nhóm con chuẩn tắc của nhóm tuyến tính tổng quát trên vành chính quy von neumann
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CHUaNG
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4
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2: NHOM CON CHUAN TAC CUA NHOM
K
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TUYEN TINH TONG QUAT TREN VANH
CHINH QUI VON NEUMANN
Vanh A dU<;1c la chinh qui von Neumann
gqi
\j x E A , 3 YEA:
nC'u :
xyx = x
NC'u A la vanh chinh qui von Neumann thl mqi ideal va mqi vanh
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<;1poi don vi 1 ma A/Rad(A) la vanh chinh qui von
v
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 :
NQidung 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<;1c inh nghIa , mo ta cac ph§n ti1'va nhung Hnh chit co ban clla
d
chung nham ph\lc V\lcho ph§n tiC'ptheo.
2.1.a MQt s6 Hoh cha't cua traosvectioo
sd c;1p :
Xet x,y E A , cac transvecsion so cip thoa nhling Hnh chit sau :
l)xij .yij~(X+y)ij,
l~i=l=j~n
2) (Xij rl = (-X)ij
3) [x~i , ykI] = In nC'u j =1= va i =1=
k
l
Sur ra:
..
xl]
.y
kI
xij . ykl
4)
Xik. yjk
=Yk I .xlj..
(- X)ij = ykl
j =1=k va
j =1=k va
i , j , k khac
i , j , k khac
i , j , k khac
= yjk.xik
xki.ykj=ykj.Xki
5) [Xij ,yjk]
= (xy)ik
i =1=l
i =1= l
nhau d6i mQt.
nhau d6i mQt.
nhau doi mQt.
6) VgE GLn(A) , Vy E A, V k, l E {I, 2, ... ,n }
kl -I
'
g. y .g = 1n + gk .y. g I
voi gk la cQt k cua g, g'tla dong l cua g-l.
15
Chung minh :
C~c tinh cha't ta I dSn 5 de dang suy ra ta dinh nghla, ta trlnh bay
chung minh tinh cha't 6 :
Vdi g=(gij),
g-I=(g'ij)
~
-
g e
g - L
i,j=1
k l -1
g.y g
1J 1J
= g ( I n + Yekl
= g.g
EGLn(A),YEA,tac6:
, g-1 - L
- ~
g ' t e t ' yk l - I n + y.ekl
S S
s,t=1
)g
n
-1
.
n
-1
+ c.~ gijeij)' yek[ .
I,J=1
I
g'stesl
s,I=1
n
=
In
n
+ ( .:?: gijyeijekl ) (
1,J=1
n
= In
g'stest)
n
+ (?:gikyeikekl)
(
1=1
= In
I
s,I=1
I
s,I=1
+ (t gikyeil) (~
1=1
s,t-l
g'steSl)
g'stest)
n
= In
+
L
i,s,t=l
n
= In
+
L
i,t=l
n
L
i,t=l
I
gik Yg steilest
'
gik yglteileZt
'
gikyg lteit .
= In
+
= In
,
+ gk.y. g l
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
cac transvection so ca'p bij, b E B .
M~nh d~ 5 : ( xem [1] , 1.2.26
trang 36 )
.Vdi n C:: , i :;z!:va x E A, nhom can ehuctn tde sinh b(JixU
3
j
trang E,lA) La E,l
2.1.c Nhom con Gn(A,B) :
Ki hi~u Gn(A,B) chi anh ngu'<;1c Him GLn(A/B) qua d6ng ca'u
cua
chinh t~c :
q>: GLn(A)
) GLn(A/B)
(aij)
~
(ai)
Ta co : q> la d6ng ca'u nhom .
Nh~n xet : Cell (GLn(A/B)) la nhom con chu~n t~c cua GLn(A/B)
nen Gn(A,B) la nhom con 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 ::: 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 - 2 , ta co :
."
[
O:::O} .
=
anI
( a~j"J
dong
i
t
cQtj
a..
11
=>
= a.. J
J
f ak i = 0
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 = 0 ,. V k ::;c
l
all
g =
[
0
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 do : a E Cen(A)*
M~nh df 6 da chang minh xong.
Ap dl,mg m~nh de 6 cho tam
)
.
= 1 =>
a kha nghich trong A
(GLn(AIB)), ta co :
1VI~nhd~ 7 :
gECenGL,lA/B)<:::::>
g=a.I
Trang d6 :
ra E (Cen(A/B)) *
-
,
0
11
')
I
I=lod
Til do ta co thS mo ta Gn(A,B) bc3im~nh de saD :
M~nh d~ 8 :
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 con chu§'n t~c cua En(A) sinh bdi cac
transvecsion sd dip xi.j trong 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 ~ 3 , 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,
1 . 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
.k
*-j va I *-i :
[ yk l, xij ]
.k
= In ~ h = Xij
En(B) c E
E
*-j va I = i:
= (YX)kj~
[yki , xij ]
h=
(YX)kj :Xij
x E B ~ yx E B ~ (yx)kj . xij
. k =j
h
h
~
h
E
E.
va I *- i:
=yjl
= xij (-X)ijyjl
xij (_y)jl
= xij.
. k =j
E
E
[(-x)ij, yjl]
xij (_y)jl
= xij.(-xy)il
EE
va I = i:
= y.ii Xij (- Y)ji
E E
V~y h E E trong mQi tntC1ngh
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~n
hoan vi p tlidng ung 1 phep the" (j
E
Sn co dC;lng: p
= (8j ,cr (j))
Ta co : p -1 = «>i,o--I(j))'
Th~t v~y , d~t q = ( <>i,o--I(j)),
ajj
= 1;1
i ,0-(0- -}(j)'
.
k
=
(j
-IU)
.
<> ,0-(k)
i
= (aij)
p.q
k ,CT -I (j)
0--1 U) ,0--1(j) ,
( so' hC;lng
khac 0 duy nha't ung voi
).
V~y : ajj
Suy fa:
= bij
p.q
, '\! i,j
= In
Xet : p .Xkt.P -1
-1
~
=
p
= ( <>i ,0--1(j)
'
)
In + (8 i,O"U»)k ,X.(
(CQt k)
i,CT -I (j) )t
(dong
t)
81,0- (k)
- I
= In+
X
(8
t,o-
-I
(1),...,8
t,o-
(n))
80 ,0-(k)
0
=
+
In
x ( 0 ... 1 ... 0)
!rI
dong cr(k)
CQt cr -I (t)
0
0
I
= 10 +
I
.'rIL~£!!!L(j(i<:)
I
0......
-fO
cQ't cr -let)
= X er(k), er-
I (I)
M~nh d~ 10: vdi n ~ 3 :
En(A,B) dli(/Cchu6n hod biJi tqp cdc lna trqn hodn vi.
Chung minh .
X6t ph~n tit sinh cua En(A,B) dc;wg :
kt
y.x.-y
ij
GQi p = (8 k ,er())
I
kt
P y.X
ij
kt
-1-
(
kt
..
) ,XE B ,YE A ,1*-J, k *-t
la ma tr~n hoan vi , ta co :
kt
-I
ij
-I
kt
-y ) P - P . Y .p .p.X P .p.( -y ) p
C
=yo-
(k) 0- -let)
.xo- (i)o- -I
-I
(j) ,(-y)o-
(k)o- -let)
EEnCA,B).
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.
2.2.a Dinh If 1 :
Gid sa A/Rad(A) la vanh ehinh qui van Neumann, n 22 va B la
ideal eua A, fa e6 ..
(a) EnCA,B) ehua ede ma lr(in d(lng In + vu , lrong d6 v la n- d)1
frong A, u la n - dong trong B va u v = 0 .
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 .
Chung minh
1.
a :
(v )
v=(Vj)=
U=(Uj)=(U',
, ujEB,j=l,...,n
un)
Truong h(jp 1:
f)~t : d
,VjEA,i=l,...,n
lvJ
1 + Vn Un E GL1(B)
= 1 + Vn Un
d' = 1 + UnVn => d' - 1
UV
=0
II
=>
"Uovo
L..,
1
E
B
=0
1
I
=> u'v'
=> d'
+ UnVn
=0
= 1 - u'v'
Taco:
( 1 - Und-1vn ) d'=
( 1 - Und-I Vn) ( 1 + UnVn)
= 1 + Un Vn - Un d-I Vn - Und-I (Vn Un ) Vn
= 1 + UnVn - Und-l Vn - UIId-I ( d -1 )Vn
-I
-I
=. 1 + Un Vn - Un d Vn - Un VII + Un d VII
= 1
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
=
1+
-I
-1
UnVn - Und Vn - UnVn + Und VII
=1
V~y ( 1 - und-I vn)d'
Do d6 d'
Khi d6 :
E
~ d'( 1 - und-I Vn)= 1 , Suy fa d' kha nghich.
GI1(B) va (d' rl = 1 - 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 - 1 + v u - V un d VnU
= 111- 1 + v'(1
=111-1
- Un d-1 Vn) U'
+ v'(d'ylu'
0
V,~"d')
Ta ch1?ng minh : In + vu ~ (l"~
XetO v€ phai
0
VIUO
d)
l
l
)
d
v 0 u'
v
( 10-1
.
ld-Ivo
d
0
-I
v
0
u
v'u
d
0
+,V'U'
0
01
I
1
OJ
v>J
U
V~y : In +vu
1)
u'
( ln--I
0
10-1
=( o
V';"d-')
(~
d
)
Voi :
'
n-I
(1
o
V Un d
1
ld-Ivn H' 1)
01
1)
(lool
(a +V'U d-Iv u'
=
(10-1
V.~"d-') (~
= (a 0
d)
:
10-1
g:= ( o
(~
(10-1
-I
(1n-I
)
'
l
d-Iv 0 u'
a
In + vu E En(A,B)
01
(o
)
1
E E'
B
n( )
0
d) En(A,B).
l
d -I V n U I
01
d
01
a
Ta c~n chung minh : 0
.
(
~)
E
En(A ,B)
111-1
0
Tnloc he't, ta phan tich :
-
(~UI
~) = (~"
yldl-I
) ( - u'
1
1)
0
,
d,-J
111_1
j
:') (~"'
(0
Tht vy , xet vS phcH:
h := ("'
) (n-J
1
-- V'd") C u'
("-'
')
d'--J)
y
=
.
- v'r'l
1
0
Co-'
)
C n-J
:- U
I
n (h
d
,
y' d 1-\
=
("-'
(1
In-I u I
- Y'd'-I,
=
- vor')
1
('"
(1
1
)
In-I
u'
(1
l
=
-Vt'd"'J
-u
e,,-:
=
-
1+V1dl-IUI
(-
(-:'
U I
~J
~)
nU I
- I
,
1-\
(-llIV'+l)dl')
y Id
1
-V'd'.')
=
- u'yldl-J+d,-1)
v'd
= ("-'
('"
'
0
n-I
d '. d'
v ' dJ ,-
'J
1- \
,
J)
I
d
1- I
,..1,
)
V~y :
1n-I
-Vldl-I
C:I
OJ
~
) (-
1
(~"'
Ta cung co:
Suy fa :
d,-1 -
= d'
(1 + u 'y 'd' -1). d'
-V'dl-I
1
~I'-'
J
=1
+ U 'y , = 1 - u 'y , + U 'y ,
1 + U'y' d,-l
1n-I
-
(0
- v' (1 + "; y' d ,-I )J
= (~"-'
(~"'
n (~"'
~) (~n-I
u'
- V'-V't:'V'd'-']
J
-
-
1n-I
(0
-
-
111-1
VIUIVldl_l
-
111_1
VIU
- (0
-
v;]
] (0
1
1
'Vld'-I)
j
1
.
: ')'
( 0"-1
Nen:
--!
a
(-u'
VI
J= (~~'
-V'U~V'd")
1)
(~"
111_1
(
111-1,
- U
0
I
) (0
V
1
.
I
In-1
) (0
Vdi:
1n-I
- VIUIV'd'-I
(~"'
1
J '
(-H'
0
1)
EE
In-I
(B )
VI
(0
n,
-I
In-1
Suy fa:
(0
VI
1
In-I
)
(-H'
.
~) (~n-I
~')
E
En(A,B)
1)
E
En(A)
0
d,-I J
V~y :
0
1 E En(A,B)
)
(-:'
0
10-1
(0
')
d 1-.1)
Nen:
O
(~
~)
J(~I
=C:
Ta cling co : (~
~)
10-1
(0
1 E En(A,B)
J
= (~
~) (~"
0
d'-I
10-1
J
( u'
~J
(:)
Nen:
0
10-1
d ). E En(A,B).
(~
0
(0
1n--1
d,-I J
( u'
OJ (~"'
~J
O)(
(
d
0) EEn(A,B)
0
(1,_,
d,-I
d
OJ (O,
d'
0
0 ) (I",
0
d'-I) (1,
on-')
Ta co:
("-I
d,-I
0)
u'
(1"_-
OJ ("-'
d'
0
0 ) = (1"-J
,,)
(:t
ad')
In-1
( (1'--1
0
(d"
d'-I)
Suy fa: (~
Tlido
') 1
;
(
l1'
~) E En (B)
.
0 ')
1
~)
= (~"-'
~) E En(A,B).
: ]0 + vu E En(A,B)
.
d'-l d) E En(A,B) , ( Ke't qua tU [5], ~2)
J
Truong hqp 2: ::3i E { 1, 2 , ... , n} : ViE Rad(A).
Ntu i =n :
Vn E Rad(A)
=>
=> 1 +
VnUn E Rad(A)
VnUn
kha nghich trong A
Soy ra : 1 + VnUnE GL1(B) => In + V U EEn(A,B) (tnf
Ntu
i < n:
xet phep
the- (j
=(
n)
i
Va ma tr~n hO
= ( 0 j ,0 (k) ).
p
£)~t :
(VI)
(OI.a(l)Vl +OI.a(2)V,
I
°
v
I
O'2,u (1) V 1
+ ...+OI,a(,,)V"
+ 52 ,crC V 2 + ... + O'2,cr) V n
2)
C
n
I
I
I
.
I
I
.
I
I
I
I
= pv =
=
I
V II
I.
l~","(l)Vl +O","CZ)VZ+ ...+o"."c,,)V,,)
I
I
l~J
( ma tr~n co du'Qcb~ng cach hoan vi cac dong' i & n cua v)
=
UO U P
= ( UIOlp(l) +:..+ullb;l,a(l)'Ul~,a(2) +"'+Ullb;1,O"(2)' U1O1,acn)
...,
+...+Unb;],acn))
= (U1
,. . . , Un ,...
Ui
)
( ma tr~n co du'<;1c
b~ng cach hoan vi cac cOt i & n cua u)
Ta co: uOvO= up.pv
vnO
= Vi E
In + v u
= u(p.p)v = 0
Rad(A)
=p.p-l +
=> In + va UOE En(A,B)
p.pv.up .p-I = P On + va DO)p-l E En(A,B)
Truong hqp tang quat:
Voi A / Rad(A) la vanh chinh qui vorl Neumann, ta co :
-
-
::3xEA:vn.x'Vn=Vn
vn - Vn . X . Vn E Rad (A)
(vn - Vn . X . Vn ). Un E Rad (A)
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:
t = (- Vn-l.xt-1,n ( In + V X VnU) ( Vn-l:Xt-1,n
.
= In
+ ( - Vn-l.X ) n-l,n . ( V X VnU )(
.
=. In
+
D~t
U1
1
V
= (-
V
Vn-l.X )
U
n-l,n
. ( Vn-I.X)
n-l,n
;.
(
V2
=I
l
l
.
v
n-I
V
-.
.V X Vn - U v X Vn -
I
I
U
I XVn
XV
V nn-I
n
J
= V n-l
( 1 - X. Vn )
= Vn-l
VI
n-l,n
VI
I
n- cQt trong A )
II V21
. I X.Vn
-V"l'.xJ
~.
n- dong trong B )
'\ (VI'\
.
.
l~
(U(Vn-l.xt-1,n)
(]a
( - Vn-l.X)
0
= I'
V \1-1
X Vn )
'.V
.V X Vn
1
\0
VI
n-I n
n-l,n
(la
0 ...
(1
VI
Vn-l.X )
= U .( Vn-l.xt-I,n
1 1-
U
(-
Vn-l.X )
( XVn
Vn
-
X Vn
- X . Vn X Vn) =
Vn X Vn E Rad(A)
lien
Vn-I X
(
Vn
-
Vn X Vn)
vln-I E Rad(A),
ta SHY ra :
0
t
= In + Ul
VI
E
h - (Vn-1.X )n-l,n t ( g.h
= (1n + v(1
(TnfOnghQp2 voi i = n-I)
En(A,B)
n-l,n
Vn-l.X )
E
En,
(A B)
- X Vn ) U ) .( In + V X Vn U )
= In + v(1 - X Vn) U + V X Vn U + v(l - X Vn) U . V X Vn U
= In + v(1 - X Vn)
= In +
U
+
V X Vn U
vu - V X Vn U + V X Vn U =.In + vu
g , h E En(A,B)
=> In + VU E En(A,B)
Ph§n tie'ptheo , 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),
i 7=j , ta co : X E B.
En (A,B) du'Qcchu§'n boa bdi t~p cac ma tr~n hoan vi Hen co th~ ghi
sU' (i, j ) = (1 , n ). Ta co :
g xl,n g-l = In + v X w
(VI'
I v21
voi
V
=I. I
Ia
cQt 1 Clla g .
lvJ
W =
( WI
,
'
... W n ) 1a d ong n cua g .
?-1
W2
WVla ph~n ti'ta dong n , cQt 1 clla
£>~t
-1
g g
u = XW , ta co Ui E B ,'\1i = 1,2, ..., n.
UV= (xw)v = X (wv ) = x. 0 = 0
Suy fa : g.x1,n.g-l = In + vu
E En(A,B)
= 1n
"
Den
WV
=0
2. Tie'p then , ta xet cac ph§.n ti't g E GIn(A), h
= yi i 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)
n-l
3. Cu6i cling, xet
vdi
h=
gE Gln(A) , h E En(A,B) ta co :
hi la ph§.n tit sinh d<;1ng y-iixij (-yi
Il
hi ,
i=l
cua En(A,B).
n-l
g h g-1 = g (
Il
i=1
hi) g-l
n-l
=TI
i=1
g hi g -I E En(A,B)
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)
. .
..
..
I
[11,.I,g]
=11,.I.g(-I)l,.I.g-
,\/ g E Gn(A,B).
En(A,B) c1t(Qcchu~n boa bdi t~p cac ma lr~n hoan vi Den co the
gia siT (i, j ) = ( 1 , n ), khi do :
[11,Il,g]=
11,Il.g.(-1)I,n.g-I
= 11,
=
(
:J
n
( 1n - VW )
H\CQt 1 cua g,
= ( W'
ydi
y
w.v
la ph§.ntud()ng n , cQt 1 cua g-l:g = In
A/Rad(A)
W
la vanh chinh qui van Neumann
W,,) Iii di,
~
ng n cua g
wv
~
I
= O.
lien :
.3 X E A : Vn X Vn - Vn E Rad(A)
Ta chung minh
Ta co:
(xvn )I,n va g giao hoan theo modulo En(A,B).
h:= In - vxvnW E En(A,B)
(Tntong hQp t6ng qUCtt iu (a), ap dvng cho u =
d
-
w)
gE Gn(A,B) va Vn la ph§.n tiTthuQc d()ng n , cOt 1 cua g nen VIlEB
Do do: X'Vn E B
(Xvn) I.il E En(B)
[ (xvn )1.0, g ]
= (xvn )1.n (1n - VXVnW)
E En(A,B)
Tiep theo ta chung minh (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
= - vn(1
VllUn
Suy fa:
£)~t :
- XVn) Wn
= (Vn
- Vn)W
X Vn
E Rad(A)
d := 1 + VnUn E GLI(B)
",
-I
,
,a .- 1n-1+ v U - V Un d VnU
Chung minh tu'dng tv Call la , ta co :
(1 -XVn )
1,n
(1 n + vu ) = (1 -XVn )
1,n
(
(1
(l-xv,,)
0
.
-
lo .
- -
,
.
-
(1,..
I I'OJ..,
II I -
-
1
I
)
1..
v2u"d-I
l'O.: :. :. . . 1 l:"d',
; <,
g
E
Gn(A,B) => V2, v3 ,
=>V2Ull
, D6ng thai,
g
=
d
-I
,
,v3un
Vn - 1 E
-I
d
B
,...,vn-IUn
.
d
ol
xet trong GLI1(A/B) va GLj(A/B),
-
a
.',
,
I
; g-l
=
f3
E,
I
j
B
ta co :
,
'
-. , . vur,1
v,u"d'
.
,. ,,(
I
I
.1
"n
= 10
d
-
vdi
a
II
I
0
1
l-- . . . . vn-Illnd,1)
vlud-'+I--xv)
I
0)
(1
°
.
(1,,1
d) ld--'vn u'
) (0
1
. . . I-xv,,)
V'U"do') 1°'
I
=1--
I,n (1"
0
a
V'l lnd-Ij
In-I
,
f3
E Cen(A/B)*
.
g
Vn E
u
.
g
-
=
-I
-
I
=>
=
B => xvn
= - (1 - xvn)
a
.
.
1
=>
=
-
1
0
-
=
d-I
Wn
VI
=> un = (1 - XVn) WII =>
W
.
U"
-
-
d E GL1(B) => d-l E GL1(B) =>
VI
=
B
Wn
-
~
==
.( - w "
VI
=-
U"
). I
1
=-
-
1
Nen : vlund -1+ 1 E B
vlund -1+ 1 - XVn E B
1n
(1-XVn)
In-I
(
,
0
V'~"d-')
E
(a
In
(1-XVn)
,
E,,(B)
(1n + Vu) E En(B) \0
0
d) En(B)
.
0
Ta co:
d) E En(A,B)
(~
[( 1 - xvn)l,n, g]
= (1-xvn)l,n
(Chang
minh 1a )
(1n+ vu) E En(A,B)
Tom l~i , ta co [(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,n g (-1 + XVn)1,n (- XVn)1,11
g-I
= (xvn)l,n. (1 - xvn)l,n g (-1 + xvn)l,n g-1 . g (-x.vlI)l.n g-1
= (Xvn)I,n. [(1-xvn)l,n,
g ](-xvll)I,II(xvn)I,n.g.(-
xvn)l,n g-l
= (xvn )1,n[(1- XVn)I,n ,g] (- XVIII,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.
[yi.j,g]=Zkll1.n(_Z)kl g
.
= Zkl.
11,n .
£)~t h = (- Z)kl .g.
E
(- Z)kl g
(_I)I,n
Zkl
(-Z)kl
g-1
. (-1 )l.n . (- Z)kl g-1 Zkl .(- Z)kl
Zk l
g
Zkl
E
Gn(A,B) => h
Gn(A,B)
Tli chung minh ph~n (a), ta co : [11,n,h] E En(A,B).
[yi,j,g]=
Zkl
11,nh(-1)1,n
= Zkl
[11,n,h]
h-1 (-Z)kl
(-Z)kl
EEn(A,B)
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
Ma:
[En(A),
. h-1
E
t En(A) , Gn(A,B) ]
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
V~y H chu~n boa bdi En(A).
Chung minh loc :
Chung minh En(A,ll)
c [En(A),
En(ll) ] :
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 )i j
= [yji,xij].[1ik,xkj]
Sur fa yji xij (_y)ji E [En(A),
En(A,B) c [En(A),
(ChQnk-:(::i,j)
En(B)]
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)
En(A,B) la nhom con chu§'n t~c cua GLII(A),( chang minh la) nen :
g h g-l E En(A,B) ~
Do do : [ GLII(A), En(A,B) ]
[ g , h ] = g h g-l.h-1 E 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't thue ph~n Ie, ta ehu'ng minh En(A,H)
= [ En(A),
H) :
Ta co : 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 ]
D6ng thoi :
[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 da co: En(B) c En(A,B) , chI c~n chung minh :
En(A,B)
c En(B)
X6t ph~n tti sinh bat ki cua En(A,B)
YE A, 1 ~ i =I: ~ n ) . Vi
j
q<;lng (- yi i) Zij yi i, ( ZE B ,
En(B) dl«;1C
chu~n hoa hdi t?P cac ma tr?n .
hoan vi, ta co th€ giil sU' (i, j ) = (1 , 2). Ta chang minh
h:=(_y)12z21y12EEn(A,B),
(zEB,YEA)
:
Vi A chinh qui van Neumann nen t6n t~i phfin tU' x
E
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.
( a 1
1
=
I
I
a12b21 (-a)12
I
I
b (1 -a 0 ...
In +
0)
l )
I
(CQt 2 cua a12)
(ab
= In +
I
I
I
l
b
0
-aba
-ba
(D()ng
0
0
...
...
0
01
0\
.\
0
Ap dl,lng cong thlic tren cho a
1 cua (-a)12
)
= (xz-I)y
E
B,b
ab
=
ba
= z (xz - 1) y = (zxz)y - zy = zy - zy = 0
-aha
(xz-l)y.z
= -a (ba) = 0
Ta co: g
= a12b21
(-a)12
=z
)
A thoa :
0
0
...
z
0
0
... 0 I
0
0
.
.
0
0
(xz - l)yz
0\
I
I
g
= In +
I
l
=
~j
0
z
0
0
...
1
(1 + (xz - 1)yz
I
'1
0
...
0
0
\
I
I
I
l
0
Ij
0
Ta co th€ pMin tich g thanh d<;lng:
g
=rl~
l0
. (xz-\)yzx
= «xz
0
01 (I
".
0
I
:..::::::::.. O I~
: I
J ~O
~
01 (I
:::::::.o II~
(I-xz)yzx 0 ... 01
'
~
'.'~~~.'..'.
O
1J ~O ..."
1 J
-1)yzx )12. Z21.« 1 - xz )yzx )12 E En(B)
Th~t v~y , xet :
(1
t :=
~
I
(xz-l)yzx
~
0 ... 0\ (1
0
o I! ~
1
::::::::..
I0
(1+(xz-l)yzxz
=
I
~
I
1
0\ (1
(l-xz)yzx 0 ... 0 \
l
::::::::.
()
I
~~""""'~"""":"'::::::~:"~"
I J l0
J ~O
(xz-l)yzx
...
0\ (1
1.
...::::::... O
1
J
(I-xz)yzx 0 ... 0\
I
I
~
1 J ~o '
l"."".""::::::::"~
I.
J
. .
(1+(xz-l)yz
=
z
I
...
(xz-l)yzx
.
01 (1
1...
0 110
j
l0
I
1
0I
1
)
1 ~O
(1 + (xz - l)yz (l + (xz -l)yz)(l-
=
\
(l-xi)yzx 0 ... 01
z
xz)yzx + (xz -l)yzx
01
z(l- xz)yzx+ 1
0
,
I
j
l0
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-]
)yz(xz-] )yzx
= (X7,-])(yzxz
= (xz-l
- ] ) (xz-l)yzx
- yz) yzx
)(yz - yz) yzx
=0
/.
Do do : g
= t = «xz -1)yzx)
12
21
. z . « I - xz )yzx)
J2
E En(B)
Suy fa h = (- xzy)12 g (XZy)12 E En(B)
V~y En(A,B) c 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)
Nen
En(B) c [En(B),En(B)]
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 0 . H La nhom con
dla
cua GLn(A)
= (gij)
g
ij 1 <.
- l
a,
chwin hod biJi tqp cac aU, i :;rj
thod
:;rJ
gn]
=0
va t6n tc;zii, j
thod tlnh chat H chaa
sao cho g kh6ng giao hoan vcJi
'< n,
-
Khido , H chaa mQt transvection sa cap khac ill'
Chung minh :
n-I
Il
Tru'ong hqp 1: g = i=1b;,n voi bi E A va t6n t~i j < n thoa bj -:f= .
()
ChQns6 nguyen du'dng k thai: k < n va k -:f= ta co :
j,
-I
kj
kj
kj
- kj
kj
a .g. ( - a ) E.H => [ a, g ] - a .g.( - a ) .g E H
.
n-I,n
n-I,n
kj
j,n
- kj I,n
kj
( -a )(b- n-I)
[ a , g ] - a b 1 ... bj ... b n-I
... (b) j
-
N Su j
=n
j.n
... (b) I
-
I,n
- 1 , ta co :
[akj, g]
=akj
bIl,n
... bjj,n
(-a)kj(-bj)j,n...
(- bI) I,n
NSu j < n -1 :
bn-In-I,n (-akj).(- bn-dn-I,n = (-a)kj [aki, bll_tll-I,n] (-a )ki
=
Dodo:
I,n
n-2,n
n-2,n
kj
kj
j,n
(- .
)
[ a , g ] = a kj b I "'.1b .j,n ... b n-2 (- a ) (-b n-2
"'.1'" b )
= akjblI,n ... bjj,n (-a)kj(-bj)j,n...
( - b t)
(- bt) I,ll
V~y :
[akj, g] = akj blI,n ... bjj,n (-a)kj(-bj)j,n... (- bI) I,n, V j S; n - 1
kj
.
= a kj.b I,n "'.1-1. j-I,n .(- a) kj .a,.1.' b .j,n ] (-b.1-1) j-J,n ... ( _b)I
b
[
.
.
.
= akj b I,n ... b.I-Ij-I,n (-a)kj (ab .1' k,n ( - b.I-I ) j-I,n ... ( - b 1) I,n
)
= kj b I,n b. j-I,n (-a) kj ( -b . ) j-I,n ( -b ) I,n (ab .) k,n
a
[akj, g]=
... .1-1
.
.1-1
...
I
.
I,n
.1
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
V~y H chlia (abj )k,n-:f=In
= akj
(-a)ki. (abj
/,n
= (abj
)k,n
I,n
