CARTAN

OF

SUBALGEBRAS

W n AND

SIMPLE

LIE

p-ALGEBRAS

Sn S.

P.

UDC 512.9,519.46

Demushkin

C l a s s e s of C a r t a n subalgebras of simple n c n c l a s s i c a l Wn and Sn Lie p - a l g e b r a s over an a l g e b r a i c a l l y closed field k of c h a r a c t e r i s t i c p are d e t e r m i n e d up to conjugacies. All Cartan subalgebras of these algebras are Abelian. w 1.

Introduction

A s u b a l g e b r a H of a Lie algebra G is called a C a f t a n subalgebra if !) H is nilpotent, 2) the n o r m a l i z e r of H in G coincides with H. F o r a finite-dimensional algebra G over an infinite field k, t h e r e is always a C a f t a n subalgebra ( T h e o r e m 1, Chapter III of [1]). It is known that any two Cartan subalgebras of a finitedimensional Lie algebra G over an algebraically closed field of c h a r a c t e r i s t i c z e r o are adjoint ( T h e o r e m 3, Chapter IX of [1]). M o r e o v e r for simple algebras G of this type, the C a f t a n subalgebras are Abelian ( P r o position IV, Chapter IV of [1])o The same r e m a r k applies to the case in which the b a s e field is of c h a r a c t e r i s t i c p but the Lie algebra is of c l a s s i c a l type [2]. We will also prove that, for simple Wn and Sn Lie p - a l g e b r a s of n o n c l a s s i c a l type o v e r an a l g e b r a i c a l ly closed field k of c h a r a c t e r i s t i c p, the situation is not completely the same as in the p r e c e d i n g c a s e s : C a r t a n subalgebras a r e Abelian but the n u m b e r of conjugacy c l a s s e s of C a r t a n subalgebras is not equal to one. w 2.

Maximal

Tori

Let G be a simple Lie p-algebra over an algebraically closed field k of characteristic p and let H be a Cartan subalgebra. A torus in G is a commutative subalgebra T for which the p-mapping is one-to-one. In a torus T there is a basis (al,... , an) such that nip = aio The elements a = aP will be called toroidal elements.

LEMMA 1.

The toroidal elements of a C a f t a n s u b a l g e b r a H are in the c e n t e r of H. 2

P r o o f . L e t a P = a EH. i . e . , a is in the c e n t e r of H.

Then [a, h] =[aP, h ] = [ a p , h ] = . .

Hence the set of toroidal elements in H g e n e r a t e s a t o r u s .

.=0foranyh

E HsinceHisnilpotent,

We denote this t o r u s by the symbol T (H).

LEMMA 2. Some power of an element h E H belongs to T (H). P r o o f . Since H is nilpotent, some power h y of h is in the c e n t e r of H, which is the d i r e c t sum of T (H) and a subalgebra whose p(~-map is the zero element. Raising h v to the power p~, we find that hVP ~ E T (H). LEMMA 3. The c e n t r a l i z e r of T (H) in G coincides with H. P r o o f . We h a v e H c _ C(T(H)). L e t g E C(T(H)). T h e n g ( a d h ) k = 0 f o r h E Handksuchthat(adh) k E T (H) (from Lemma 2). But elements g possessing this property belong to H (Proposition I, Chapter Ill [1]). LE MMA

4. The torus T (H) is maximal

in G.

Translated from Sibirskii Matematicheskii Original article submitted June 11, 1969.

1970.

Zhurnal,

Vol. Ii, No. 2, pp. 310-325,

March-April~

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233

P r o o f . L e t T (H) cA_- T, w h e r e T i s the m a x i m a l t o r u s . T h e n T b e l o n g s to C (T (H)) s i n c e T i s a c o m m u t a t i v e s u b a l g e b r a , i . e . , T ~ H. But T i s g e n e r a t e d b y t o r o i d a l e l e m e n t s , i . e . , T = T (H). L E M M A 5. The C a f t a n s u b a l g e b r a s H t and H2T a r e a d j o i n t i f and o n l y i f t h e i r t o r i T (HI) and T (H2) are adjoint. P r o o f . L e t H 1 = I-I2T, w h e r e T i s an a u t o m o r p h i s m of G.

T h e n T (Hi.) = T (H2) T.

If T (Ht) = T (It2) T,

then

H~ = C(T(H~)) = C(T(H~) ~) -= ( C T ( H ~ ) ) ~ = H ~ "~. We c o n c l u d e f r o m t h e p r e c e d i n g t h a t t h e c l a s s i f i c a t i o n of C a r t a n s u b a l g e b r a s in s i m p l e L i e a l g e b r a s G i s e q u i v a l e n t to t h e c l a s s i f i c a t i o n o f m a x i m a l t o r i in G. We c o n s i d e r the p r o b l e m in t h i s f o r m u l a t i o n b e low. COROLLARY. it i s n i l p o t e n t .

w 3.

The

Algebra

T h e c e n t r a l i z e r of the m a x i m a l t o r u s in G i s a C a f t a n s u b a l g e b r a in G if and o n l y i f

W n

Let O = k[xl,... , x n] be the ring of polynomials over a field of characteristic p such that xi p = 0, and let W n be the Lie derivation algebra of the ring O. The algebra W n is realized in the form of operators fi0/Oxi, where fi E Q, and is a simple Lie p-algebra of dimension np n. Multiplication in W n is defined by the relation

i,k

T h e a u t o m o r p h i s m s of W n a r e t h e t r a n s f o r m a t i o n s ~ x i = ~ o i ( x t , . . . , x-n) , q ~ i ( O , . . . , O) = O, w h e r e t h e l i n e a r p a r t of g0i ( x l , . . . , x n) i s n o n d e g e n e r a t e , a n d 9 o p e r a t e s on D = E r i o / O x i a c c o r d i n g to t h e r u l e

z O~xk \ ~ i,h

where f~ =f(~-lx1,...

, ~ - l x n) f o r f E 0 [3].

In W n t h e r e is t h e u n i q u e f i l t r a t i o n

z~ = {Y,/?lOxjld~g/j LEMMA Proof.

6. All toroidal elements Let

(~,

>i

+

:',l (i = -

:, o, I ....

).

of G not belonging to ~f0 are adjoint.

]~O/Ox~)p= ~fiO/Oxi(~.%fo.

Thenfv(0,...,

0) ~ 0 for somev

=l,...,n.

Ifv ~I,

then application of the automorphism 9 xl=x,,

Cx~---z~,

4)xi=x~

(i4=t,v),

yields

axi ) a/axh

w h e r e fv ~ (0, . . 9 , O) = fu ( 0 , . . .

, O) ~ O, i . e . ,

we m a y a s s u m e f r o m the b e g i n n i n g t h a t f l ( O , . . .

W e now c o n s t r u c t the a u t o m o r p h i s m ~ x 1 = ~ x l , ~ x i = x i (i ~ 1):

( E f ~ O / O x ~ ) ~ =[~ r a a / O x , + E ]

~r O/Ox,.

i=/=t

S e t t i n g c~ = 1 / f ~ ( O , . . .

234

, 0), we m a y a s s u m e t h a t fl(O, . . . , O) = 1.

, O) ~ O.

We a l s o c o n s t r u c t the a u t o m o r p h i s m ~ x I = xl, ~ x i = x i + ~ i x l (i ~ 1)o We h a v e 9

(2

o

="

2,ox ,

A.,,J (f~ + Icl ai)a/axi.

s i~/=i

i~P-i

S e t t i n g ~ i = - f i (0, " " " , 0), we m a y a s s u m e t h a t f i ( 0 , . . .

, 0) = 5il.

To r e d u c e e l e m e n t s o u t s i d e ~ 0 to a s i m p l e r f o r m , we s e t ~x i = x i + 99i, w h e r e ,~i a r e f o r m s of the s a m e d e g r e e ~ - 2. We h a v e

H e n c e a n y e l e m e n t o u t s i d e ~ 0 is a d j o i n t to the e l e m e n t D = a/ax~ + x~ -~

where r

~k [x2,...

2 ~a/ax.

, xn].

We now c a l c u l a t e the p - t h p o w e r of a n e l e m e n t D of this f o r m .

Dx~--~t§

p-t

We h a v e

~,

D~xi = (p -- i) x~ -~ r . . . . . D"x~ ~ - ( p - - t ) . . . ( p - - v + i)xV~-" r

(2~>v~>p--i)

and D~'xt = (p

-

-

l) 1(t + xt p-I r

r --

-

-

(t + z~ -i ~ ) r

( H e r e we a s s u m e t h a t p -~ 3.) Similarly D2xi ~ ( p - - t)x~v - 2 r . . . . . D~z~ = ( p -- ~)... (p - - ~ + i)x~v-*~p~(2 > I .

>i p -

i)

and DPxi~(p--t)!(t-{-xt

p-I

~ i ) ~ p i ~ - ~ - - ( t - b x ~-i ~ ) ~ ,

i.e., Dv __--- - (t + xi~-t~pi) 2

~,O/Oz~.

It f o l l o w s that f o r a t o r o i d a l e l e m e n t we h a v e

p-t i.e.,

r

=-1,

r

v-i %E ~a/axi,

= 0 (i ~ 1).

H e n c e all t o r o i d a l e l e m e n t s o u t s i d e ~f0 a r e a d j o i n t to the e l e m e n t (1 - x~l - l) O / 8 x 1 and so b e l o n g to one c l a s s , and the l e m m a i s p r o v e d . R e m a r k . The e l e m e n t ~e0 is a l s o a t o r o i d a l e l e m e n t o u t s i d e (1 + xl) 8 / 8x 1. T H E O R E M 1. In the a l g e b r a Wn t h e r e a r e n + 1 c l a s s e s of m a x i m a l t o r i of d i m e n s i o n n . t i v e s of t h e s e c l a s s e s a r e

Representa-

r r = {(i + x,) a / az, . . . . . (1 + xr) a~ axr, zr+,a / axr+, . . . . . x~a / axe}

(r = O, t . . . . . n).

235

Proof.

Plainly T

is a torus.

We

will prove

that it is maximal.

To do this we find its centralizer.

r

Let

[Ef~O/Ox~,

(i

+xl)O/Ox~]----O,

i.e.,

fi--(l+x~)O--ifi----O, oxt

T h i s m e a n s t h a t f 1 = (1 + x l ) f l w i t h O f ~ / O x 1 = O. Similarly,

Oln

f.-- x ~ - - = O ,

Oxn

let [ E]~O/Oxi, x~cVbxn]-=O ,

i.e.,

Oh --0 ( ~ n ) .

Oxn

T h e n fn = X n f n with O f n / 0 X n = 0, and s o C (Tr) = T r . On t h e o t h e r h a n d t h e r a n k of t h e m a t r i x fij ( 0 , . . .

, 0) of s o m e s e t of e l e m e n t s

Di = E ]i'~O/c~x~i s

v a r i a n t u n d e r a u t o m o r p h i s m s of the a l g e b r a Wn, and f o r e v e r y t o r u s T r t h i s r a n k i s r . a r e not a d j o i n t .

in-

H e n c e the t o r i T r

C O R O L L A R Y 1. In t h e a l g e b r a W n t h e r e a r e at l e a s t n + 1 c l a s s e s of C a r t a n s u b a l g e b r a s . We

now prove that any maximal

First let T~0 is an automorphism

torus T in W n is adjoint to one of the tori T r.

and let the basis (Di,...

,Dp) inTbe

chosen

so that DiP =D i. We prove that there

in W n for which the basis of the torus T takes the form

D~ ~ E u~jxj~/~xj

Since Di,... , Dy commuSe with one another the matrices for the linear parts of DI, . . . , Dp commute with one another, and we can conshnaet an autornorphism in W n such that the linear parts of Di,... Dp become diagonal. Let

,

D~= E a~ixjO/c~x~§ D{, w h e r e D ( fi ~.~ (~t > i ) . C o n s t r u c t i n g t h e a u t o m o r p h i s m ~ x i = x i + q~i' w h e r e t h e q~i a r e a r b i t r a r y d e g r e e t~ + 1, we o b t a i n

xj - - - - 5~j~j) ~xj

#/Oxk § D~'.

( c o n g r u e n c e s a r e m o d u l o ~e~+x ). Let

D( ~-- E,q)~jO/ax~rood s where

the r

are forms

To reduce

of degree/z

D i modulo

~+,

+ I.

to the form

E

a~jxja/#xj, it is

l oqc~ VJT;;x (for all i and k).

Since k i s f i x e d h e r e , we a s s u m e t h a t k = 1. EaO

C o n s i d e r this relation for i = i:

236

\

(xj Oqoi

o%-7- 6ij t) +,.

necessary

0 Hence

= o.

and sufficient that

f o r m s of

Oxi H e n c e r n c o n t a i n s no m o n o m i a l s x a t . . .

xnan f o r w h i c h - - 6~) =/= O.

Za~

and we

may

assume

that ~li contains

only monomials

x al. . . x n

ai~(a~- 61;)= 0.

for which

values of ~i (which do not alter eli) also contain only these monomialso Now x I (we calculate terms of degree ~t + i): for D1x I we have terms r of degree are

~-~

w h e r e D~ = Z Let ~

~_~

0

0

...

=

~D~l

-~ a~

~li-]-

~,

~,a. . . . . n X t l . . . X n . We h a v e

Z

a~ja~ = a~

and so we

must

have

[~;i + ....

~'a i . . . a n

0, o r

ai~a~- a~ = 0 (i = i,...

not alter Di, . . . , D s modulo

~ - - ~] ~ . . . . . .

~ = 0,

s + l ~ 0. C o n s i d e r w h i c h m o n o m i a l s o c c u r in r s + l . It f o l l o w s f r o m [Ds, D s + l ] = 0 t h a t r s +1 c a n c o n t a i n o n l y t h e m o n o m i a l s • x for , s).

~+l.

But these Hence

and t h e o t h e r t e r m s do not o c c u r in r

are just the permissible

el, s +i contains

Z H e n c e if D 1 , . . .

+

eli =0.

Let r = 0,... , r = 0, r t h e c o n d i t i o n s [D1, Ds + t ] = 0 , . . . , whichZ

~-~

apply the condition Di p = D 1 to # + i, for D!Px I these terms

a~x~O/Ox~.

But in our case

i.e.,

p-i

Permissible

no terms

transformations x a~. . .

do

for which

(~s+l, j aj - - qs+~, s+~ =fi=0,

s + 1 s i n c e DPs + 1 = Ds + t.

, D v E~0, t h e y c a n b e e x p r e s s e d s i m u l t a n e o u s l y in t h e f o r m Z

But t h e e l e m e n t s x j 0 / a x j this case

for ~i which

e~:xjO/Ozj.

= ( x j 0 / ~ x j ) P c e n t r a l i z e T and a r e t o r o i d a l , and s o x j a / 0 •

E T.

Hence

in

T = { x 9 / Oxj; i = i . . . . . n} ---- To. Now

let ~r~

~0

that D~ ~ ~0 9 Lemma have

and let the basis of T consist of the toroidai 6 and the remark following it imply that we

C (D 0 = { (1

elements. Suppose that D i = D p E T but may assume that D I = (I + xl}0/ax i. We

@ x~)]~O/Ox~-}- Z]iO/Ox~}, i~%2

where 0fj/0x i =0.

LetD

E C(Di) and l e t

237

D = DP =

( l ~-

D =

(t +

xi)l,O/Oxi q- s

We have

where

D'P

= D'.

Let

D' @ ~0.

Then

there

xl)flO [c)xl -}- D ~,

is an automorphism

in W n leaving

D I invariant

while

D takes

the form

D= Now constructing

the automorphism

(i + x,)kO / Ox, + (l + x~)O l Ox~.

6 x 1 = x 1 + (1 + x , ) r

~ x i = x i (i = 2 , . . . ,

n), w h e r e

aq~/Ox 1 - - 0 , d e g

-> 2, w e o b t a i n Dl ~ :

D ~ ~--~(J. -[- xl)flO/Oxt &p

DI,

+ (t + .~) (i + x~) ~

O/Ox~ + (l + x~)O/Ox~

= (t § x~)f(O/Ox~ + ( 1-4- x2) O/Ox2, where using

f I, =fl +(l +x2)O,(o/0x2. It follows that fl canbe assumed the condition DP = D we find that r = 0, k an integer. Let D 1 = (I + x0O/0xl,

be toroidal.

This

implies

. . . ,Dm

= (i + Xm)O/ax

m

to take the

and let the element

formfi=X+•

~, X Ek.

D centralize

DI, . . . , D m

Now

and

that rn

D = ~, ('1.47 xl)/i O]Oxl + - ~"L I~ O]Ox~, w h e r e fj E k [ x m + * , .

9 9 ,x n].

Let

D' =

,~, /iO/Oxd~ &. i-.>.mq-1

Then since D 'p = D' and the trm~sformation reduced

t o t h e f o r m (1 + x m + i) 8 / 0 x m .

fi (i ~ m) Now

p-i

take the form consider

ki q- x,~+i~,

the case

k i E k, and the condition

and all the remaining

D' is

Ox i = x i + ~ai (i -< m + 1), w e f i n d t h a t

D p = D implies

that r

= 0, and k i integers.

in which

D, =

implies

of X m + ,, 9 9 9 , x n d o n o t a l t e r t h e f o r m of D, t h e e l e m e n t

Applying the automorphisms

elements

~- xi)c9 10xl . . . . .

(1

of t h e t o r u s

a r e i n s 0.

D~- =

(1 q-

xr)O / Oxr,

L e t D tiC ( D 1 , . . .

, D r ) , D E s 0.

The first condition

that

D=

(t ~7 xl)/,O]Oxl ~ . . . + (l -}- xr)/rO/OXr-/

~, /jO/Oxi, O/~[Oxj=O ( ] = 1 . . . . . r). j~r+:

The representation

of D i n t h e f o r m D' + D", w h e r e D" =

~

[~OlOxj, p o s s e s s e s

the same properties:

j>r+l

i) D P = D = > D " P = D " ; Hence the set of parts of D" also forms affecting only x r + I, 9 9 9 , Xn and leaving torus

2)

[D,C]=O:=>-[D",C"] = 0 .

a torus in Wn_ r over xl, . . . , x r invariant,

k[xr + i," 9 9 , Xn]. By using the automorphism we can reduce an element D of the maximal

to the form r

D=~, * As

238

in Russian

original

- Consultants

(i +x~)/jO/Oxj+

Bureau.

~, c~jx~O/a j.

Now c o n s t r u c t t h e a u t o m o r p h i s m @xj = x i + (1 + x~)qoj, w h e r e d e g ~o~ --- 2, Coj d e p e n d s only on Xr + ~ , 9 . . >x n, w h i c h l e a v e s D 1 , . . . , D r i n v a r i a n t . T h i s a u t o m o r p h ~ s m r e d u c e s D to a f o r m s u c h t h a t fj (j ~ r) c o n t a i n s o n l y m o n o m i a l s x a l . . , xnn f o r w h i c h E a j a j = 0. Now u s i n g t h e c o n d i t i o n DP = D we f i n d t h a t fj = 0. By c o n t i n u i n g t h i s p r o c e s s we o b t a i n the p r o o f of the t h e o r e m . 9

~

COROLLARY 2. In the algebra algebras are of dimension n.

w 4.

The The

Algebra algebra

S

J

W n there

.

J

are exactly

n + 1 classes

of Cartan

subalgebras.

All these

( n > 2)

n

S n (n > 2) is defined

to be the second

commutative

algebra

n+~ ~ f~O/Ox~ of

of all derivations

i~i

, X n + i ] w i t h d i v f = y~Of~/Oxi = O. A s a v e c t o r s p a c e , Sn i s s p a n n e d b y t h e d e r i v a t i o n s

the r i n g k [ x ~ , . . .

D~, j {~} ---~(c?u I Oxj) 01 oz~ - (Ou I axe) a I oxs, u 6 k [x: . . . . . z,,+:]. and h a s d i m e n s i o n n (pn + 1 _ 1). T h e a l g e b r a S n i s s i m p l e and i s a p - a l g e b r a , and it i s n a t u r a l to i n t r o d u c e i n i t f i l t r a t i o n with r e s p e c t to the d e g r e e of t h e e l e m e n t u. T h e a u t o m o r p h i s m s of Sn a r e a l l t r a n s f o r m a t i o n s x i - - @ x i f o r w h i c h t h e d e t e r m i n a n t of the m a t r i x (0 @(xi)/Sxj) i s c o n s t a n t and d i f f e r e n t f r o m z e r o . L E M M A 7. T h e t r a n s f o r m a t i o n @xi = x i + ~oi, w h e r e t h e ~ i a r e f o r m s of d e g r e e ~ <-- 2, e x t e n d s to an a u t o m o r p h i s m of S n i f and only if d i v q0 = 0. P r o o f . L e t @xi = x i + ~ o i + r wheredegr ~ u + 1 , be a n a u t o m o r p h i s m o f S n . T h e n i t S ( x i + (Pi + r ) / S x iu = 1. C a l c u l a t i n g m o d u l o of t h e f o r m s of d e g r e e Ur, we o b t a i n d i v qo = 0. Now l e t d i v go = 0. W e p r o v e t h a ( t h e r e a r e ~i (deg r ~ v + 1) s u c h t h a t ll0(xi + qo~ + r = 1. L e t the r b e c h o s e n s o t h a t US(x~ + goi + r ~- 1 r o o d ~ nlU'(u ' ~ u, nl i s the i d e a l i n k [ x ~ , . . ~. , X n + l ] generate~d b y x l , . . . , ~%t+~)" T h e n forx i+go i+r

+pi, wheredegpi

~

u' + 2 , we h a v e

II~(x~ + m~+ r + p~) IOx~lf ~- II~(x~ + ~ + , , ) l~x~ + ~p~/Ox~ll II0(x~ -t- mi + r Since

ficient to prove of degree and s o

ui"

~n + i

O~ I

all elements

x i , . . . , Xn+

I

, except

that II0(x i + gai + r We

must

show

To obtain -

-

(xl...Xn+1)

~

r~ ~+, =

LEMMA Proof.

O,

p-I,

contain

that [I8ai/Oxjl[ does

it is necessary

"

elements

"

'

"

(x I.

r-el

"

fi' = fi + 8~i/Sxi"

'

"

.....

Assume

that a i is a monomia[

We

n + l

it is suf-

have

a, = x~ i~ .

x ~i,'~

"

2Fi,~+i--~:p--~,

i.e.,

~ril

0.

of S n outside but

r 0 are adjoint. (~ :~r

Since

in this ease,

linear part of the automorphism W n is arbitrary but nondegenerat% fi(0'" . . ,0) =0 (i =2,... , n+ 1). Applying the automorphism the new values div f =0that

of the divergence,

Xn + i)p - t

~rliM1~p--~

that

fiO/Oxi----- f~O/Oxi ~S~,

are in the image

(x I. . . Xn+i) p-~.

not contain

2

which implies that llrijN :

8. All toroidal Let

(x I. . . Xn+l) p-I

cannot

9

+ div p rood m"'+'.

Let fl = x~.P-lfi

+ gi, where

as in the case

of ~xi, the

we may assume that fl (0, . . . , 0) = i, ~x i =x i + gPi, deg ~Pi ~ 2, we obtain gi does

not contain

x p-i.

It follows

from

os i x,+ 7~.-

i . e. div]~O,

divg~xi

p-2-

fi.

239

Setting Xl

Or~/Ox~= O, we have 9c!

_ _ __ i'( ~gi~xi x p--~f~) dx~

0

0

Og~ div*=--(gt--t)--~x~dXt+divF'-~--(gi--i)+

+ div F = 1 - - gl (0, x2. . . . . x.+~) + xiV-1 f i + div r . S i n c e g~ (0, x 2 , . . . elements

, x n + 1) d o e s n o t c o n t a i n (x 2. . . X n + l) p - 1 [ t h i s f o l l o w s f r o m t h e r e p r e s e n t a t i o n

o f t h e f o r m D i j (u)], F c a n b e c h o s e n s o t h a t d i v F = g l ( 0 , x 2 , . . .

t i o n o f Sn b y e l e m e n t s

o f t h e f o r m D i j (u) i m p l i e s

,Xn+l) -

1.

of S n b y

But the representa-

that

n+t auj j=2

Hence,

setting xj

Jo Oxj we have n+t

div q~= div ~ - - xi

P--t

).a..2 . j=2

i.e.,

the automorphisms

Ou = z~-' f, - x:-~7, = o, Oxj

@xj = xj + q~j a r e p e r m i s s i b l e .

T h e n e w v a l u e s o f fi' a r e a s f o l l o w s :

h ' = x3-17t + gt(O, x 2 . . . . .

xn+O,

xj

o

(ir

6tx'j

i. e. ,

w h e r e 71, 9 9 9 , T n + l , There

g2, 9 9 9 , g n + 1 a r e i n d e p e n d e n t

is an element

o f x 1.

D with fi = x f - t d i v g + l + 71,

fj =

div f =- O, Application (j ~ 1). L e t

of the automorphism

U s i n g t h e c o n d i t i o n D p = D, w e o b t a i n f j

apply the transformation

div y = O.

Ox 1 = x l , Oxj = xj - x l T j (j ~ 1) s h o w s t h a t w e m a y a s s u m e 7 j = 0

f~=x~-idivg+l+yl,

Now

xlV-tfj + x~ -z g~ + y~(j =# i ) ,

= 0 (j = 2 , . . .

fj=x~ -- ifj+x~ p--2 gj , n + 1).

with la~j j=2

240

(] =/= 1).

[this is possible since the representation of S n in the form (x 2. . . Xn+i)P-I ]. We find that D has the form

of derivations

Dij (u)shows that7~ does not contain

n~ki

(~ + x~ -~ div ~) a/ax~+ ~-~ ~

~9/ax~.

~2

C a l c u l a t i n g D p = D a s a p p l i e d to x l , we o b t a i n d i v g = - 1 . ~2 = x p - l ( g 2 +x2), ~0j = x p-2gj- (j ~ i , 2) we h a v e

Now a p p l y i n g t h e t r a n s f o r m a t i o n w i t h ~ l = 0,

p--2

D ~ ( i - - x~~-~) 8/Ox~ - - x~

x~O/Ox:.

and t h e I e m m a i s p r o v e d . W e g i v e a n o t h e r p r o o f of L e m m a 8. L e t D @~f0 b e t o r o i d a l and ~ Sn. Sn i s i m b e d d e d , s u c h t h a t

L e m m a 6 i m p l i e s t h a t t h e r e i s an a u t o m o r p h i s m

5 of W n + l, in w h i c h

D +-i = (i § x O a / Ox~ - - x~+~a / az~+~,

i.e., ((i + xl)a/0x I - Xn+10/DXn+1) that ((I +x1)a/0x I- Xn+10/Dxn+1)

r E S n. Hence to prove @ E S n implies @ EC(DI) DI

=

Lemma 8 it is necessary G(Sn) , where

and sufficient to prove

(t d- xj ) O / Ozl - - x~+,0/ Oz~+~.

H e r e C d e n o t e s t h e c e n t r a l i z e r and G (Sn) d e n o t e s t h e g r o u p of a n t o m o r p h i s m s of t h e a l g e b r a Sn. S i n c e t h e l i n e a r p a r t of an a u t o m o r p h i s m of Sn i s a r b i t r a r y b u t n o n d e g e n e r a t e , we m a y a s s u m e t h a t .+xi = x i + ~i, w h e r e d e g ~Pi ~ 2. T h e c o n d i t i o n D~ + ~ Sn i m p l i e s t h a t d i v D~ r = O, i . e . , f o r f o r m s of l o w e r d e g r e e we h a v e ~ div ~ / 0 x ~ = 0, o r div q i s i n d e p e n d e n t of x t. On t h e o t h e r hand t h e c e n t r a l i z e r of D~ c o n s i s t s of a u t o m o r p h i s m s +x i = x i + r where

*i = Z A~(l q-xi)j+i x~+~, Xnq-i~

'~ = 2 ,

oj

(l + x0 x.+~

Hence

a vr

A (j +

+

It is thus sufficient that the system

+

of equations

q: divr

Remark.

~

U/~j

+

j

or that the system

(m) OBo Ox~= = div %

(Aj+:B:)(]+I)+)'~--~-~

be solvable.

n ~D(rn)

+:

div ~ be solvable,

A o § Bo §

m==2

j

--~0

(]=t ..... p--i).

OXra

But t h i s s y s t e m i s c l e a r l y a l w a y s s o l v a b l e . The e l e m e n t (1 + x I ) 0 / 0 x 1 - x 2 O / 0 x 2 E Sn i s t o r o i d a l .

THEOREM 2. There are n + I classes of maximal tives Of these classes can be taken to be Qr =

tori of dimension

n in the algebra S n-

Representa-

{ ( i § z~) a / az~ - x=+~a / ax~+, ..... (I + z,) a / axr -- x~+:a / az~+~, 32r+1~ / ~Xr+l - - Xn~-I8 /OXn+l . . . . .

x~O/

ax,~ - - x,~+~a / ox~+,}

(1"= o, 1 . . . . . n).

241

Proof.

The e l e m e n t s g e n e r a t i n g Q r a r e t o r o i d a l and c o m m u t e with one a n o t h e r .

D --~ ~, f~O/Ox~~ C(Q~). It

We find C (Qr).

Let

follows f r o m the r e l a t i o n

that li =

Aj.(t + x,)J+l x.+i,

Jr-.,

],+i=2Bi(l

,+i,

(m)

w h e r e Aj, Bj, BI m) 6 k [ x 2 , . . .

, Xn].

.....

.),

T h e condition

[ 2 S,O/Ox,, x,+~O/Ox~+,--x,+~O/Ox,~+,]=O i m p l i e s that

I,.-~-) ~j x~+ixr,+t j B i(m)' 6 k [ x l , . . . w h e r e Aj' B.'

(rn=i ..... r--i,

, Xr, x r + 2 , . 9 9 , Xn].

w h e r e Xij 6 k , I I = ( l + x l ) . . . ( l + x

r + i . . . . . n),

Hence the condition D E C (Qr) i m p l i e s t h a t

r) x r + t . . . x n + t .

W e n o w p r o v e t h a t the t o r u s Q r is m a x i m a l . Let D 6C (Qr) and let DP = D. T h e e l e m e n t D h a s the f o r m indicated above, and we m a y a s s u m e that hi0 = 0, c a l c u l a t i n g the c o r r e s p o n d i n g l i n e a r c o m b i n a t i o n of e l e m e n t s of the t o r u s Q r f r o m D. U s i n g the r e l a t i o n d i v D = 0, we have

~ . ~ ( ] + 1 ) + ~ 2 ; ( S + i) + . . . + ~.+,.J(i + i) = 0, and we c o n c l u d e that Xlj +k~j + . . . + h n + l , j = 0, s i n c e in all s u m s f o r fl, 9 9 9 , f n + t the s u m m a t i o n with r e s p e c t to j is only to j = p - 2, as follows f r o m the r e p r e s e n t a t i o n in the f o r m of l i n e a r c o m b i n a t i o n s of the e l e m e n t s Dij(u). The e l e m e n t D thus has the f o r m D = ~X~;HJ ( (4 +

xt) O/Ox~ x~,+,O/Ox~+~)~- E -

-

~jHJ(

(i -~ x~)O/Oz~-- xn+iO/Oxn+,)

All t h e s e s u m s c o m m u t e with one a n o t h e r and so it is sufficient to p r o v e that the e l e m e n t

j=l

is not t o r o i d a l . Only one t e r m hljlIJ0((1 + x i ) a / a x 1 - X n + t a / a X n + l ) (i ~ ] 0 ~ p - - 2) of this s u m need be i n v e s t i g a t e d . But it is e a s i l y s e e n that the p - t h p o w e r of this e l e m e n t is equal to 0. H e n c e Q r is the m a x i m a l t o r n s r e d u c i n g to the Abelian C a f t a n s u b a l g e b r a of d i m e n s i o n n(p - 1).

242

C O R O L L A R Y 1. T h e r e a r e at l e a s t n + 1 c l a s s e s of A b e l i a n C a r t a n s u b a l g e b r a s of d i m e n s i o n n ( p - 1) in the a l g e b r a S n ( n > 2). W e now p r o v e t h a t any m a x i m a l t o r u s Q in Sn i s a d j o i n t to s o m e t o r u s Qr" F i r s t l e t (2 ~ ~e0. Since in t h i s c a s e , a s in the c a s e of Wn, t h e l i n e a r p a r t of a p e r m i s s i b l e a u t o m o r p h i s m 9 i s a r b i t r a r y , we m a y a s s u m e t h a t t h e e l e m e n t s D 1 , . . . , D u of Q a r e of t h e f o r m

Di = ~

tti~x~O/Oxj 3c D(,

where

D ( ~ c~o.

We now apply the reasoning we used for Wn, ~4th the additional condition (imposed the automorphisms) concerning the vanishing of the divergence. Let Q ~_ S% and let D i = DiP ~ Q, but @ ~fo . Then by virtue of Lemma D I is adjoint to the element (I + xl)0/0x I - Xn+iO/0Xn+ I. The centralizer

{ 2Aj(t

of this element

both on the terms

8 and the subsequent

Di~ m~d

remark,

is the set

+ xt)i+~ x~+lO/Oxl @ Z

B~(i ~- xl)jxn+lO/Ox~+i + L-4 Bij

x,~+iO/~x~}.

i~2

The a u t o m o r p h i s m s ~xj = xj + qj a l s o c e n t r a l i z e D b w h e r e qi=

i -~ x~)i+l Z~zTi,

~,q)i(t

q~+~

-----2 r

•

~'~ (m)

q,.

?,~j

9~i) Xn+h ~.,~

(t+xd

J+~

. ,i

3 Xn+t

(we a s s u m e t h a t a l l s u m s r e p r e s e n t e l e m e n t s w h o s e l o w e r d e g r e e s a r e g r e a t e r t h a n o r e q u a l to 2). A c o n d i t i o n f o r t h i s a u t o m o r p h i s m to b e l o n g to Sn i s t h a t the d i v e r g e n c e v a n i s h f o r the l o w e s t t e r m s , and in t h e present case this condition is n

Forj

~ (i)

+ t)-4- z._.zOX~ Oxi = o

(pj(j+ '1)+r

for all

j.

(I)

=p-lwehave ~

(i)

atop-1 __ Ox~

O.

i=2

For the last condition to be satisfied it is sufficient that 3:7 0

X2, p-i ---- -- ] B2, ~-I dx2 + B2 p-1 (0, x3 . . . . x,~)x2 0

if Bi, p _ t d o e s not c o n t a i n x~2- i.

New v a l u e s f o r Bi, p _ ~ a r e (~----3. . . . . n),

B~, p - l = 0

B2.~-~=B2,~_I(O, x3 . . . . . x~,).

H e n c e the Bi, p _ i r e d u c e to t h e f o r m p--I

Bi, p-, = x2 w h e r e 7 i s i n d e p e n d e n t of x 2.

--

B~, p-1

(~ = 3 . . . . . n),

P--I

B2, p-i = x2

--

B2, p-1 -}- ~1,

F o r j ~ p - 1, c o n d i t i o n (1) i m p l i e s t h a t we m a y t a k e

243

(o 0

and t h e r j a r e o b t a i n e d f r o m (1) if n e i t h e r Bi~ n o r A,~ c o n t a i n x~22t r a n s f o r m a t i o n s show t h a t

i.

It f o l l o w s t h a t if D

s

permissible

n

+

_ ~(l+ ~p - , r)~, B

x ,) ~x~+~O/O~§ ~+'

§247247

~-~ ~ ( ~ + x l )

§

.

~x~+,O/Ox~ ~

~-~

~ ~+1

w h e r e T i s i n d e p e n d e n t of x 2. (The r e p r e s e n t a t i o n of Bj f o l l o w s f r o m t h e c o n d i t i o n d i v D = 0.) s u m e t h a t 7 = 0 b y a p p l y i n g , if n e c e s s a r y , t h e t r a n s f o r m a t i o n

agx2=x2

(i§

--

-1 Xn+1 ~1

Cxi = xi -~ ~< (i § ~ j

\ p--i

We m a y a s -

X2, p--i

x~+i,

where

2

a(p~

/=3

[this i s p o s s i b l e s i n c e T d o e s not c o n t a i n (x 3. . . Xn) p - 1]. -

j

Now c a l c u l a t i n g DPx 2 = Dx 2 we o b t a i n ~ , B2j(l § x~) J x~+, ~---- - i . D ~ x~ -1 p--2

--

-~- (~

F A5 (t +

xi) j+l x~+lO/Ox:

XJ-~I (~

-

-

X2

Hence

p--2

~ ~j

) ~/OX2 ~- XP2- t

j§

j J i~3

The condition div D = 0 implies

~ i.e.,

F 0=-1,

Fj =0(j

=1,...,p-2).

x2

§

Fj(] § t) (t + x~) j X nJ§ The elementD

2 j (t § x:) 3+1x~+l O/Oxl § x2

~-- O ~

r e d u c e s to

8j (i § zl) 5z;+l O/Ox~+:

p -: 2 B I j ( t - _ ~ X l ) j Xn+OIO/X~i_I(_X2TO)I-O/X2__X2J

p--2 X n + l O / O X n + l .

i=3

Now calculatingD p as applied toxl, x3,... ,Xn+1, we find thatAj =0, Bj =0, Bij =0, i.e., D ~ (t -- x2P-i ) O/~x2 -- x2P-2x~+l~/Ox~+l. We t h u s find t h a t a l l t o r o i d a i e l e m e n t s o u t s i d e ~f0 h a v i n g fl ( 0 , . . . , 0) =0 and c e n t r a l i z i n g D 1 a r e a d j o i n t . An e l e m e n t of t h i s t y p e i s (1 + x 2 ) 0 / 0 x 2 - x n + 1 0 / 0 x n + ~. T h e p r o o f of t h e t h e o r e m i s o b t a i n e d b y c o n t i n u i n g

this process.

244

In conclusion I wish to thank A. I. Kostrikin for his valuable advice~

LITERATURE 1.

2. 3. 4.

CITED

N. Jacobson, Lie Algebras, Interscience Publishers, New York (1962). G. B. Seligman, "On Lie algebras of prime characteristic," Memoirs of the Amer. Math. Soc., No. 19. A~ Rudakov, "Automorphisms of infinite-dimensional simple Lie p-algebras," Izv. Akad. Nauk SSSR, Seriya Matem., 33, No.4, 748-764 (1969). A. i. Kostrikin and I. R. Shafarevich, "Graded Lie algebras of finite characteristic, ~ Izv. Akad. Nauk SSSR, Seriya Matem., 33, No.2, 251-322 (1969).

245