta co: (~ ~) (~1 01) (~ ~) = (~1 \) (~ IJ ~ (:1 10J Suyra : f= (~ \) (~1 OJ (~ \) D~t: (~ 1 ) ( 1 0 ) ... 1 ' -1 1 Hin It(9tla f] , f 2 Ta co: f =fl . f 2 . fl E E2 (A) D~t f' =G ~J Taco: =11,2(-1)2,1 11,2E En(A) D~t: g:= f' .h.(f'rl , taco g E H. Ne'ug E CeDeGLn(A)) thl g.f' =f '.g f '.h=f 'g Den h =g , (mau thu~nvoi h ~CeDeGLIl(A) )) V~y: g ~CeDeGLn(A) ) D6ngthai: ( 0) ( 0 Htl 0)= (flf2 O) (fl J1n-2 0 In-2 0 111-2 0 111-2 0 - (flf,f, 0) - '0 1 f11-2 Vy: f' =( 0 )(f2 lJ ( 0)In-2 0 1n-2 f' = 1 ~ (f'r1 = 0 -1 . . o l1 0 . 1 -I 0 .. 1) 0 1 . . 0 -1 0 0 - - . 1 0 1 . . 0 -1 0 (0 -1 . . 0 I 1 0 g = 1 . I .h. 1 l~ . . . 1 0 . - 1 ( \ (0 -1 . . 0\ I' . . . . .. 'I 0 I I. . . . . .. I . I = I. . . . . .. , .~ 1 '1 l~,: 'h,,', '..".. 'hJ l~ , , : ;) ( tachiquailt~mdSndongcu6iclIng) Dodo: gnI =hn,2=0 Suy fa H va g thmicaediSu kit$ncua tfu'ongh<;1p3 . Truong hQpt6ngquat: H *-Cen(GLn(A))~:3 h=(hjj)EH: h ~Cen(GLn(A)) A/Rad(A)lavanhchlnhquivonNeumannDen: :3X E A/Rad(A): hn2 =hn2' ~ .hn2 Suy fa : z:= hn2.x.hn2- hn2 E Rad(A) Bi;}t p := 1 - hn2 X . ~ Ntu p. hill =0 thl (1 - hn2.X ).hn I =0 Suy fa hn1- hn2.(x.hn1)=0 0 NSu hn2 =0 :H vah thoacaediSukit$ncuatru'ongh<;1p5. 51 0 Ne'uhn2"* 0 :d~ty =x~hn1=>hn 1-hn2 Y=O. H va h thai!cacdi~uki~ncuatnI'Cingh<jp2. Ntu p. h"l ;r:O:taco p"*O. E> ~ t - ( ) .- h-I I,n hC;l g- gij.- .p. Ne'u g E Cell GLn(A) thl g.h=h.g D d~ h In h . in0 O:g. =p'. ,suyra:g=p' Khi do: pi,n E Cell GLn(A) nen pI,n la matr~nvo hadng,di~u naymallthu~nvdip "*0 . V~y, g ~CenGLn(A). . Ta co: g = In+(h-1)IP hn Vdi (h-t)I la cQt 1 cua h-I , hn .la dong n cua h. g22= 1 + (h-1h1.P.hn2 = 1 + (h-1h 1( 1- hn2.x) hn2 = 1 - (h-1h1 (hn2.x.hn2-hn2) = 1 - (h-1h 1.z Vdi Z E Rad(A) => (h-1h 1.ZE Rad(A) => g22 E GL1(A). H va g thoacacdi~uki~ncuatraCinghqp4. B6d~2 dachungminhxong. 2.2.dB6 d~3: Cho H la nhomcancila GL,lA) vachwinhodbiJiE,lA) , n 2 3 . Neu H chaa xU, trongdo x E A, 1s i ;r:j s nva B la ideal2 phia cila A sinhbiJi x thi: H:::J E,lA,B). Chung luinh : Xet tEA, 1~k ;r:I sn, taco : [x ii, t kl] =xii. t kl (- x)ii (- t kl) E H [tkl,xij]=tkIXii(-t)kl .(-X)iiE H Ta chungminh XkI E H , vdi k,l ba'tkl thai! Ta cocactraCingh<jpsail: *k;r:i,j: -z ;r:i,j: l~k;r:t~n -l=i: XkI =(1.x / I =[ 1k i, X it ] X if =( x.I ) i I =[ X i.i, 1i I] E H ::::> Xk I E H XkI =Xki =(x.Oki=[ Xk.i,1j i ] k' k' k'" kl X .f =(1.x)] =[I I, K I]] E H ::::>x E H kl k' k' k'" X =X ] =(1.x) .f =[I I, X I]] E H I ;r:i. Xkl =Xii =(x.I)it =[x ii, Iii] E H Xk I =xii E H -Z=j: * k =i: taco -l;r:j: -l=j: 52 * k =.i" taco l;r .i. . . l;r i .. Xkl =Xj l =(1.x)j l = [ 11i, X it ] X it = (x.1 ) it = [ x i.i, 111] E H ~ Xk I E H. . l = i .. Xk I =Xj i=(1.x)J i =[ 11P,Xpi] ,( chQn p;r i ,j ) X pi = (X.1)pi = [xpJ, 1Ji] X pj = (1.x)p1 = [ 1pi, X i1] E H ~ X piE H ~ XkI E H V~y, trong mQi tru'ongh<;1p,ta CO XkI E H . Xet phfintusinhbfftki cuaEn(A,B) codCJ.ng: y I k II (-y) I k , YEA, Z E B, 1s k ;r l S n B =xA =>Z =xt , tEA. Zkl = (xt)kl = [Xkp, tPI] = XkP. tPI (-X)kp(-t )pl E H, (chQn p ;;tk,l) Soy fa : y I k II (-y) I k E H . V~y: En(A,B) c H. B6 de 3 dachungminhxong. 2.2.e Bjnh Ii 2: Giil sa n;:::3 va AIRad(A) la vanhchinhqui van Neumann.Khid6,mQinh6mcanH cuaGLiA) chwinhodbcJiEiA) d~u la nh6mconnlllc B, nghfala.. EiA,B) c H c G,lA,B),V(ji B lelideal cuaA. Chung minh : GQi H lanhomcancua GLn(A) chugnboabdiEo(A). Voi phfintIt X bfftki thuQCA, gQi Bxla idealcuaA sinhbdi x, taco Bx=xA =Ax. B~t B = {X E A / Eo(A,Bx)c H }, tasethvchi~ncacbu'ocsan: a) ChungmillhB faideal cua A: 0,,2=1 E H =>Eo(A,Bo) c H (bB de 3) .'" V~y: 0 E B =>B;;t 0 GQi x,y la2 phfintubfftki thuQcB, tachungminh : 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: Phfintusinhbfftkl ctla Eo(A,Bx_y)co dCJ.ng: h =tji (( x-y) Z)ij (-t )ji ,voi t, z E' A, i;;tj =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 chungminh \/ x E B, \/y E A, xy E B vayx E B Ta co En(A,Bx) c H, din chung minh En(A,Bxy) c H va 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 =tji [ Xik, (YZ)kj](-t )ji , (ChQn k ,-:f:.i, j ) =tjixik (YZ)kj(-X)ik (_YZ)kj(-t)ji =tiixik (-ti. tji (YZ)kj(-X)ik (_YZ)kj(-t)ji Voi : ~iXik(-t i E En(A,Bx)c H (YZ)kj(_X)i\ _YZ)kjE En(A,Bx)c H~ tji.(YZ)kj(-x)ik(_YZ)kj.(-t ).iiE H Suy fa: h E H En(A,Bxy) C H ChungminhEn(A,Byx)c H hoantoaDtu'dngtl!. V~y,B la ideal2phiacuaA. b) Chu1lg11li1lhEn(A,B) C H : Phfintusinhba'"tkl cua En(A,B)cod?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 D6ngthai: x E Bx~ yjixij(_y)jiE En(A,Bx)c H ... ~hEH V~y: En(A,B)c H c) Chu1lgmi1lhH c GII(A,B): 54 c.l. TntochSt,ta co A 1avanhchlnhquiVallNeumann.nen AIB Ia . vanhchfIlhquivonNeumann.Do do (AIB)/ Rad(A/B) clingIa vanh chinhquivanNeumann. c.2. GQi H'=q>(H), q>(En(A)) =En(A/B). Ta chungminh H' du'Qc chuffnboabdi En(AIB): La"yba"tkl h' E H' , (x)ij E En( AIB ) , ( i:f=j), ta chungminh : (x)ij h'(-x)q E H'. Ta co : h' =q>(h), h E H. (X)ij h' (-X)ij =q>(Xij ).q>(h ).q>«-X)ij) =q> ( (X)ij . h .(-X)ij) h E H ~ (X)ij. h .(-X)ij E H Do do (H) (X)ij .h' (-X)ij E H' V~yH' du'Qchuffnboabdl En(AIB). c.3.Chungminh H' c CenGLn(AIB) : Ta changminhbangphilochung: giastYH' r:r.Cell GLn(AIB). Khi do, AIB va H' thoamancacdiSuki~ncuab6dS 2 DenH' chuamQttransvectionsdca"p(X,)ij khac'matr~nddnvi cua GLn(AIB). Do do t6nt(;liph§nttY x ~B d~: r - 1 0 0 1 (X,)ij = l> 0\ X "' J '... ~. . 1 ( i, j ) Dodo : (x,)ij =q>(h)voi: 55 bl h=. . 0 °1 0 I (Dong i ) bJ bi ... X ... t CQtj Caethanhph~nbi eua h thml: bi -1 E B , V i = 1,..., n f)~t g =(-x)ij.h,taeo : r~1 g = (-xij I. l~ (1 I. - I. -, l~ bi ... X 01 (11 .1 I. I ... -x ... O J I. t t l . °1 °, 1 a (i,i) (i,j) 56 01 . I ... 0 I bn) (i,i) 0) I q ... x ... ~ I bJt I f (i,j) rbl ' I : = I l~ bi ... x-xb.J t t I' r 01 I ... ~ I bJ (i,i) (i,j ) (bl I . I = I. I I . l~ °l b; on x(1- bj) 0 I . I bJ x( 1 - bj ) E B => g E Gn(A,B). D6ngthai: xij.g= h E H X6t [xi.ig,J.ik]=xi.ig.1jkg-1(-x)i.i(-1).ik, (chQn k :;t:i,j) Xii g E H=>g -1(-x)ijE H:::::>l.ikg -1(-X)ij(_l)jk E H =>[xi.ig,ljk]E H Ta clingco : [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 =Xik , Ijk xij [(_1)jk,g ] (-X)ij (-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 suyra : xij[(-l)jk,g](-x)ijE En(A,B) ]jk .xij [(_1)jk, g] (-X)ij (_1)jk E En(A,B) [Xijg, Ijk] E XikEn(A,B)c XikH [Xijg, ljk] =Xik.t , t E H Xik = [ xij g , Ij k] . t -1E H Do b6d~ 3, taco : En(A,Bx) c H => X E B , di~unay mall thua"nvoi X ~ B . V~y H' c CenGI.iA/B) Suyra H c Gn(A,B) Do do E.iA,B) c H c Gn(A,B) Dinh Ii 2 dadlf<;Jcchungminh. D~cbi<;t,ne"uA Ia vanhchlnhqui vanNeumannva n;?:3 , til dinh 11I.d, En(B)=En(A,B) nenco th~suyra : 2.2.fHf:qua: Ne"uA IavanhchinhquivanNeumannva n;?:3,taco H la nhomcancuaGLn(A) chua"nboabdi E,iA) khivachikhi : En(B)c H c Gn(A,B) 58 Ke'tlu(ln: De ma ta cac nhomcan cua nhom'tuy€n Hnht6ngqmH GLn(A) chu§nho~bdi En(A), tad~tdtiQcacke"tquasau: * Ne"un vah~n: H la nhomcan cua GL(A) chuffnboabdi E(A) khivachIkhi t6nt~iduynha'tideal B cua A saocho: E(B) c H c GL(B) Khi do, H cungla nhomcanchu§nt~ccuaGL(A) va k€t qua naydungchovanhA ba'tk1. . * Ne"un hullh~nva n ~3 , d6ngthai A/Rad(A) la vanh chinhqui vonNeumann,taco H la nhomcancuaGLn(A)chuffn boa bdi En(A) khivachIkhi H la nhomcanmilc B, nghlaIa t6nt~iduynha'tidealB cua A saocho: En(A,B) c H c Gn(A,B). * N€u n hull h~n va n,~ 3, A la vanhchinhqui van Neumann,H la nhomcancuaGLn(A)chu~nboa bdi En(A) khi va chI khi t6nt~iduynha'tideal B cua A saocho: En(B) c H c Gn(A,B). 59