Äîêóìåíò âçÿò èç êýøà ïîèñêîâîé ìàøèíû. Àäðåñ îðèãèíàëüíîãî äîêóìåíòà : http://halgebra.math.msu.su/wiki/lib/exe/fetch.php/staff:bunina:aut_chevalley_rus.pdf Äàòà èçìåíåíèÿ: Wed Feb 13 11:26:37 2013 Äàòà èíäåêñèðîâàíèÿ: Sun Apr 10 00:14:40 2016 Êîäèðîâêà:

Al , Dl , El 1/21
. .
. , Al , Dl El GL (V ) (V ). G ad ­, Al (l 2), Dl (l 4), El (l = 6, 7, 8); G ad (, R) G ad R; E ad (, R) G ad (, R), R . E ad (, R) 1/2. . [55] . [44] . , ­ [25], ­ [28], [29]­[33]. . [1] , ( 2 ), [1] , . , 11 , ad (x )2 = 0 , . E8 , , ad (x )2 = 0 , E8 . , , 1/2. 1 MK-2530.2008.1 08-01-00693.
1


2

. . [14]. , Al , Dl , El , Al . [63], .. [13], -- [47], . .. .. [8] Cl , . .. , .. , .. , .. , , . 1. . , Al (l 2), Dl (l 4), El (l = 6, 7, 8), , () + (- ), W . , . |+ | = m. [45], [26]. , L H ( [45]). L L = H L ,
=0

L := {x L | [h, x] = (h)x h H}, L = 0, dim L = 1, H , L = 0, . L C ( ) . L (x, y ) = tr ( ad x ad y ), H. , H H . {h1 , . . . , hl } H x L , {hi ; x } L . . (., , [54]). L ( C) , : L gl(V ) ( n). H L, H , v V ( ) , h H (h)v = (h)v . V (x )k /k ! k N () . . R . n â n R, (x )k /k ! , k N Mn (R).


3

Rn exp(tx ) = x (t) = 1 + t (x ) + t2 (x )2 /2 + · · · + tk (x )k /k ! + . . . (x ) , . x (t) . Aut (Rn ), x (t), , t R, (: E (, R)). : w (t) = x (t)x- (-t-1 )x (t), , t R ; h (t) = w (t)w (1)-1 ; N w (t), , t R ; H h (t), , t R ; x (t) [27], [62]. , N H , N/H W (). ( ) ( , Z- C- H ), . , . , , R . : , , ( ad ) , ( sc ). ad sc . , . . : (R1) t, u R x (t)x (u) = x (t + u); (R2) , t, u R + = 0 [x (t), x (u)] = x (t)x (u)x (-t)x (-u) = x
i+j

(cij ti uj ),

i, j , i + j , ; cij , t u, . [x (t), x (u)] = x+ (±tu). (R3) (R4) (R5) (R6) , , , w = w (1); - t R w h (t)w 1 = hw ( ) (t); - t R w x (t)w 1 = xw ( ) (ct), c = c(, ) = ±1; t R u R h (t)x (u)h (t)-1 = x (t , u).


4

X , x (t) t R. E ad (, R). . : R R R. x (x) E ad (, R) E ad (, R), E (, R). , t R x (t) x ((t)). -. V E ad (, R), C GL (V ) , : CE
ad

(, R)C

-1

=E

ad

(P hi, R).

x C xC -1 E (, R) , i - E (R), C GL (V ). 1. E ad (, R) Al (l 2), Dl (l 4), El (l = 6, 7, 8), R 1/2. E ad (, R) -. 1. 2. . , R c 1/2, , . [14]. J () R, k R/J . EJ = E ad (, R, J ) E ad (, R) (. [18]). , EJ . , : E ad (, R) E ad (, R) :E
ad

(, R)/EJ = E

ad

(, k ) E

ad

(, k ).

E ad (, k ) , , (. [54]), . . = ig , g N (E
ad

(, k )),

, k . , g GL n (R) , R J g . , g N (E ad (, R)). = ig-1 . E ad (, R) GL n (R) GL n (R), , R J .


5

1. A E ad (, R) R R, , A · GL n (R, J ) = {B GL n (R) | A - B Mn (J )}. a E ad (, R), a2 = 1. e = 1 (1 + a) 2 Mn (R). e R- V Rn : = V = eV (1 - e)V = V0 V1 ( V0 , V1 , [49]). V = V 0 V 1 k - ( ) k n a, e = 1 (1 + a). V= 2 2. ( ) V 0 , V 1 V0 , V J . . . V0 = {x V |ex = x}, 1 (1 + a(x)) = 1 (1 + a(x)) = e(x). V0 V 2 2 x = x0 + x1 , x0 V0 , x1 V1 . x = x0 .
1

V0 , V1 J V0 , V1 , V1 = {x V |ex = 0}, e(x) = 1 (1 + a)(x) = 2 , V1 V 1 . 0 e(x) = e(x0 ) + e(x1 ) = x0 . x V0 ,

b = (a). b2 = 1 b a J . V = V0 V1 b. 3. , a, b E (, R), a2 = b2 = 1, a R R, , b a J , V a, V = V0 V1 V dim V0 = dim V0 , dim V1 = dim V1 .

. R- V {e1 , . . . , en } , {e1 , . . . , ek } V0 , {ek+1 , . . . , en } V1 . ,
n n

aei = aei = (
j =1

aij ej ) =
j =1

aij ej .

k - () V V = V 0 V 1 , V = V 0 V 1 a b. , V 0 = V 0 , V 1 = V 1 . , 2 V0 V0 , V1 V1 J . {f1 , . . . , fk } V0 , {fk+1 , . . . , fn } V1 , f i = ei , i = 1, . . . , n. {e1 , . . . , en } {f1 , . . . , fn } ( J ), {f1 , . . . , fn } R- R- V . , {f1 , . . . , fk } R- V0 , {vk+1 , . . . , vn } V1 .


6

3. w

i

E = E ad (, R) Al (l 2), Dl (l 4), E6 , E7 E8 , GL n (R) (n = l + 2m, m ), v1 = x1 , v-1 = x-1 , . . . , vn = xn , v-n = x-n , V1 = h1 , . . . , Vl = hl , . , 2. h1 (-1), . . . , hl (-1) . hi (-1) = diag [±1, . . . , ±1, 1, . . . , 1],
l

(2j - 1)- (2j )- -1 , i , j = -1. , i hi (-1)2 = 1. 3 , hi = (hi (-1)) ±1 , 1 -1 hi (-1). hi , , hi , hi (-1) . , g1 . , g1 GL n (R, J ). 1 = i-1 . g1 E GL n (R) , R J , 1 (hi (-1)) = hi (-1) i = 1, . . . , l. 1 . hk (a) = diag [a1 , 1/a1 , a2 , 1/a2 , . . . , am , 1/am , 1, . . . , 1] hi (-1), , 1 hi (-1). , C1 0 0 C2 . . . . . . 00 00 . . . . . . .. .. . . 00 00 . . . .. . . . Cn 0 .0C , Ci (ai ) 0 0 (1/ai ) mod J, C GL l (R, J ).

wi = wi (1) () , - . 1 (wi ) wi mod J , 1 (wi ) , wi . , , , e. W , i w(i ) W , w(i ) 1 = i . e1 , . . . , e2m , e2m+1 , . . . , e2m+l , e1 = e, ei = 1 (w(i ) )e; 2m < i 2m + 1 ei . , 1 J . , .


7

, 1 (wi ) (i = 1, . . . , l) {e1 , . . . , e2m } wi . hi (-1) wi , . , , 1 (wi ) - 2m l . , , l , . wi 1 (wi ) wi 1 (wi ), . , -1 . V = V0i V1i 1 (wi ). 1. 1 (wi ) 1 (wj ), i = j , , V1i V0j V1j V0i . . 1 (wi ) 1 (wj ) , ( ) V1i 1 (wj ) ( ) V V1i V1j , , V0i = V1i V0j ,
j 1

1 (wi ). , V1i V1j , V1i V0j . V1i = V1j . V0i 1 (wj ), V0i V0j , V0j , , , 1 (wi ) = 1 (wj ) . , , , V1j V0i .

2. V , 1 (w1 ) , w1 , .. -1 1 0 0 1 0 . 0 0 El-2 . w1 , V11 1, {e1 , e2 , . . . , el }, 1 (w1 ) diag [-1, 1, . . . , 1]. {e1 , e2 - 1/2e1 , e3 , . . . , el } 1 (w1 ) . 3. A2 , 1 (w1 ) 1 (w2 ) , w1 w2 , .. -1 1 01 . . 2 V , 1 (w1 ) , w1 . 1 (w2 ) ab . cd c (1 - c)/2 0 1 10 , 1 -1


8

( , c 1 mod J ). 1 (w1 ) , 1 (w2 ) ab 1d .

, a + d = 0, a 2 + b = 1. , d = -a . -1 1 01 a b 1 -a
2

=

a b 1 -a

-1 1 . 01

( , ) 1 - 2a = -1, , a = 1. a 2 + b = 1 b = 0. 4. = A2 V , 1 (w1 ) 1 (w2 ) , w1 w2 , . . V 0 l - 3. , , 1 (w1 ) 1 (w2 ) . 0 El-3 , 2 , 1 (w1 ) , w1 . V , . a1 a2 a3 1 (w1 ) = b1 b2 b3 . c1 c2 c3 b1 (1 - b1 )/2 0 0 1 0 , 0 0 1 1 (w1 ), 1 (w2 ) a1 a2 1 (w1 ) = 1 b2 c1 c2 a3 b3 . c3
1 0

V

2 0

, . 1 (w2 )2 - 2 1 = 0 (. 1), (1 (w1 )1 (w2 )) - 1 (w2 )1 (w1 ) = 0 (. 2). 1 2, ( 2, 1) a1 = 1, ( 2, 2) a2 = 0, . 1, 1, 3, a3 (1 + c3 ) = 0. c3 1 mod J , a3 = 0. ( 2, 3) b3 (b2 + c3 ) = 0, b3 R , c3 = -b2 .


9

, 100 0 1 0 , 0 -c1 1 1 (w1 ), 1 (w2 ) 10 0 1 (w1 ) = 1 b2 b3 . 0 c2 -b2

1 b2 = -1, c2 = 0, diag [1, 1, b3 ] 1 (w1 ) 1 (w2 ). 5. D4 1 (w4 ) , w1 , 1 0 00 10 0 -1 1 0 0 0 1 0 0 1 - 1 1 1 0 1 0 0 0 1 0 , 0 0 1 0 , 0 1 - 1 00 0 0 001 0 0 01 . , w2 , w3 , w4 10 0 0 1 0 0 0 1 1 00
1

(w1 ), 1 (w2 ), 1 (w3 ) , .. 00 0 0 , 1 0 1 -1

. , 1 (w1 ), 1 (w3 ), 1 (w4 ) , w1 , w3 , w4 . , w1 , w3 , w4 , , 1 (w1 ), 1 (w3 ), 1 (w4 ) diag [-1, 1, 1, 1], diag [1, 1, -1, 1], diag [1, 1, 1, -1], . , 1 -1/2 0 0 0 1 0 0 0 -1/2 1 0 , 0 -1/2 0 1 . 2 : w2 = 1 (. 1), (w1 w2 )2 = 2 (w4 w2 ) = w2 w4 (. 4). a1 b1 1 (w2 ) = c1 d1 1 (w2 ). w2 w1 (. 2), (w3 w2 )2 = w2 w3 (. 3), a2 b2 c2 d2 a3 b3 c3 d3 a4 b4 . c4 d4

1 0 0 0 0 1 0 0 0 0 1 0 , d1 b 0 2d1 -b1 0 b1 -1 d1 2


10

1 (w1 ), 1 (w3 ), 1 a1 b1 c1 0

(w4 ), 1 (w2 ) a2 a3 a4 b2 b3 b4 c2 c3 c4 d2 d3 d4 2 d2 = d3 = 0, d4 = 1. b4 (b4 - 1) = 0. b4 R , 0 0 , 0 1

( ). 4- 1 2, 4 1 4 b4 = 1. , b a3 3 0 b3 -2a3 2a3 -b3 0 1 0 0 0 1 0 0 0 1 (w1 ), 1 (w3 ), 1 (w4 ), 1 a1 b1 c1 0

(w2 ) a2 0 a4 b2 b3 1 c2 c3 c4 001

( ). 1, 3 1 a2 b3 = 0 a2 = 0, 1, 1 1 a2 = 1 a1 = 1, 1, 4 2a4 = 0 a4 = 0. 2, 1 4 1 2 b1 = 1, 3, 4 c4 = c1 . 2, 3 1 c3 = -b2 . , 10 00 0 1 1-b 0 0 , 3 0 b3 0 2 00 01 1 (w1 ), 1 (w3 ), 1 (w4 ), 1 1 c1 0 1 (w2 ) 0 0 a4 b2 1 1 . c2 -b2 c1 0 0 1

2, 3 3 b2 = -1, . 1 c1 = c2 = 0 . 6. , 1 (wi1 ) = wi1 ,. . . , 1 (wik ) = wik , 1 (wik+1 ), :


11

) = Al , l 3, i1 = 1, i2 = 2,. . . , ik = k , ik+1 = k + 1, k + ) = Dl , l > 4, i1 = l, i2 = l - 1, i3 = l - 2, . . . , ik = l - k ) = E6 , E7 E8 , i1 = 1, i2 = 2,. . . , ik = k , ik+1 = k + 1 V , 1 (wi1 ) = wi1

1 < l; + 1, ik+1 = l - k , 4 < k < l; , 4 k < l - 1. ,. . . , 1 (wik+1 ) = wik+1 .

. 1 (wik+1 ) 1 (wi1 ) = wi1 ,. . . , 1 (wik-1 ) = i wik-1 , (. 1), j = i1 , . . . , ik-1 V1j V0 k+1 . V1i1 · · · i V1 k-1 = ei1 , . . . , eik-1 , , 1 (wik+1 ) k - 1 . 4, , 1 (wik+1 ) l - k - 2 . , 1 (wik+1 ) {ek , ek+1 , ek+2 } ( ). 4. 4. = Al , Dl , El V , 1 (w1 ),. . . , 1 (wl ) w1 ,. . . , wl , . . A2 3. = Al , l 3, 4, 6 l - 3 , , , , 3, 1 (wl ). = Dl , 5 l-3 , . . . , l , 6 l - 5 l-4 , . . . , 2 , , , , 3, 1 (w1 ). = El , 5 2 , . . . , 5 , 6 6 , . . . , l-1 , , , , 3, 1 (w1 ) 1 (wl ). , 2 1 , , 2 (wi ) = wi i = 1, . . . , l. , 2 . 4. xi (1) . xi (t). 2 (x1 (1)) = x1 . x1 hi (-1), i = 1, 3, . . . , l, , x1 : 2 â 2 {vi , v-i }, i > 1 i , 1 = 0; 4 â 4 {vi , v-i , vj , v-j }, i > 1, i = j ± 1 ; , {v1 , v-1 , V1 , . . . , Vl }. h2 (-1) , h2 (-1)x1 h2 (-1) = x-1 . 2 â 2, 1 , x1 ab , cd 1, a2 + bc = d2 + bc = 1, b(a + d) = c(a + d) = 0. a + d 2 mod J , a + d R , . . b = c = 0. a2 = d2 = 1 a, d 1 mod J , a = d = 1. , 2 â 2 x1 (.. x1 (1) ), , .


12
- wi x1 wi 1 = x1 i 3. , 4 â 4 , wi , i 3. - wi x1 wi 1 = x1 i 3 -1 -1 h2 x1 h2 = x1 ,

{v1 , v-1 , V1 , V2 , V3 , . . . , Vl } .. ... ... .. 0 0 0 0 1 ... 0 . . . . . ... . . . . . . . . . . . . . . . 0 0 0 0 , 0 . . . 1

5, . . . , l . 4 â 4. , x1 a1 a2 a3 a4 b1 b2 b3 b4 c1 c2 c3 c4 , d1 d2 d3 d4 v2 , v-2 , v
1+2

,v

-1-2

e1 f1 g1 h1 e2 f2 g2 h2 e3 f3 g3 h3 e4 f4 . g4 h4

{v1 , v-1 , v2 , v-2 , v1+2 , v-1-2 , V1 , V2 }. - ,
b 1 - a4 4 b - a4 1 4

{vi , v-i } ( , b4 J ), {V1 , . . . , Vl }. wi , hi , x1 b4 = 0. , , 2 , 2 (x1 (1)) b4 = 0. , a14 · E {v1 , v-1 , . . . , vm , v-m }, {V1 , . . . , Vl }. , wi , hi , a4 1. , , 2 (x1 (1)) b4 = 0 a4 = 1.


13

w1 = 0 -1 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 00 00 00 10 00 00 0 -1 00 0 0 0 0 0 0 1 1 , w2 = 0 0 0 0 1 0 0 0 00 0 -1 0 00 0 0 -1 0 0 -1 0 0 0 -1 0 0 0 00 0 0 0 10 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0 0 0 . 0 0 0 0 1 0 1 -1

1 -1 0 0 1 0 1 0 0 0 0 0 x1 (1) = 0 0 -1 0 0 0 0 1 0 00 0 a1 b1 0 0 x1 = 2 (x1 (1)) = 0 0 c 1 d1 a2 b2 0 0 0 0 c2 d2 0 0 e1 f1 g1 h1 0 0 0 0 e2 f2 g2 h2 0 0 0 0 e3 f3 g3 h3 0 0 0 0 e4 f4 g4 h4 0 0 a3 1 b3 0 00 00 00 00 c3 c4 d3 d4 , 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 -2 00 00 10 00 10 01 00 1 0 0 0 0 0 0 1 ,

a1 , b2 , e1 , f2 , f4 , g3 , h4 , c2 , c3 , d4 1 mod J , a2 , g1 -1 mod J , a3 -2 mod J , J . g3 h3 f3 e = 3 0 0 0 0 g4 h4 f4 e4 0 0 0 0 g2 h2 f2 e2 0 0 0 0 g1 h1 f1 e1 0 0 0 0c 0 0 0 0 a1 b1 c1 - d1 c 0 0 0 0 0 0 0 0 0 0 0 0 a2 1 + a3 -1 b2 b3 0 c2 c3 + c4 -c4 - d2 c3 - d3 + c4 - d4 -c4 + d4

x

1+2

= 2 (x

1 +

2

(1)) = w2 x1 w

-1 2

1

2


14



=

x2 = 2 (x2 (1)) = w1 x

1+2

w

-1 1

h4 g4 0 0 f4 e4 0 0

h3 0 g3 0 0 a1 0 b1 f3 0 e3 0 0 -d1 0 c1 - d1 c

0 0 a2 b2 0 0 -d2 2 - d2

h2 g2 0 0 f2 e2 0 0

h1 0 0 g1 0 0 0 -1 - a3 a3 0 -b3 b3 f1 0 0 e1 0 0 0 d3 + d4 -d3 0 -c3 + d3 - c4 + d4 c3 - d3

.

:

Con1 := (x1 x

1+2

-x

1+2 x1

= 0),

Con2 := (h2 x1 h2 x1 - 1 = 0).

(3,8) . Con1 f3 = -e3 , (2,8) . Con1 h3 = -b3 c4 , (2,8) . Con2 b1 = b3 c4 , , h3 = -b1 . (1,1) . Con1 b1 (a2 + g4 ) = 0, b1 = 0, a2 + g4 R . :

Con3 := (x1 w1 x1 w1-1 - w1 h2 x1 h2 = 0),

Con4 := (x2 x1 - x

1+2 x1 x2

= 0).

y1 = a1 - 1, y2 = a2 + 1, y3 = a3 + 2, y4 = b2 - 1, y5 = b3 , y6 = c1 , y7 = c2 - 1, y8 = c3 - 1, y9 = c4 , y10 = d1 , y11 = d2 , y12 = d3 , y13 = d4 - 1, y14 = e1 - 1, y15 = e2 , y16 = e3 , y17 = e4 , y18 = f1 , y19 = f2 - 1, y20 = f4 - 1, y21 = g1 + 1, y22 = g2 , y23 = g3 - 1, y24 = g4 , y25 = h1 , y26 = h2 , y27 = h4 - 1. yi J . 1­4 27 ( yi ):


15

y23 (-a2 ) + y24 (a1 - b2 ) + y27 a2 = 0, y18 (-g1 ) + y22 (a1 - e1 ) + y26 (a2 ) = 0, y1 g1 + y15 (-g2 ) + y19 (-g1 ) + y25 a2 = 0, y6 (a3 + 1) + y10 (-1) + y16 (g1 + g2 ) = 0, y3 (c2 ) + y7 (-1) + y11 (-1) + y20 + y21 (-f4 ) + y22 (-e4 ) = 0, y3 (c3 + c4 - g3 ) + y8 (-1) + y9 (-1) + y13 (-1) + y14 (-1) + y23 2+ + y24 (-b3 ) = 0, y9 (-a3 - 1) + y13 + y23 (-1) = 0, y5 c2 + y25 (-f4 ) + y26 (-e4 ) = 0, y16 a2 + y17 (b2 - f2 ) + y18 (-f4 ) = 0, y5 (e4 - f4 ) + y16 (1 + 2a3 ) = 0,
2 y15 (-f1 f4 - f2 e3 ) + y16 (a1 - a2 + a1 h2 + f1 f2 - f2 ) + y22 (e3 a2 - f4 b2 ) = 0,

Con1, . (1,2), Con1, . (1,3), Con1, . (1,4), Con1, . (1,5), Con1, . (1,6), Con1, . (1,7), Con1, . (1,8), Con1, . (2,6), Con1, . (3,6), Con1, . (3,7), Con4, . (3,5), Con3, . (8,2), Con2, . (1,2), Con4, . (6,2), Con3, . (3,3), Con2, . (2,2), Con2, . (3,3), Con2, . (3,4), Con4, . (7,3), Con2, . (4,6), Con2, . (5,3), Con2, . (5,6), Con2, . (7,2), Con2, . (7,8), Con3, . (1,2),
27

y10 (-d3 - d4 ) + y11 (a1 + 1) + y12 c1 = 0, y1 (-1) + y2 (a1 + b2 ) + y3 (-c2 ) + y4 (-1) + y7 2 + y11 (-1) = 0, y5 (-c2 g3 ) + y16 (b2 - b2 h4 ) + y17 a2 = 0, y14 g3 + y16 (e4 - e3 ) + y21 + y23 = 0, y4 (b2 + 1) + y5 (-c2 ) = 0, y14 (e1 + 1) + y15 f1 + y16 (-g1 ) + y17 (-h1 ) = 0, y15 (e1 + f2 ) + y16 (-g2 ) + y17 (-h2 ) = 0, y6 (-g1 a1 ) + y9 (c3 + c4 - d4 )(c1 - d1 ) + y10 (c2 + c3 c4 - d3 c4 - e1 )+ 3 + y11 (-f1 ) + y25 c2 a1 = 0, y16 g4 + y18 e4 + y19 + y20 (f2 - h4 ) + y27 (-1) = 0, y14 + y21 (g3 - e1 ) + y22 f1 + y23 (-1) + y24 h1 = 0, y17 (-g1 ) + y22 (-f4 ) + y24 (g3 + h4 ) = 0, y4 (-c2 ) + y6 (-a2 ) + y8 c2 + y9 d2 = 0, y6 (-1) + y9 (c3 + d4 ) = 0, y1 a2 + y2 + y4 + y6 a3 + y10 (-1 - a3 ) = 0, y19 h4 + y20 (-1) + y25 (-g2 ) + y26 (-h2 ) + y = 0, y6 (-g3 f2 ) + y15 (-c1 g4 - c2 h4 ) + y16 (d2 - d1 )+ + y22 (c4 d2 - c3 c2 - c2 c4 ) + y26 (c4 d1 - c3 c1 - c4 c1 ) = 0,

Con3, . (6,6), Con4, . (7,5).


16

J
0 0 -1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 2 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 0 0 1 0 -1 0 0 0 1 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 -1 -2 0 1 0 0 0 0 -1 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 -1 0 0 0 0 0 0 2 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 0 1 0 0 0 -1 0 0 0 0 1 -3 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 1 1 0 0 0 0 -2 -1 0 0 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 0 0 0 0 -1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1 0 0 0 0 0 1 0 .

28 , , R. , y1 = · · · = y27 = 0. , x1 = x1 (1) . W , x1 = x1 (1) . , x2 = x2 (1). dt = 2 (h1 (t)). h1 (t) diag [t2 , 1/t2 , 1/t, t, t, 1/t, 1, 1] . 7. dt h1 (s) s R .

. dt dt w3 = w3 dt , . . . , dt wl = wl dt . l > 2 11 12 . . . 1l 21 22 . . . 2l dt = . . .. . . . . . . . . l 1 l 2 . . . l l V . dt wi = wi dt , i > 2, 1i = · · · = i-1,i = i+1,i = · · · = - 2 dt w1 dt w1 1 = 1 33 = · · · = l2l = 1 33 = · · · 1. dt wl = wl dt , l-1,j l,j , j = 1 dt w3 = w3 dt , 2,j l,j , j = 1, 2. dt v1 , v-1 , v2 , v-2 , v1+2 , v-1-2 , V1 , V2 , x1 . l i = 0 . = l l = , 2, . . . ,


17

dt k1 k3 0 0 dt = 0 0 0 0

hi , , k2 0 0 0 000 k4 0 0 0 0 0 0 0 l1 l2 0 0 0 0 0 l3 l4 0 0 0 0 . 0 0 0 m1 m2 0 0 0 0 0 m3 m4 0 0 0000 0 n1 n2 0000 0 n3 n4

- , x2 = x2 (1) xt = 2 (x2 (t)) = dt x2 d-1 = dt x2 w1 dt w1 1 . t 2 , xt dt . 2 Con5 := (xt x2 - x2 xt = 0). (1,6) k1 (k2 + k3 ) = 0 k2 = -k3 , 2 2 (2,1) k4 (l3 - m3 ) = 0 m3 = l3 , . (2,5) l3 (l1 + m4 ) = 0 l3 = - 0, . (5,6) k2 (l4 + m1 ) = 0 k2 = 0. Con6 := (dt w1 dt w1 1 - 1 = 0) - - - k1 k4 = l1 m1 = l4 m4 = 1. Con7 := (w2 dt w2 1 - dt w1 w2 dt w2 1 w1 1 = 0), m2 = l2 = 0 ( (1,2) (4,3)) l4 = l1 k1 (. (1,1)). (7,7) Con6 n2 - (n1 + n2 )n3 = 1, (3,7) Con5 n2 - 1 1 (3n1 + n2 - 2n3 - n4 )n3 = 1. , n3 = 0, n2 = 1. 1 2 n1 = n4 = 1, n2 = 0. , (3,4) Con5 k1 = 1/l1 . 1/l1 s. , wi , i = 3, . . . l, . , 1 , k = p, (h1 (t))vk = sp ·vk , (h1 (t))v-k = s-p · v-k . (h1 (t)) = h1 (s).

5. xi (t), . , 2 (hk (t)) = hk (s), k = 1, . . . , n. t s : R R . , t R 2 (x1 (t)) = 2 (h2 (t-1 )x1 (1)h2 (t)) = h2 (s-1 )x1 (1)h2 (s) = x1 (s). t R , t J , . . t = 1 + t1 , t1 R . 2 (x1 (t)) = 2 (x1 (1)x1 (t1 )) = / x1 (1)x1 ((t1 )) = x1 (1 + (t1 )). , R ( (t) := 1 + (t - 1), t R), 2 (x1 (t)) = x1 ((t)) t R. , , , . R , , R R . , C E (, R)C -1 = E (, R ) C GL (V ). , R = R. Eij .


8. E (, R) Mn (R). . (x1 (1) - 1)2 -2 · E12 . , · E12 ( -2 R R R). , .. k w W , w(1 ) = k , E12 · w E1,2k , w-1 · E12


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E2k-1,2 . , , , E2,1 . , Eij , 1 i, j 2m. , M2m (R). x1 (1) - 1 , E2m+1,2 - 2E1,2m+1 + E1,2m+2 . () E2,i , 1 i 2m, E2m+1,i , 1 i 2m. Ei,j , 2m < i 2m + l, 1 j 2n. -2E1,2m+1 + E1,2m+2 . () E2m+1,1 , E2m+1,2m+1 . E1,2m+1 , , , Ei,j , 1 i 2m, 2m < j 2m + l. , , .. Mn (R). A2 . (x1 (1) - 1)2 = -2E12 , h2 (t)(-2E12 ) = -2tE12 , . . E12 . w1 E12 w1 (1)-1 = E21 , E12 E21 = E11 , E21 E12 = E22 , w2 (1)E12 = E52 , w2 (1)E21 = E61 , E52 E21 = E51 , E61 E12 = E62 , E12 w2 (1) = E16 , E21 w2 (1) = E25 , E21 E16 = E26 , E12 E25 = E15 , E51 E15 = E55 , E61 E16 = E66 , E51 E16 = E56 , E61 E15 = E65 , Ei5 w1 (1) = Ei3 , i = 1, 2, 5, 6, Ei6 w1 (1) = Ei4 , i = 1, 2, 5, 6, w1 (1)E5i = E3i , i = 1, 2, 5, 6, w1 (1)E6i = E4i , i = 1, 2, 5, 6, E41 E13 = E43 , E41 E14 = E44 , E31 E13 = E33 , E31 E14 = E34 , , M6 (R). y = x1 (1) - 1 = -E12 - 2E17 + E18 + E46 - E53 + E73 , y = y + E12 - E46 + E53 = E18 - 2E17 + E72 , (E18 - 2E17 + E72 ) · E2i = E7i , i = 1, . . . , 6, (w2 (1) - 1)E7i = E8i , i = 1, . . . , 6, y = y - E72 = E18 - 2E17 , E81 y = E88 , E71 y = -2E77 , y E88 = E18 , y E77 = -2E17 , Ei1 E17 = Ei7 , Ei1 E18 = Ei8 , M8 (R). 9. C GL (V ) C E (, R)C R R, R = R.
-1

= E (, R ),

. , R R. C Mn (R)C -1 = Mn (R ), E (, R) Mn (R), E (, R ) = C E (, R)C -1 Mn (R ). , C GL n (R). , , R. , C GL n (R), ( E (, R) ) . 1.
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