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On generalized Witt algebras Ree, Rimhak 1955

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0 N • G E N E R A L I Z E D  WITT  A L G E B R A S  by Rimhak Ree A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of MATHEMATICS  We accept t h i s t h e s i s as conforming to the standard required from candidates f o r the degree of Doctor of Philosophy.  Members of the Department of Mathematics  TEE  U N I V E R S I T Y  OF  B R I T I S H  A p r i l , 1955  C O L U M B I A  Abstract  01  Let over a f i e l d  $  be a commutative associative algebra of c h a r a c t e r i s t i c 01 .  the d e r i v a t i o n algebra of friOl) and  ^  , ... , D  i s expressed uniquely as 01 , then »m } D  iL  of  f o r any  . For a regular subalgebra  i f there exist  e  A subalgebra  i s c a l l e d regular i f f D e D e  m  e £  D =  &{0l)  p > 0 , and  f e 01  of  ,$-(<?L) ,  such that every  D e srl  + •.. + f m » where D  m  i s said to be defined by the system a n c  *  i  s  denoted by the notation  D]_, ... , D ) . m  In t h i s d i s s e r t a t i o n , the family  of L i e algebras of the type  S£(t7l; D i , • •• , D ) m  is  studied. It i s shown that i f Ot algebras i n  ^  i s a f i e l d then a l l  are simple except when  It i s also shown that i f <& then every simple algebra i n  p = 2, m = 1 .  i s a l g e b r a i c a l l y closed ^  i s a generalized Witt  algebra of the type defined by I . Kaplansky [ B u l l . Amer. Math. Soc. v o l . 10 (1943), pp. 107-12l], and, conversely, that every generalized Witt algebra belongs to  ^  . A  simpler form of the generalized Witt algebras i s given. By using t h i s form, the problem of whether every generalized Witt algebra can be defined over  GF(p)  i s partly  solved. ^  I t i s shown also that a subfamily  ^  !  of  , consisting f o r the most part of non-simple alge-  bras, has a remarkable property:  every algebra i n  ^  f  has the same i d e a l theory as that of a commutative associative algebra.  Jacobson's result on automorphisms  of the d e r i v a t i o n algebras of the group algebras of commutative groups of the type  (p, ... , p)  i s extended  to generalized Witt algebras, and, f i n a l l y , i t i s shown that  m  i s an invariant of the algebra  = - £ ( 0 1 ; D i , ... , D ) m  i f -sC i s normal simple.  THE UNIVERSITY OF BRITISH COLUMBIA Faculty of- Graduate Studies PROGRAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSFRT of RIMHAK REE B.A. (Seoul) 1944  WEDNESDAY, MAY 4th, 1955, at 10:00 A.M. HI HUT MI3 COMMITTEE IN CHARGE H.F. Angus,. Chairman D. Derry S.;A. Jennings F.A. Kaempffer D.C. Murdoch  R.M. G.M.. A.P. F.-H.  Clark Shrum Maslow Soward  External Examiner - Hans Zassenhaus, McGill University.;  Abstract On Generalized Witt Algebras Let A be a commutative associative alegebra over a f i e l d K of characteristic p different from zero, and D(A) the derivation algebra of A . A subalgebra L of D(A) i s called regular i f f D i s i n L f o r f -in A and D i n L . f o r a. regular subalgebra L.~ of D(A) , i f there exist D,,  1*  such that every  ...  , D  •  '  in L  m  D i n L i s expressed uniquely as  D = f ^ + ... f TJ , +  where f± i s i n A , then by the system ^D-L,  m  m  L i s said to be defined  ,  ...  and i s denoted by the notation L(A:' D1*, . . . ' , D ). nr v  In this dissertation, the family F of Lie algebras of the type L(Aj D , . . , D J 1  F  i s studiedi It i s shown that i f A i s a f i e l d then a l l algebras i n F are simple- except when p = 2, m = 1. I t i s also shown that i f K i s algebraically closed then every simple algebra i n F i s a generalized Witt algebra of the type defined by I. Kaplansky (Bull. Amer.. Kath. Soc. v o l . 10 (1943), pp. 107-121), and, conversely, that every generalized Witt algebra belongs to F. A simpler form of the generalized Witt algebras i s given. By using this form, the problem of whether every generalized Witt  alegebra can be defined over GF(p) i s partly solved. It i s shown also that a subfamily ,F' of F , consisting f o r the most part of non-simple algebras, has a remarkable property:,, every algebra i n F' has. the same, ideal theory as that, of a. commutative associative algebra.-. • Jacobson?s result, on automorphisms of the derivation algebrasjof the group algebras of commutative groups of the type (p, ....... , ;) i s extended to generalized Witt, algebras, and,' f i n a l l y , it- i s shown that m i s an invariant of the algebra. L-= L(A; D , ...... , D ) 1  if  L  w  i s normal simple'.-  GRADUATE STUDIES Field of Study: Mathematics Modern Algebra -.- D.C. Murdoch Fourier Series and Integrals — F.M.C. Goodspeed Other Studies: Electromagnetic Theory -- W. Opechowski Special R e l a t i v i t y Theory ~  F.A. Kaempffer  PUBLICATIONS 1.  On a problem of Max A. Zorn, B u l l . Amer. Math."Soc. v o l . 55 (1949), pp. 575-576.  2.  On ordered, f i n i t e l y generated, solvable groups, Trans. Royal Soc. Canada, Third Ser. v o l . 48 (l954), pp. 39-42.  3.  On projective geometry over f u l l matrix rings, Proc. Amer. Math. Soc. v o l . 6 (1955), pp. 144-150.  0  Table of Contents  Introduction The algebra  .  .  -£(01',  D  . lf  .  .  ... , D ) M  .  . .  1  . .  .  4  Generalized Witt algebras Reduction of algebras  -£.(01,;  1  D-^, ... , D ) m  7  to orthogonal form Some lemmas  13  Derivations of a f i e l d  16  Simple derivation algebras when  ® is 19  a l g e b r a i c a l l y closed Nilpotent systems (1) Nilpotent systems (2).  The case  m = 1 .  •  31  .  45  Normal systems  50  The case  54  $ = GF(p)  57  Non-simple algebras Automorphisms of References  ~C (01; D]_, ... , D ) m  .  .  63  82  Introduction  QL  An a l g e b r a space o v e r  $  over a f i e l d  $  i s a vector  i n w h i c h a (not n e c e s s a r i l y a s s o c i a t i v e )  bilinear multiplication  xy  i sdefined.  an a l g e b r a i s c a l l e d a s u b a l g e b r a  A subspace o f  i f i t i s c l o s e d under  the m u l t i p l i c a t i o n defined i n the algebra.  0  of  01  yx  belong t o 0  algebra  01  i s c a l l e d an i d e a l o f  01  x e 3  f o r any  o n l y i d e a l s o f Ot • if  x y +-yx = 0  for  a l l elements  subalgebra  i f both  xy  y e 01 •  and  i s c a l l e d simple i f 0  A  01  and  and An  are the  An a l g e b r a i s c a l l e d a L i e a l g e b r a x(yz) + y(zx) + z(xy) = 0  and  x , y,  and  z  hold  i n the algebra.  In  t h i s d i s s e r t a t i o n , by a n a l g e b r a we s h a l l a l w a y s mean a finite-dimensional algebra, unless the contrary i s specified. A derivation  D  t o be a l i n e a r t r a n s f o r m a t i o n o f that  D(xy) = (Dx)y + x(Dy)  any two d e r i v a t i o n s  D]., 2  l i n e a r transformations w h i c h a r e d e f i n e d by •» a f D ^ x ) ,  and  D  01  of an algebra  OL  i s defined  into i t s e l f  such  f o r a l l - x , y e Ol . F o r  °£ ^  an<  *  D i + D2, aD^,  a  e  and  ® » ^ Di*  ( D j + D2)x • Dpc + D2X,  (D]_« D2)x - Di(D2x) - ^ ( D j x )  t i v e l y , a r e a l s o d e r i v a t i o n s o f Oi  e  ,  (aDi)x respec-  :  D ) ( x y ) = Di((D2x)y + x ( D y ) ) - D ((Diac)y + x ( D ) ) 2  2  2  1 7  . i i„ = ( D i ( D x ) ) y + x ( D i ( D y ) ) - (D (D]x))y - x ( D ( D i y ) ) 2  -  If  01  ((D3/  2  D ) x ) y + x{(B c 2  D )y)  1  2  2  .  i s a commutative a s s o c i a t i v e algebra then f o r any  derivation ping  2  aD  D  of  OL  and any element  which i s d e f i n e d by  d e r i v a t i o n of  OL .  of an algebra Ol the m u l t i p l i c a t i o n algebra of  a e 01  (aD)x = a(Dx)  The t o t a l i t y  & ( Ol)  the mapi s also a  of d e r i v a t i o n s  forms a L i e algebra w i t h respect t o «• . $r  (OL)  i s c a l l e d the  derivation  OL . From the p o i n t of view of s t r u c t u r e t h e o r y , a  c r u c i a l problem i s the determination of a l l simple algebras. Simple L i e algebras over the f i e l d of complex numbers were completely determined by W. K i l l i n g ' and E. Cartan,  and  those over a r b i t r a r y a l g e b r a i c a l l y c l o s e d f i e l d s of chara c t e r i s t i c zero by W. Landherr and N. Jacobson.  Roughly  speaking, these simple L i e algebras f a l l i n t o f o u r w e l l known f a m i l i e s , w i t h f i v e e x c e p t i o n a l algebras.  However,  simple L i e algebras over a l g e b r a i c a l l y c l o s e d f i e l d s o f characteristic mined.  p > 0  have not been completely d e t e r -  Four f a m i l i e s , which correspond t o those i n the  case of c h a r a c t e r i s t i c zero, are known. For c h a r a c t e r i s t i c  p > 0 , however, there i s  a f a m i l y of simple L i e a l g e b r a s , (the f i r s t examples were discovered by E. W i t t ) which has no counterpart i n the case of c h a r a c t e r i s t i c zero, although, as we s h a l l  - i i isee l a t e r , there are some f a m i l i e s of i n f i n i t e dimensiona l simple L i e algebras of c h a r a c t e r i s t i c zero which may be regarded as the zero c h a r a c t e r i s t i c counterparts of the above family.  The Witt algebras are algebras over  a f i e l d of c h a r a c t e r i s t i c e  0> l» ••• » p - l e  e  where  i + j  a n c  p > 2  with basis ej. o e j - ( j - ije^+j ,  * relations  i s t o be calculated modulo  p .  H. Zassen-  haus ( [ 5 ] , P» 47) generalized the Witt algebras to algebras with basis | ^ j , where  a  runs over a subgroup  of the additive group of the ground f i e l d , and with the relations  e » e^ a  • ( p - a)e |g a+  •  Another generaliza-  t i o n of W i t t s algebra was obtained by J a c o b s o n [ 3 j . T  his  In  investigations Witt [ 1 ] used i m p l i c i t l y the f a c t  that the Witt algebras defined above are the d e r i v a t i o n algebras of the group algebras of c y c l i c groups of order p •  In the paper c i t e d above, Jacobson proved that the  derivation algebra of the group algebra of an elementary p-group, by which we s h a l l mean throughout t h i s d i s s e r t a t i o n an abelian group of the type  (p, p, ... , p) , i s  simple i f the order of the group i s >  2 .  Recently, I . Kaplansky ( [ 4 ] , p. 471) gave an ingenious generalization of the Witt algebras, which i n cludes the generalizations obtained by Zassenhaus and  -  the L i e algebra^over  IV -  #  with basis { ( i , o")} , where  i e I, o" e Ofy , and the m u l t i p l i c a t i o n (0.0.1)  ( i , o > ( j , t ) = u ( i ) ( j , cr + <t) - <r(j)(i, cr +T) .  It appears that  s£  i s simple except when  of a single element and Zassenhaus  1  I consists  3? i s of c h a r a c t e r i s t i c 2.  algebra i s the case when  I  consists of a  single element, while Jacobson s i s the case where (fy T  consists of a l l integer-valued f u n c t i o n a l s . We s h a l l c a l l the above algebra In order that  -JL a generalized Witt algebra.  be f i n i t e dimensional i t i s necessary  and s u f f i c i e n t that both i s f i n i t e , then and  Cfy  $  I  and  Cfy be f i n i t e .  must be of c h a r a c t e r i s t i c  I f Cfy  p > 0 ,  i s an elementary p-group. In t h i s d i s s e r t a t i o n , we study a family of  Lie algebras, defined by means of derivations, which includes a l l the generalized Witt algebras.  One of our  main r e s u l t s i s that i f the ground f i e l d i s a l g e b r a i c a l l y closed then any simple algebra i n our f a m i l y . i s a genera l i z e d Witt algebra.  However, i f the ground f i e l d i s  not a l g e b r a i c a l l y closed then i n general our family cont a i n s many simple L i e algebras, i n p a r t i c u l a r many new. f i n i t e simple L i e algebras, which cannot be constructed by Kaplansky s method. f  Moreover, our formulation enables  us t o derive i n a natural fashion some of the remarkable properties of the generalized Witt algebras.  The author would l i k e to express h i s gratitude t o Professor S. A, Jennings, under whose d i r e c t i o n t h i s d i s s e r t a t i o n was written, f o r h i s assistance and encouragement.  -  1.  1  -  The algebra  £  Throughout t h i s paper,  $  (01;  t i v e algebra over  m  w i l l denote a f i e l d of  p > 0 , and  characteristic  ... , D ) .  OL  a commutative associa-  $ , with a unit element.  Suppose there exist derivations of  OL  Dj_, ... , D  m  such that m  (1.0.1)  D Dj - DjDi = Z I a ±  i j k  D  k  0  i , j = 1,  for set  ... , m , where  (Ol; D^, ... , D )  of a l l derivations of  m  of the form  f i D ^ + ... + f D m  a subalgebra of  , where  m  Ol  f ^ e Oi , forms  (01) . More generally, the set of  a l l derivations of Ol where  a j j k e OL • Then the  of the form  fj[ runs over an ideal  Q~  f ] ^ + ... + f D m  m  ,  of Ol , forms a sub-  algebra of ^9- (01) . For, m f i ^ i gjDj = f i ( D  i g j  )Dj - g^DjfiJDi + 2 1 figja  where a l l the c o e f f i c i e n t s of the right-hand to  0" .  Dk ,  side belong  In the following we s h a l l r e s t r i c t our L i e  algebra •£, ( 0 1 ; D^, ... , D ) m  by imposing the following  condition: (1.0.2)  i j k  fB 1  1  + ...  +  implies  « 0  f]_=... = f  m  = 0 .  - 2The number £  (OL  ;  D  X  ,  m w i l l be c a l l e d the D-dimension of . . .  D  ,  M  )  .  Because of the condition (1.0.2) there e x i s t s a one-one correspondence f ^  +  . . . +  f  M  D  M  « - » (t  . . .  lt  ,  fm)  between the elements of J C (OL ; D ] _ , . . . , D )  and the  set of a l l vectors  runs over  M  OL,  (fl,  I f we i d e n t i f y . . . , fm)  (f]_, . . . , f m ) f ]_D;L + . . . +  M  D  M  . . . , fm) + ( g  (fl,  l f  ( f l + SI,  s  hi -  . . . , gm)  . . . , gm)  ( h l r . . . , h m ) , where 21 ( f s ( D s g i ) s  +  a e $ .  , f m + Sm) ,  . . . , fm)°(gi, -  Z. £ s S t a s t i s,t  gs(Dsfi))  •  Conversely, we may define a L i e algebra ing with derivations  i£*  D]_, . . . , D m s a t i s f y i n g  but not necessarily (1.0.2) and the set vectors  with  . . . , fm) = ( a f i , . . . , afm) for  (fl,  f  fj[  then  o(flf  (1.0.3)  , where  (f]_, . . . , f m )  by s t a r t (1.0.1)  s £ * of a l l  and defining scalar m u l t i p l i c a -  t i o n , a d d i t i o n , and m u l t i p l i c a t i o n according to (1.0.3).  < C * i s d i f f e r e n t from  In general,  It i s e a s i l y seen that the set (fl,  ... , f ) m  an i d e a l of  satisfying and that  0  •£ (OX ; D]_, ... , D ) . m  of a l l vectors  f]_Di + ... + f D m  •£* /0  m  = H (01 ; D  • 0 lf  forms  ... , D ) m  Since we are mainly interested i n simple L i e algebras, we prefer t o work with "£* . bras  •£ (OX ; D]_, ... , D ) m  rather than  In what follows we study the properties of the alge £(01*, D^, ... , D ) m  always assuming  (1.0.2).  - 4 2.  Generalized Witt algebras*  any generalized Witt algebras  £.  the form aC (OL ; D i , ... , Dm) .  We show that  can be written i n Let it  be defined  with respect to the set I * { l , 2, ... , m j of i n d i c e s and the t o t a l additive group with values i n ft .  Let ?5j  (ty of functional s on =  u^; ... ] be a  •[UQ-,  m u l t i p l i c a t i v e group isomorphic t o pondence  UQ- «—* a •  For each  O^: <J -*• ft by  mapping  ®1> ••• t ®m  a  r  e  Cfy v i a the corres-  i e I  we define the  Oi(uQ-) • cr(i) .  homomorphisms' of  I  ^  Then  into the addi-  t i v e group of ft , such that (2.0.1)  eifuoO • ... • 9 (u ) - 0  (2.0.2)  a ^ e i + ... + a 9  m  m  implies Now l e t Ol  m  = 0  by  be the group algebra of  Di^o- s ei(u -)u - . Then  Oi ,  with  a]_ = ... = a  and define the l i n e a r mapping o  0  D^  UQ- = 1 ,  implies  ff  e ft m  = 0 . ^  of Ol  over ft , into i t s e l f  Dj_ i s a d e r i v a t i o n of  since Di(uo-ut) - Di(uo-+t) - 6i(uo-+t)ucr+t - SitUo-Ju^UT; + eifutJUtfUfc •  (DiUo-)ut + Uo-(DiUfc),  It i s c l e a r that (1.0.1) i s s a t i s f i e d f o r D^, ... , D since  DiDj - DjDi = 0  m  ,  f o r a l l i , j . We w i l l show that  (1.0.2) i s also s a t i s f i e d .  Let  f i D j + ... + fmDm - 0 ,  -  with  .  f±E0l  5  -  Then we have  fi6i(ucr)  X  for  0  -  At  all JEL it  .  Let  f± «* - 0  a.^(X)Q^(u ) a  it follows that f]_ = . . . » f  a±(X) » 0 for a l l  • 0 .  m  . Then we have and cr . From  d^ftOu^  for a l l t  and t  . Thus  Therefore we can define the Lie  algebra •£ (01 ; DT_, . . . , D ) .  The set { u^DiJ , where  m  i £ I , ff E ^  i  (2.0.1)  is a basis of this Lie algebra, and we  have u  c r i t j • Ucr(DiUt)Dj - ut^DjU^Di D  -u  D  -  U ( i ) u  f  f  Comparing the above with  +  c  D  j  "  0  < J ) <r+t i u  (0.0.1),  •  D  we see easily that  the given generalized Witt algebra is isomorphic with H  (OX  ; D]_, . . . , D ) . m  We note that  (2.0.1)  is equiva-  lent to the following property of £ (Ol\ D]_, . . . , D ) : m  (2.0.3)  D]_f - . . . = Pmf •  0  implies  f e * .  Conversely, for any elementary p-group Cfr , i f there exist homomorphisms 8 i , ... , ^ into the additive group of (2.0.2)  algebra  $ such that  of 0J  (2.0.1)  and  hold, then we can construct a generalized Witt (01 ; D]_, . . . , D ) m  by the above method.  Suppose now that homomorphisms 0 i , . . . , $ satisfy p  n  and  (2.0.1)  , and let  erators of cj  .(2.0.2)•  xi, ... , x .  n  Let the order of  (ft  m  be  be a set of independent gen-  We set 0j.(xj) = a^j e $ .  Then  (2.0.1)  «- 6 ••» and  (2.0.2) are r e s p e c t i v e l y equivalent to the following  conditions: (2.0.4)  if ^  k^, a  then (2.0.5)  ... , k  are integers such that  n  ij j - 0 , i - l ,  ...  k  ki s  ... s k  n  s. 0 (mod  the rank pf the matrix j = 1, ... , n , i s  ,m, p ) , and  ( i j ) , i = l j •••  ,  a  m .  Thus a generalized Witt algebra whose dimension i s i s completely  characterized by  s a t i s f y i n g (2.0.4) and immediately that that  ft  i s of rank  mn elements  (2.0.5).  m^n. £ n  If over  m e 1 , and ft = GF(p), then  algebra.  o  mp  n  a^j e ft  From (2.0.5) i t follows  m = 1 GF(p)  then (2.0.4) implies .  Therefore i f  n = 1 , so that the only  generalized Witt algebra of D-dimension i s the Witt  »  m  1  over  GF(p)  -  3.  7 -  H((X\ l ,  Reduction of the algebras  to orthogonal form.  D  In t h i s section, we show that any-  simple algebra of the form  =  Dj, ... , D ) m  = i (Oi; Vj.t ... , D ) , where  can be written as ^ D^DJ - Dj-Dj = 0  ••• , m)  D  ffl  for a l l  i , j .  An ordered set { Dl, ... , D J - of derivations m  Ot w i l l be c a l l e d  of a commutative associative algebra  a system of derivations of OL or simply a system i f i t s a t i s f i e s (1.0.1) and (1.0.2). We s h a l l say that the Lie {Oi; D^, ... , D )  algebra { l» D  i s defined by the system  m  ••• i % ] • A system { D J L , ... , D  c a l l e d orthogonal i f DjDj - DjDj_ - 0 that i s , i f i n (1.0.1)  aijk - 0  ort ho normal i f there exist that  D^fj o 8 j j  m  m  j - w i l l be  for.all  for a l l  elements  (Kronecker d e l t a ) .  i , j,  i , . j , k,  f ^ e OL such  An orthonormal  system i s always orthogonal. Two. systems { B± ... D j t  and. | i , D  ... , Djjj } o f  i  - TL  and such that { l» D  cijDj  ( i = 1 , ... , m)  det(cij)  > °m }  a  n  d  D  D  lt  Lemma 3.1.  i s a unit of OL .  { i» ••• » m }  and only i f •£ (01 ; B  tions of  Ot w i l l be c a l l e d equivalent  c i j e Ot such that  i f there exist D  m  a  r  e  equivalent i f  ... , D ) = -£ (OL; B{, ... , D ) . m  A system {^l  m  t  ••• » ^m}  of deriva-  OL i s equivalent to an orthonormal system i f  -  8  and only i f there exist det ( D i f j )  -  £]_, ... , f  e 01 such that  i s a unit i n Ol .  Proof.  Suppose that { DT_, ... , D  equivalent to an orthonormal and l e t  m  system { D^j ... , D }• m  Dj[ = 2Z.CijDj , D j f j = 8^j , where  i s a unit i n Ol  Djf -j = C J . J  • Then we have  Conversely, suppose that unit i n Ol  . Thus  f o r some  det(Difj)  £]_, ... , fm  D^ = «Z1 C i j D j . Then c ^ "to 1^1» ••• » m J a D  ... ,  n dw  t  have  i s equivalent  m  • °ij »  s  o  that  ^ i s orthonormal, which proves the lemma. - £ (Ol 5 D  For a given algebra we denote by  Ifl the set of a l l elements  that  for a l l D e £  Dc = 0  (cij)  (D-jfj) . We set  Dji, ... , D e  is a  Ol . Let  e  be the inverse matrix of the matrix  •  A  lf  ... , D ) m  c E Ol such  i s a subalgebra of  w i l l be c a l l e d the algebra of constants of *C . OL i s always assumed t o have a u n i t y element, we  Since have  det(cij)  i s a unit i n OL .  det(Difj)  Ol •  ]• i s  m  c e  i f and only i f  some d e f i n i n g system  Die = ... = D c • 0 f o r m  { D i , ... , D  m  ^ of  -s£ •  The following lemma i s u s e f u l . Lemma 3.2.  I f the algebra  a d i v i s o r of zero, then simple.  £.(011  D  l,  of constants has ••• , °m) i s not  - 9 ~ Proof 3  The set  Let c e jpl be a d i v i s o r of zero.  of a l l cD , where  D e  , forms an  i d e a l of s& . For, (cD)oD* - c(D«D')e 3 m0  then from (1.0.2) i t follows that I f $01  contradiction. for  $ . If  some  f ^ , ... , f  follows that  m  then  c • 0 ,a  = c(fiDi  + ... + f D ) m  m  e Ol . Then again from (1.0.2) i t  1 «= c f i , which i s impossible i f c divides  0 , and therefore  s£ i s not simple. Ol i s  A commutative associative algebra  completely primary i f the set o f a l l non-units coincides with the r a d i c a l of Ol . I f -JL (Ol \• D^, ... , D )  Lemma 3.3. then  Ol  m  i s completely primary. Proof  Since  (Ol; Di, ... , D ) m  i s simple,  from Lemma 3.2 i t follows that the algebra stants has no d i v i s o r of zero. and  Let f e Ol  D fP - pfP-^D-jf - 0 ±  fP £ 0  then  i s also a u n i t . f  p  Since  Ol  of coni s commutative  i s f i n i t e dimensional over the ground f i e l d , is a field.  If  i s simple  • 0  for a l l  be a non-unit. i , we have  Since  fP e  f P i s a unit i n 01 , and hence This i s a contradiction.  f o r a l l non-units  f . Thus  . f  Therefore  OL i s completely  primary. Lemma 3.4.  Let Ot be completely primary.  If  fi,  ... , f  f e Ol  with  10  be such that  n  implies  f f i = ... = f f • 0 n  f - 0 , then at. l e a s t one  Ol •  i s a unit i n  Proof.  Assume that a l l f i are non-units.  Then there e x i s t s a positive integer fl  k  * ... • f  q  f ^ l ... f  + ... + r  negative  r  such that  r i , ... , r  are  n  non-  Suppose, therefore, that (3.4*1)  r i + ... + r  r i + ... + r  = 0  n  n  a nk , where  n  integers.  holds whenever Let  k  = 0 , and hence  k n  (3.4.1) if  f^  > r , a p o s i t i v e integer.  n  • r , f - fi l  ... f  r  n  f f 1 • ... - f f • 0 , and hence n  r n  n .  f = 0 .  Then  Using complete o  induction with respect to (3.4-1) holds whenever lar,  f i = ... = f  Then we have Therefore  n  r , we can conclude that  r i + ... + r  • 0 .  We are now  > 0 .  Take a non-zero  f f i = ... = f f  at least one  n  n  In p a r t i c u f e Ol  .  = 0 , a contradiction.  f i must be a u n i t .  i n a p o s i t i o n to prove the follow-  ing: Theorem 3.5. then any system  If  Ol  Di, ... , D  m  i s completely primary, of derivations of  i s equivalent to an orthonormal system. any simple Lie algebra of the type i s defined by an orthonormal system.  In p a r t i c u l a r ,  ( 0l\  Di, ...  D ^ m  - 11 Proof.  Let u i , ... , u  over the ground f i e l d  < E > . We set D  l*<  l t.  D  u  f .  (3.5.1)  D  where  be a basis of Ol  n  r  i ,  u  •••  D  r  t  u  r  1 i> r £ m . We s h a l l prove by using Lemma 3*4  that, f <  '  i s a unit f o r some choice of f e Ol i s  Suppose, therefore, that  il: such that f f  •m.  If  = 0  for a l l  .  (3.5.2)  ff .  0  i s true f o r some  r , and a l l i]_, ±2*  by expanding the determinant  . ,i  f•  r  , then  along the r - t h  column, we have ftj  (3.5.3) where all  c i  r  r  i  • f/  . V  , we have  (1.0.2) we have ff^  ... i  1  - U c i D i + ... + f e p D p J u ^ - - 0 ,  r  =0  fCT_DI  .  Since (3«5.3) i s true f o r  + '... +  f c i = ... = f c  r  f c  R  D  R  «  0 • Then from  = 0 , and i n p a r t i c u l a r  f o r a l l i]_, ... , i „i • Proceeding r  by induction with respect t o r , we can conclude that (3.5.2) holds f o r a l l r • Taking the case have  fD]U^ - 0  f o r a l l i i . Therefore  Hence from (1.0.2) we have  r - 1 , we fDi = 0 .  f • 0 . Therefore, by Lemma  - 12 3 »4  f •  „•  i s a u n i t f o r some  f r o m Lemma 3.1 i t f o l l o w s t h a t  ii  f  ... i  £ D i , ... , D  m  m  .  Then  j- i s e q u i v a -  l e n t t o a n o r t h o n o r m a l system. The second p a r t o f t h e theorem fo-llows i m m e d i a t e l y f r o m t h e above r e s u l t and Lemma 3.3•  -  4.  13 -  Some lemmas.  We establish here a number  of r e s u l t s we w i l l need l a t e r .  We assume throughout  t h i s section that { D i , ... , x i , ... , x 3  m  e Ol are such that  i s an i d e a l of  -  Lemma 4.1. then  f  k  D e  Do(xkD) a f D e k  J  S1 1 + ••• + gm^m g^ - 0  Since  k  a  m  m  m  e ^7 >  Dx = f k  k  m  e  , we have  .  If D = f  1  e  f o r any i  f  t  (01 j D i , ... , D ) .  i s a unit i n Ol  D  - $£j n d that  I f D = f ^ D i + ... + f D  Since  Lemma 4.2. k  % X J  f o r any k .  Proof.  and i f f  ]• i s orthonormal, that  t  where  such that  ^ + ... + f D m  , then g  k  = 1  and where  f± = 0 .  Proof.  Consider the element  D e J  by Lemma 4 . 1 , we have also  U e 7 , where  V e $ ,  where  *  J  4  %  - 14 Then we have  Setting  V - 2U e Cf , where  V «- 2U = g l D i + ... + gmDm , we have  4  g, - -  ^  ^  4  ,  +  -  I,  4  and f o r i / k ,  Therefore, i f f ± = 0  then  g^ * 0 , completing the  proof. +  Lemma 4.3. algebra  of constants of  1 1 + • • • + fm m D  D  then  I f f]_, ... , f  Di e J  e  ^7 >  a  n  d  and are such that i  f  some  fk i s a unit,  f o r a l l i • 1, ... , m .  Proof.  Suppose that  f ^ i s a unit.  ( f l D i + ... + f m D ) » ^ *J - Di e ^ D  m  i  s  l ,  Then  for a l l  ... ,m. Lemma 4.4.  when  belong to the  m  p • 2 ,m = 1 .  Dj_ e  implies  $ = j£  except  - 15 Proof.  If  Dj[ e  follows that  a r b i t r a r y element Dji»(fD ) = (D-jfjDi i  (4.4.D  V  D]_ e  3  f o r i = 1, ... , m .  f e Ol . we have  (DjfjDi e  (4.4.I) we have xiDj[ e Cf> •  have  (4.4.2)  Take an  Then from  ?  for a l l i , j . p / 2 .  F i r s t we consider the case from  then from Lemma 4.3 i t  2x D i  e  i  D j j x i ) = 2x^ ,  Since  t7 •  2  p / 2 , we  Since  Hence  (fDiJ^XiDi) = fD  On the other hand., since  - X i t D i f j D i e Cf .  ±  D j j x i f ) « f + Xj[(Dj[f) , from  (4.4.1) we have (4.4.3)  3 .  fDi + XifDifjDi e  From ( 4 . 4 . 2 ) and (4-4.3) we have p / 2  we have  we have  $  fD^ e $  •  2fDi e 3  Since  f  D J _ ( X J  L  i X J )  we may take  are a r b i t r a r y ,  p = 2 , m y 1 • For  j  such that  = Xj , from (4.4.I) we have  (fDj)o(xjDi) = f D i - X j ( D i f ) D j e  ^.  j ^ i . X J D ^  Since, f  and  U - dL , completing.the  i  Since  e -J •  Therefore  are a r b i t r a r y we have  proof.  Then  However, we have  x j ( D i f ) D j - D i t x j f j D j e jf from ( 4 . 4 . 1 ) . fD^ e jf .  i  Since  - £ •  Now we consider the case given  and  .  - 16 ^ 5.  Derivations of a f i e l d .  of the d e r i v a t i o n algebra & (Ol)  Ol  w i l l be fD e -£  .  $-{01)  .  i t s e l f i s a regular subalgebra of  i s i t s e l f a f i e l d , any regular subalgebra  &(Ol)  Ol  , since -JL  f-P+ f»D» e £ Dj,  ... , D  l»  ••• » m Ol  Ol  over  , that i s , i f  f , f» e Ol .  , where  m  If  -£_(0l\  , i t i s e a s i l y seen that Therefore,  D  l,  ••• , m) D  » &  nd  Let  Then any regular subalgebra OL m  over  3?  -  that  £  I D]_, ... ,  f ^ D i + ... + f D m  p = 2, m = 1 ,  can be written i n the form .  By Theorem 3.5 we may assume 0  be a non-zero element i n  that the number of non-zero fk £ 0  § •  •  be a non-zero i d e a l of m  •  -JL of the d e r i v a t i o n alge-  J i s orthonormal.  ^  m  be a f i e l d over  i s simple except when  (Ol; Di, ... , 1^)  Let  If  Oh  i s the D-dimension of Proof.  ^  .£-(01)  we c a l l  the D-dimension of the regular subalgebra Theorem 5.1.  ,  has a basis  i s a f i e l d , any regular subalgebra of  i s of the type  bra of  D, D» e  *L  s a t i s f y (1.0.1) a'nd (1.0.2).  D  where  of  i s closed under the scalar multi-  p l i c a t i o n by elements of Ol  if  If  may be considered as a vector space over the  field  D  Ol  of  , f e Ol imply  c a l l e d regular i f D e 3r(0l)  A subalgebra  then by Lemma 4*2  f^  and such  i s as small as p o s s i b l e . ^  contains an element  - 17 S l l + ••• gm m D  +  g± - 0  D  whenever  set that  fk = 1  we have  B±e(fiDi  s  u  c  that  n  g  = 1  k  f i = 0 , so we may assume at the outf o r some  k . Since m  m  f  k  J  i s an i d e a l ,  + ... + f D ) m  = (D^f-j^D^ + ... + ( L \ j f ) D e Since  and such that  J f o r i = 1, ... , m .  m  • 1 , the number of non-zero c o e f f i c i e n t s i n  ( D i f i ) D i + ... + ( D i f ) D i s l e s s than that of m  f ^  + ... + fjiP  m  and hence we have constants of £  m  . Therefore f i , ... , f . Since  from Lemma 4.3 we have and  =  D-jfj - 0 m  for a l l i , j ,  e $C , the algebra of  $C i s a sub-field of Qt , e  $  sC from Lemma 4*4.  f o r i = 1 , ... , m ,  Therefore  i s simple.  The method used i n the proof of Theorem 5.1 can also be applied to the case of a f i e l d of characteristic  0 , i f we start with an orthonormal  example, consider the f i e l d functions i n m  ft(x]_,  0 , and l e t  dimensional extension f i e l d of Oi  ... , x )  over a f i e l d  ft(xi,  ... , x ) . Then m  i s an i n f i n i t e dimensional algebra over ft . I t i s  01 over ft such that  every d e r i v a t i o n  ... , f  bra £  -x. a .  •  r-,  •• • . z~z  , and that  of OX. over ft i s w r i t t e n uniquely 2 o V frz+---+fT? > where ~  i n the form f^,  m  Oi be a f i n i t e  well known that there exist derivations of  of r a t i o n a l  m  variables x i , ... , x  ft of c h a r a c t e r i s t i c  system. For  D  S!  m  e Ol • I n other words, the d e r i v a t i o n alge*  [Ol) of Ol  over ft can be w r i t t e n as  - lo = -L(Ol\  ~  > • •• j ^ )  enables us to prove that  .  The above method  ^ ( t f t ) ; i s an i n f i n i t e dimen-  s i o n a l simple L i e algebra of c h a r a c t e r i s t i c zero. I f we consider the polynomial domain Ol = <&[xi, ... , x  an algebra over  m  J , instead of  * ( x i , ... , x ) , as m  $ , then again we may prove that  &  (Ol)  i s simple. The above two classes of i n f i n i t e dimensional simple L i e algebras, together with i n f i n i t e dimensional algebras constructed by Kaplansky^  method, may be r e -  garded as analogues of the Witt algebra i n the case of characteristic  0 •  -  6.  19 -  Simple derivation algebras when  a l g e b r a i c a l l y closed. i s that i f Ol  $ is  The main r e s u l t of t h i s section  i s a commutative algebra over an alge-  b r a i c a l l y closed ground f i e l d  <£ , then any simple  dL(Ol; D]_, ... , D )  algebra of the type  m  i s a gener-  a l i z e d Witt algebra. Lemma 6 . 1 . £  Suppose that  - £, (Ol; D , ... , D ) X  such that or  f  i s a unit i n  .  If f  elements of the form  i s as above, the set  fD , where  £, . For, i f X. S i i D  e  £  then  D e £  0  of a l l  , i s an i d e a l of  (fD). (Z g±B±)  f(Dgi)Di - Z g i ^ i f D = f Z((.Dgi)Di - g±XiD) e J .  Since If  f =0  Dif - X±f ., X± e $ , f o r a l l i , then  • Proof.  = L  I f f e Ol i s  i s simple.  m  3  i s assumed to be simple, - 0 then  again by ( 1 . 0 . 2 )  f - 0 by ( 1 . 0 . 2 ) . j  $ = 0 or If  0 -  3=01. 01 then  i s a unit i n Ol , as required.  By Theorem 3*5, any simple algebra of the form £(01;  D^, ... , D ) m  i s defined by an orthonormal  Moreover, by Lemma 3«2, the algebra  fpt of constants f o r  the simple algebra £, [01; D]_, ... , D ) m  $ , and i f $  system.  i s a f i e l d over  i s a l g e b r a i c a l l y closed, we have  $? » $ .  Since we are mainly interested i n t h i s section i n simple algebras, we s h a l l assume that the conditions (6.1.1) - ( 6 . 1 . 1 ) , below hold.  The l a s t two of these are  - 20 necessary i f SL (Ol;  D^, ... , D )  i s simple, as i s seen  m  from Lemma 6.1 and the above remark.  The ground f i e l d .ft  i s assumed a l g e b r a i c a l l y closed. (6.1.1)  The system  {Di, ... , D j  (6.1.2)  I f f e Ol  i s such that  \± e ft f o r a l l  Dif - \$f  i , then  f = 0  with  or  f  is  Ol •  a unit i n (6.1.3)  i s orthogonal.  m  Dif - ... = Dmf - 0  implies  These conditions and the fact that  ft  closed w i l l enable us to prove that  f e ft . i s algebraically  Ol  i s the group  algebra of an elementary p-group. Ol  We consider operator domain  fi  as an ft -module, where the  consists of m u l t i p l i c a t i o n s by  elements i n ft and the l i n e a r mappings (of fL  Ol  into i t s e l f ) .  D i , ... , D  m  Since every two operators i n  are commutative, and since ft i s a l g e b r a i c a l l y  closed, a l l the factor modules i n any composition series of the fi -module  Oi  are one-dimensional vector spaces  over ft . We decompose  Ol  into a d i r e c t sum  of d i r e c t l y indecomposable fi -submodules. D  l»  ••• i m D  are commutative, each  c h a r a c t e r i s t i c root  Xi  i n Oi  v  v  as a l i n e a r mapping of Ot  v  Di  2L fo  u  Then, since  has exactly one  , when we consider  °into i t s e l f , and there  Di  - 21 exists a non-zero all  i  and  Since  v .  u{ e 5  u  v  e 01v  such that  D^u  =  v  By condition ( 6 . 1 . 2 ) ,  u  A.J[V V U  i s a unit.  v  by ( 6 . 1 . 3 ) , and since ft i s a l g e b r a i c a l l y  closed, we may assume (6.1.4)  u£ = 1  for a l l v .  We s h a l l prove that a l l the  u  forms an elementary p-  v  Ol •  group with respect to the m u l t i p l i c a t i o n i n Lemma 6.2.  If  Djf = \ j f , \± e ft ,• f o r a l l i ,  and i f f ^ 0 , then there e x i s t s an f e Olp ,  v . v  Let  Djf - \±f  Since  Djf  v  e  f = JFf  v  , we have f  v  v  and since  V±  • \jf  Then •  D  for a l l i  v  f  and  and  v  since f  Djjfyf"" ) = 0 1  f v ^ e * • How1  Oly C\ 01^ • 0 , and thereare zero.  v  f e C%p .  Since  Thus there exists f ^ 0  i s assumed,  has only one c h a r a c t e r i s t i c root  <7Lp , we have  .  ^ 0 £ f ^ f o r two d i f f e r e n t indices  fore a l l but one of the such that  v  ZL i * " - H MA>  Then by (6.1.3) we have  ever, t h i s impossible  Olp  e0l  v  By an easy c a l c u l a t i o n we obtain  for a l l i .  in  Djf  f  u. • Then, by condition ( 6 . 1 . 2 ) ,  are u n i t s .  an  , where  v  i t follows that  Suppose that  and  such that  • ^ip • Proof.  from  Ol p  K±p  \± = Xip •  Now, f o r any two indices Di(p-v^jx) • (^iv + Xiu.)u U|j, v  v  and  for a l l i •  \i , we have Therefore,  since  - 22 u Uu. ^ 0  by ( 6 . 1 . 4 ) , i t f o l l o w s f r o m (6.2) t h a t  v  e x i s t s an OL p (6.2.1)  u u ^ e Olp  such t h a t  \-[ +  for a l l i •  p  From (6.2.1) i t f o l l o w s t h a t  aP - 1 .  the  u  Thus  Ol .  tion i n  w i t h some  V  (a - 1 ) P = 0 ,  Hence  • up .  U UJJ,  a e ft ,  Therefore a l l  O^. w i t h r e s p e c t t o t h e m u l t i p l i c a -  f o r m a group  v  for a l l i .  1  V  and t h e r e f o r e by (6.1.4) a • 1 .  D^u^^u" ) = 0  u U|j, - aup  Then by (6.1.3) we have  and we have  and s u c h t h a t  v  • \±  v  there  i s an e l e m e n t a r y p-group because o f  (6.1.4). We s h a l l show t h a t t h e r e e x i s t s o n l y one v  such t h a t  \i  u n i t y element o f  V  • 0  for a l l i •  Ol  then  If  Djf • 0  f - 1  1 e OIq  .  Suppose t h a t  Di(u ) - 0  for a l l i .  v  and hence  • 0  u  v  i s the  for a l l i .  f o r e by Lemma 6.2 t h e r e e x i s t s an i n d e x  0  index  There-  such t h a t  for a l l i •  By (6.1.3) we have  Then  u  v  e ft ,  = 1 , v = 0 .  We s h a l l prove t h a t  (71  =  v  U 0LQ v  for a l l v .  To do t h i s , we need some lemmas. Lemma 6.3. fi -submodule o f  OL .  characteristic root Proof. DJL  F o r any On  u^  v  \ i p , where  Since  0L  V  v , u^Oly  u- and , each u^u  i s an  has o n l y t h e one c h a r a c t e r i s t i c  v  i s an  Dj[ has t h e one = Up .  fi-submodule root  \j.  v  i n which  , there  - 23 exists a basis (6.3.1)  v i , ... , v  X vi , D  Djivx =  iv  o f OL  n  such that  v  = X  i V k  i v  v  +  k  ZL  a  i k s  v  ,  s  (1 < k < n) , with  ol±\hs e D  D  i^ H k) u  v  $ • Then  i ( u n v i ) - Uip, + \ i v ) u ^ v i ,  (^ijx  88  +  x  iv) M. k u  vi^Oly  Therefore  v  +  2 L aiksvs  » ( 1 < k £ n) .  i s an SI -submodule of Ot , and + K±  has the only c h a r a c t e r i s t i c root By (6.2.1) we have  v  on v^OL  v  .  + Xj_ = \±p , completing the v  proof• Lemma 6.4.  For any f i x e d  n , Ol - TL u „ 0 ^  i s a decomposition of OL into a d i r e c t sum of SI submodules, and Ujj0l  i s fi -isomorphic t o Ol p , where  v  « Up .  U^Uy  Proof.  F i r s t , since  u^  i s a u n i t , any element  f £ Ol can be written i n the form Let  g = Z. gv » where  g  v  e 0l  v  . Then  f = Z. % g v  Ujj,g e UpGlv . Moreover, i f Z_ u^g  v  then £ g  Ol = £  v  v  = 0 •  Since, however,  direct sum, we have  gv - 0  Thus we have proved that p o s i t i o n of  • 0  with  g  Ol • £ u^C^v u^  submodule, and the proof i s complete.  and e Ol  v  v  isa  f o r a l l v . Hence  . By (6.3) above  g e Ol .  f = u^g with  Up,g • 0 . v  i s a d i r e c t decomv  i s an 1^-  - 24 Since the number of d i r e c t components i n the Ol » 21 u^tfly  decomposition Ol - 21 Ol  i s the same as that of  and since each component i n 21 0l  v  d i r e c t l y indecomposable, by the Krull-Schmidt we see that each component i n 21 Uu,Ol  v  indecomposable, and that  u^^v  ±  v  s  theorem  i s also d i r e c t l y  i s fi -isomorphic t o  some Ol p . By comparing the c h a r a c t e r i s t i c roots, we see that  p  i s determined by the r e l a t i o n  U|jU • Up . v  (We have used Lemma 6.3 and the f a c t that for a l l  i  i f and only i f  Lemma 6 . 5 . for a l l i , then Proof. Djf = Z D j f for a l l i Since  If  58  cr • p) •  f e Ol  Djf e OIq  be such that  f e OIq . Let f = Z f  , where  v  6 OIq . Since  v  Mp  "k^  if v ^ 0 .  Djf  v  e Ol  v  f  v  e Ol  v  , we have  Then by (6.1.3) we have  v ^ 0 , however, we have  f  v  . Then Bf ±  f  v  v  - 0  eft.  = 0 . Thus  f = f o e OIq . Lemma 6 . 6 . such that every in  , then Proof.  every of  -ir  Di i n  I f -is  i s an Jl -submodule of Ol  Di has only the c h a r a c t e r i s t i c root zero C OIq . Since  0  i s the only c h a r a c t e r i s t i c o f  , there e x i s t s a basis  such that, f o r a l l i ,  wi, ... , w  n  -  (6.6.1) where  25 -  0 ^ - 0 , Diwfc • 21 aiks  W]_ 6 ft .  e  <*iks s , (k > 1) , w  ® • From (6.6.1) and ( 6 . 1 . 3 ) we have w^ e Ol q . Suppose that  Hence  DiWfc e 0 L Q  wi, ... , Wk-i e Ol o . Then ( 6 . 6 . 1 ) y i e l d s f o r a l l i • Then by Lemma 6 . 5 we have  w^ e Ol q . Pro-  ceeding by induction with respect to k , we have Wfc e Qlq  k . Therefore "fs £ OIq , as required.  for all  Now we are ready t o prove Lemma 6 . 4 ,  and  (71v  of 0 1 for  Let v i , ... , v  such that ( 6 . 3 . I ) holds.  v  UV(71Q  «  V  • By  as vector spaces over ft  UV01Q  have the same dimension.  Ol  We set  n  be a basis  w^ = \hr±  1 = 1, ... , n . Then by an easy c a l c u l a t i o n we see  that the basis  wj, ... , w  n  of U y l ^  Therefore by Lemma 6 . 6 we have  v  satisfies  (6.6.1).  v^^Oly C (%q . Hence  01m C u. 01q . Comparing the dimensions, we have v  (6.6.2)  - Uv01Q  0l  v  OIq  Lemma 6 . 7 . Proof. root of every w]_, ... , w  n  i s a subalgebra of  Oh.  0 i s the only c h a r a c t e r i s t i c  Since  Dj[ i n OIq , we may choose a basis of Qlo  wi e ft by ( 6 . 1 . 3 ) ,  such that (6.6.1) holds.  we may assume that  fore (6.7.1)  .  ww s  t  e QXq  Since  w^ = 1 . There-  if  s + t S 3 •  and w  s t w  t  Suppose that  such that  ~T.  (6.7.1) holds f o r a l l s  s + t < r . Now l e t  » where  f  v  e Ol  s + t = r ,  . From (6.6.1) i t follows  v  Djjw wt) = (DiW )w+; + w (D.jw ) e OIq f o r a l l  that  s  s  s  Therefore by Lemma (6.5) we have  6 OIq  w Wfc s  . Proceed-  ing by induction with respect t o r , we see that holds f o r a l l  s  i .  t  OIq  and t . Therefore  (6.7.1)  i s a subalge-  bra of Ol , as required. OIq depends on the system { D  Since  OIq = ^C*( 1>  we may write  ••• i m) •  D  D  W  M  equivalent to a given.orthogonal system tfl (Ei,  s h a l l show  - ^ E i , ... , E J -  that there e x i s t s an orthogonal system  such that  e  ... , D  lf  {D^,  D  m j  -  ... , E ) = ft . To do t h i s , i t w i l l b  0  m  s u f f i c i e n t t o show that we can always f i n d an orthogonal, ••• » m } equivalent to { ^ i , ... , D j  system  D  m  such that the dimension of ^o^ l» ••• » ®m) * D  than that of ^o^ l» ••• > m) D  D  greater than one.  Take a basis  OIq{Di,  ... , D ) ,  such that  W]_ e $  by (6.1.3) we may assume  m  whenever the l a t t e r i s W ] _ , ... , w  i s a unit.  2  By replacing  DJW  2  =  (6.1.3) we see that not a l l since  w  2  Since is  2  w P = 0 . Then 2  w  2  by 1 + w w  2  2  i . By  are zero, say  does not belong to ft . We set  i f w  i s a unit.  2  e ft f o r a l l ^  of  W]_ = 1 . I f w  i s not a u n i t , we can always assume that From (6.6.1) we have  n  (6.6.1) holds.  not a unit then by (6.1.3) we have 1 +w  less  s  x =  j#i ^ 0 ^£"'"W  2  ,  D  l  =  D  l» i D  1  D  D  1 i s an orthogonal  system equivalent to  DJX • 1 , D±x = 0 f o r i  ... , D } such that m  We set Di,  f o r i + 1 . Then  Pl i " Pi l  =  1^1> • • • > {D ,  27 -  / 1 .  Di = x D j , D| = D i f o r i £ .1 . Then  ... , D^ | is. orthogonal  and equivalent to  {Di, ... , JJ^J and hence to { D i , ... , D j , since  x  m  i s a unit.  We have  (6.6.1)  D J x = x ^ 0 , where  (6.8.2)  Di - 21 i j J c  perly contained i n Qi Q ( D I ,  (6.8.3)  of ^ o ( i »  D^vi = 0 , D i v  f o r a l l i , where  we have  D  D  *  D  s  pro-  m  a  such that  i k s  v  ,  s  (k > l j ,  From,(6.8.3) and ( 6 . 1 . 3 )  eft.  v i e #lo( l» ••• > m) • D  f  from ( 6 . 8 . 2 )  D  ... , D ) . Take a,basis  = H  k s  , m) •  ^ o ^ l » ••• > m)  vj_ eft and hence  Suppose that  (6.8.4)  cti  D  D  D  k  ••• > m) ,  c i j e #?o( l,  ••• > m)  D  r  D  » where  D  We s h a l l show that  v i , ... , v  x e ^o( l>  D  y i , ... , Vk„i e ^ o ( l » ••• » m) • Then D  D  and ( 6 . 8 . 3 ) we have  i V k  =  21  Z ij jks s • c  a  v  s<<k q-i  Since  Cij , a j  k s  » and v , belong t o  C7ZQ( l, D  s  ••• , D ) , m  by Lemma 6 . 7 we see that the right-hand side of ( 6 . 6 . 4 ) . belongs to Olo[TX±  t  ... , D ) m  from Lemma 6 . 5 i t follows that  f o r a l l i .. Therefore  v  e 67l ( l> D  k  0  ••• » m) • D  Proceeding by induction with respect to k , we have  28 vfc e ^ o ( l >  ••• » m)  D  ^•o( l>  ••• * m)  D  9 ^o( l»  D  Ol (B{,  that  D  D  o  D-Jf e OIq  implies  OIq  l i n e a r mapping of 0  > m)  D  ... , D^) - 0 l ( l >  0  f e OIq  for a l l k .  D  Therefore  ••• » m) D  into i t s e l f .  ••• » m)  01q{Vi,  •••  0to( if  ••• » m)  D  t Dm)  , and  D  S  i  n  c  e  as a  By the d e f i n i t i o n of Dj[  O^q  in  .  Thus  properly contained i n  i s  D  •  D{  i s the only c h a r a c t e r i s t i c root of  ^0( i>  Suppose  = <^0  , we can regard  However, t h i s contradicts to ( 6 . 3 . 1 ) . D  proved.  i s  i s  l  e  s  hence the s  t  h  a  n  t  h  a  t  dimension of o f  ^0^ 1» D  ••• » m) D  Repeating the above process, we obtain an orthogonal  system  f l>  • ••  E  ••• > m)"  equivalent to the given system "^D^,  E  OtQiE^,  such that  ... , E )  i s one-dimensional.  m  (01;  Since the algebras  D^,  ... ,  D) m  defined by equivalent systems are the same, we may that  Ol o  ^ZQ  $ »  =  ^o^ l>  ••• » m)  D  an(  (6.8.5)  D  » i v  u  for a l l i  one-dimensional. ,Then  i s  and  v .  D  u  e  Mv v u  ,  From (6.3.5) we see that  the group algebra of the elementary p-group by a l l u  .  v  » m) D  v  ^1>  i  s  i v  ••• » #m  is  <7J. formed  isomorphic to a generalized Witt  We define mapping  #i(u ) = \  Ol  We s h a l l show that i f ( 6 . 8 . 5 ) holds, then  jl(0l; algebra.  suppose  * from (6.6.2) we have 0? - YL® v  .  &±  of  Cfj. into  ft  Then from (6.2.1) i t follows that a r e  D  ho.momorphisms of  Ol  into the  by  m  -  29 -  ( 2 . 0 . 1 ) and  additive group of ft . We s h a l l show that (2.0.2) ^l( cr)  ^m( cr) •  u  i  u  , and hence  fied.  ^1, ••• »  are s a t i s f i e d by  cr » 0 , u  ff  •  0  T  h  e  •  Suppose  M.cr • 0 *"°  n  r  all  = 1 • Thus (2.0.1) i s s a t i s -  Suppose now that  ••• m ^ m  +  + a  s =  0 • Then  J^LctiXiv = 0 f o r a l l v , and hence from ( 6 . 6 . 5 ) we i have  a i D i + ... + a D m  a i = ...=.a = 0 . m  satisfied. &(0L\ Witt  m  = 0 .  Then (1.0.2) y i e l d s  Thus ( 2 . 0 . 2 )  i s also proved to be  Therefore by the r e s u l t i n section 2  D]_, ... , D ) m  i s isomorphic to a generalized  algebra. Thus we have proved the following: Theorem 6 . 6 . Suppose that ft i s a l g e b r a i c a l l y  closed and that the system -[Di, ... , D ]• i s orthogonal. m  Then the algebra  £, {Ol; D i , ... , D ) m  i s isomorphic to  a generalized Witt algebra i f and only i f the following conditions (6.1.2) and ( 6 . 1 . 3 ) hold: (6.1.2)  I f f e Ol is such that X i e ft , f o r a l l  Djf - X±f , where  i , then  f = 0 or f i s  a unit i n Ol . (6.1.3)  D]f - ... • D f m 0 m  In p a r t i c u l a r , i f an algebra  implies  f e ft .  of the form  •£{01; D]_, ... , D ) , where { D i , ... , D } m  necessarily orthogonal,  m  i s not  over an a l g e b r a i c a l l y closed  f i e l d ft i s simple, then  -£  i s isomorphic to a  -  30 -  generalized Witt algebra and of an elementary p-group.  Ol> t o the group algebra  - 31 7«  Nilpotent systems ( 1 ) . A system  ••• » ^mj  w i l l be c a l l e d nilpotent i f there exists  a .positive integer  k  I f the ground f i e l d  ft  •^Di, ... , D | i s  D^k  = ... = D k  = 0 .  m  i s a l g e b r a i c a l l y closed then  nilpotent i f and only i f  m  OloiB].,  such that  ••• , D )  Ol .  =  m  In the preceeding section we  have proved that i f ft i s a l g e b r a i c a l l y closed then any (01;  simple algebra of the form  D^,  ... , D )  defined by an orthogonal system f o r which OIQ  The case  » Ot  extreme cases.  and the case Oto  Now we  Theorem 7.1. closed.  ft  = ft •  - ft are  two  Suppose that  ft  i s algebraically  £^1,  , D} m  satis-  (6.1.3) i s equivalent to a nilpotent  orthogonal system. algebra over  OIQ  s h a l l prove the f o l l o w i n g :  Then any orthogonal system  f y i n g (6.1.2) and  can be  m  In p a r t i c u l a r , any generalized Witt  can be written i n the form  "£.(0i; Dj_, ... , D ) m  Ol  , where  of an elementary p-group and where  i s the group algebra {®l  t  •••  > ®mj  is  a nilpotent orthogonal system. Proof.  We s h a l l use the notations employed i n  the preceeding section.  Because of the remark i n the f i r s t  paragraph of t h i s section, i t i s s u f f i c i e n t to prove the following:  I f { ^ l , ••• , D  s a t i s f y i n g (6.1.2) and  j- i s an orthogonal system  (6.1.3) and i f  OIQ = Ol Q ( 1 > ••• t ®m) ^ Ot d  m  then there e x i s t s an  - 32 ~ orthogonal system { li  ••• » Dm}  D  in  - ^o^ i»  ••• » m)  D  & ~ Z- u '^o  D  u  fore, i f OIQ ^ Ol Xiv t 0 n  D  i  x  f o r some  B  i s properly contained y ( 6 . 6 . 2 ) we have  » where  u  i • 1 .  i r 1 i  o r  0  for i ^ 1 .  i ^ 1 .  for  Then  such that  hand, since Thus  o  v  .  Then  D^x = X , and  D[ • Dj  , and hence to  D^x = 1 , D|X = 0 f o r  with  Since  v  i s an orthogonal  ... ,  x e Ol Q  x = X'^ u  0l ^ ^0 •  1  m  m  Therefore  D{ - x - ^  {D{, ... , D }  ... , D | , such that  i ^ 1 .  = D]_ ,  x = \,„u  d  Then  i s an orthogonal system  We set  system equivalent to |Di,  n  { D £ , ... , M  -  v ^ 0 . -I  a  There-  We set  .  equivalent to {D'I, ... , D J DJX  e ft .  then there exists a  ^  D  i s a unit and  ^iv v  88  i , say  ~ Hv £ ~ ^ i v l D  •  D  > i v  v  j- equivalent to  OIQ  such that  1  OIQ  {D-[, ... ,  by Lemma 6 . 5 .  On the other  v•/>0 , we have  x i OIQ .  x e OIQ , from the above con-  s t r u c t i o n we have (7.1.1)  = I^eijDj- ,  j)[  C^.  eOlQ  .  Using (7.1.1) and proceeding the same way as i n the preceeding section, we see that in  OIQ  •  OIQ  i s properly contained  •  Remark.  A derivation  be c a l l e d normal i f Df = 0  D  of  implies  from the above proof that i f Di  Ol  over  f e ft .  ft  will  It i s clear  i s normal then  - 33 ^ 0  Xlv  normal.  v^ 0  f o r any  and hence  Therefore i f {D^,  ,D  =..x-lD]_ m  i s also  j. i s an orthogonal  system s a t i s f y i n g (6.1.2) and (6.1.3) and i f E>1 i s normal then t h e r e e x i s t s a n i l p o t e n t orthogonal system {D-[, ... ,  } equivalent t o  D{ i s normal.  ... , D } such that m  This f a c t w i l l be used l a t e r i n s e c t i o n 9«  The above r e s u l t may be r e f i n e d i f i t i s combined with the f o l l o w i n g : Theorem 7.2.  ••• »m } s a t i s f i e s (6.1.3) then t h e r e e x i s t  { l» D  D  X ] _ , ... , x where Ol  I f a n i l p o t e n t orthogonal system  n  x"' ...  e Ol such that the elements  x^  ,  < p , xQ = 1 , xj? eft, form a b a s i s of  0 t>  over ft and such that  D-jxi e f t ,  DjXfc e ft(x]_, ... , x j j . i ) , the subalgebra of  Ol gen-  erated by  i and  xj_,. ... , x _]_ over ft , f o r a l l k  k > 1 . I f , i n p a r t i c u l a r , ft i s p e r f e c t i n the sense t h a t every element i n ft i s a p-th power of an element i n ft , then  x i , ... , x  = ... = xP = 1  or  n  may be taken such t h a t e i t h e r  = ... =  =0 .  The proof f o l l o w s e a s i l y from the f o l l o w i n g two lemmas. Lemma 7.3.  Suppose t h a t { D  a n i l p o t e n t orthogonal system.  l f  ... , D } i s m  I f V]_, ... , v e Ol r  - 34 -  are l i n e a r l y independent over ft , i f D^VT_ = 0 , and if  DjV  i s a l i n e a r combination of  k  for a l l i v e 01  and  k  1 , then there exists an element  D^v  •••  J  v  V ] _ , ... , v  i s a l i n e a r combination of Ol  f o r a l l i , provided that l»  VT_, ... , v  which i s not a l i n e a r combination of  such that  v  v^, ... , v^...^  r  i s not spanned by  r • Proof.  spanned  Denote by  ^  v i , ... , v^ .  and each factor space  k  the fi -subspace of  Then  $ k/^k-1  8?2 O • • <£ # r  *  s  r  one-dimensional.  Since any increasing sequence of fi -subspaces of an (l-space of  (Jl can be refined into a composition s e r i e s  Ol , there e x i s t s a composition series o  < S # r £ $ i C ... of  Ol .  r +  Since { D , X  i s nilpotent and orthogonal, we have all Then  i .  Take an element  Dj_v e fl£  v  for a l l  r  in ^  r+  ^iRr+l  generated by  ft(xi_,  £ U2r  f° V  i , as required.  ... , x^)  X]_, ... , x  M  but not i n 7$L  -±  In the following i f x^, ... , x s h a l l denote by  ... ., ,D |  k  k  e Ol , we  the subalgebra o f  over ft .  The ground f i e l d  ft i s not necessarily a l g e b r a i c a l l y closed. Lemma 7»4»  Suppose that  {D^,  ... , D }M  is  a nilpotent orthogonal system s a t i s f y i n g (6.1.3), and that  X]_, ... , x  r  e  Ol are such that the elements  r  . . . x Ur  x"-'  0 ^ \>± < p , x9 - 1 , are l i n e a r l y  , where  independent over ft and such t h a t D  i k  x  r + l & ®( l> ... , x )  x  e  $  ^ l>  ••• > k - l )  x  x  f o r  r  x  ^ ( l i ••.• » r )  e  x  x  x * ... x^*'  ^ '  a n d  f°  r  I f  •  a l l i » then the elements  0 ^ v± < p , x? = 1 , are l i n e a r l y  , where  u  1  i s such that.  x  ^i r+l  a 1 1  independent over ft . Proof.  0 < v-^ < p , w i l l be c a l l e d a monomial and the  where  r  number  w «= w(y) =  the monomial  y •  its.weight. y  .  w  An element of the form  + v p + ... • + v p " " l r  A monomial i s uniquely determined by  A monomial of weight  I f f - a yo + a 0  w  w i l l be denoted by  + ... + a y  i y i  a^.. / 0 , then t h e weight w(f) = w .  the weight of  r  2  w  w(f)  of  , where  w  f  a  ±  e ft ,  i s defined by  I t f o l l o w s e a s i l y from our assumption t h a t  w(D f) < w(f) i  if 0^ f e  for a l l i  ft(x , x  ... , x ) . r  Any l i n e a r combination of the elements x ' ... x  can be w r i t t e n i n t h e form ' 1-1 f i x + i + ... + f p - i x with T  T+  fo  +  r  fQ, ... , f p - 1 e  ft(x].,-...  , x ) . We s h a l l prove by r  i n d u c t i o n w i t h respect t o f o , ... , f k (7.4.1)  e  that i f  ••• > r)» 0 ^ k < p  ^( l» x  f0 + f i x  k x  i + ... +-fk + "x  r +  .. .implies  r  0  1  f 0 = ... = f k = 0 .  , then  - 36 If  k • 0  then (7.4.1) i s c l e a r .  holds f o r a l l k < v  Suppose that (7.4.1)  but not f o r  k = v . Let k *» v , / f 0 + f i x + i + ... + f k r + l • 0 , f k r 0 , and. l e t f k k  x  r  be of minimal weight with respect to t h i s property. For any  1 , we have.  (7.4.2) Since  (D-jf^x^ + ((kD w(DjLf ) < w(fk) k  Then (6.1.3) y i e l d s assume  for a l l Since x  i X r + 1  )f  k  D^k • 0  , we have  f k e ft .  + D i f k - l ) * ^ , ' + ... - 0 .  Since  f k t 0 , we may  f k = 1 . Then (7.4.2) y i e l d s  i , and hence by (6.1.3)  for a l l i .  ^(kXp+i + f ^ i ) = 0  k x i + fk-1 e f t . r +  a < k < p , t h i s contradicts the assumption that  r + l i ^( l» «.. ». r ) » Thus (7*4.1) i s proved f o r a l l x  x  k , completing the proof of the Lemma. . An algebra if  -JLYL *  s  s£  over ft i s c a l l e d normal simple  simple f o r any extension  K  of ft . .  is  normal simple i f -S£K i s simple f o r any a l g e b r a i c a l l y closed extension  K  of ft . I t i s known [ 4 j that the  generalized Witt algebras are normal simple i f p > 2 if  or  p = 2 ,m > 1 . Theorem 7.5.  p • 2 ,m> 1 .  Suppose that  I f J % , ... , D  orthogonal system then  -£ =  or that  | i s a nilpotent  (01 ; D]_, ... , D ) i s  simple i f and only i f the algebra  •jC i s a f i e l d , while  m  p > 2  m  <£T' of constants of  - £ i s normal simple i f and only  if  £T= ft .  We need a general remark. algebra over ft , and ft ^  a subfield of ft •  Since  -£J over ft* . The m u l t i p l i c a t i o n  i£ i s b i l i n e a r as a m u l t i p l i c a t i o n i n  we have  (ax)y = x(ay) = a(xy)  Therefore  s£  for a e f t  i s an algebra over ft  T  necessarily f i n i t e dimensional. -s£  be an  i s a vector space over ft , s£ can be regarded as  a vector space in  T  Let s£  1  1  T  xy  , and  , x , y e f t .  , although not  I f { u^ J i s a basis o f  over ft , and i f | a j J i s a basis of ft over ft , 1  then the set {aju^}  i s a basis of -£  We r e f e r t o the algebra algebra over ft  T n  .  -£J as  n  £  f  over ft • 1  regarded as an  Lemma 7*6 below i s probably well  known, and i n any event the proof may be r e a d i l y supplied by the reader. Lemma 7«6.  £  i s simple i f and only i f £  i s simple. Lemma 7»7* over ft  1  , then  Since ft i s algebraic over ft  exists an extension  af  a . The set  , where  f e £fc  (af)g = a(fg)  > 1  •£} i s not normal simple.  Proof.  divisor  I f ft has a f i n i t e degree  K  1  , there  of ft such that ftg has a zero 1  $  of a l l elements of the form  i s an i d e a l of  f o r a l l f , ge s C j .  , since J i s d i f f e r e n t from  .- 38 -  a ^ 0 . We s h a l l show that  zero, since  ax = 0 i s a sub-  The set of a l l x eftj^such that  1 , so l e t a i , ..* a  algebra of ftg o f dimension  be a basis of t h i s subalgebra over a  r+l» ••• > s a  of u  %  l»  over  such that  a^, ... , a  i s a basis  s  r < s . Let  be a basis of •£ over ft .. Then  u  ... , s, i = 1, ... , n , form a basis of  a J U J L , j « 1,  K . Then { a a j U ^ j  i s a basis o f  K , and aa^ = ... = a a = 0 , so that  CJ over  jf £  r  fore  r  K .. Take  a ^ 0 , we have  K . Since  >n  over  %  6  ^7 ^ S^K •  -sCg i s not simple, and therefore  • Therei s not  simple, as required. Consider the algebra whose.algebra  &  of constants i s a f i e l d .  a subfield of the algebra Ot an algebra  Ol  -£.(01 J ^ l ,  over  fif  .  "3^ (01 ; Di, ... , D ) m  i f and only i f that  -£,(Ol]  "D i  ••• , m)  regarded  Therefore by Lemma 7«6  Since  $C i s  Ol  as  D^c = 0 f o r a l l  Since  it (01; ^ l ,  m  , we may consider  c e jfC , D i defines a d e r i v a t i o n e a s i l y seen that  .... , D )  D  Ol . I t i s  of i  s t  h  e  algebra  as an algebra over ft . D]_, ... , D ) m  £(01 ; D^, ... , D ) m  i s simple  i s simple, provided  £C i s a f i e l d . Lemma 7 . 8 . Let $C be the algebra of constants  of  -£_(0l\ D i , ... , D ) , and K m  an extension of ft .  Then the algebra o f constants o f ^£ (Otj^; D i , ... , D ) m  -  is  39  -  • Proof. and  Let uj_, ... , u  U]_, ... , u , ... , u r  f = 21 i i  Suppose  a  we have  e K ., belongs t o the  (#K>  algebra of constants of s h a l l show that  a basis of Ot .  n  » where  u  c t i = ... = a r +  1 » ••• > m) •  d  D  r+  W  e  = 0 . For any i ,  n  a i D j U ] _ + ... + ct DiU r+  be a basis of  r  n  n  = 0 • If  r+l>  ••• » n  exist  | 3 + i , ... , j 3 e ft , not a l l zero, such that  a  a  n  r  P r+l i r+l D  D  i j u  E  were not a l l zero, then there would  u  +  +  Pn i n " 0 D  u  o  j3 + i u  Ol • Then we have  a contradiction. Thus  f  r  r  r +  a l l i » since  i + ... + |3 u n  n  e  ,  ct ]_ = ... = ct = 0 . • Therefore r+  n  the algebra of constants f o r -£{0l%;  D]_, ... , D ) i s m  K • Proof of 7.5.  Suppose that  Then, by Lemma 3.2, <£T i s a f i e l d . i s normal simple.  Let K  •£  i s simple.  Suppose that  be an a l g e b r a i c a l l y closed  extension of ft . By Lemma 7.6 the algebra of constants of  is  K • Since  Conversely consider the case  C^K  i  s  a  field,  suppose that ft  , and l e t  8l- $ •  is a field. K  First  be an a l g e b r a i -  c a l l y closed extension of ft . Then by Lemma 7.8 the algebra of constants of  is K .  a l g e b r a i c a l l y closed, and since  Since  j ^ i , ... , D  K is m  f is  K  nilpotent and orthogonal, by Theorem 6.8 generalized Witt algebra. fore  Hence  i s normal simple.  stants of  , D )  i s always  i s normal simple, and hence  m  m  i s simple.  Corollary 7 . 9 . OL over  group algebra  The derivation algebra of the ft  p  i s simple i f and only i f  i s an elementary abelian group, provided that the  order of  is > 2 . Proof.  Suppose that  p-group with independent  Cfy i s an  Ol = ft(xi, ... , x )  ([2],  p. 2 1 7 ) that  - -£(01;  for a l l i  <rf = ft .  Since  .  m  £?, *' fe^  be the algebra of constants f o r _9_f - 0  , x  and i t i s e a s i l y seen  m  3r(0l)  elementary  xi, ...  generators  Then  Hence  0£  of an abelian group  whose order i s d i v i s i b l e by  Then  There-  Since the algebra of conm  •£.{01; Di, ... , D )  0^  i s simple.  -L(0\,\ Di, ... , D )  •£.(01', D]_, . . .  is a  | ^  J  , and l e t  1  , ... ,  ^  f e  c l e a r l y implies that  L d*l  ^  v  f e ft .  i s a nilpotent  3x 3 m  orthogonal system, the s i m p l i c i t y of  3r(dl)  follows  from Theorem 7 . 5 . Suppose now that x ^ 1  (fy  contains an element  of order r e l a t i v e l y prime to  , p) • 1 .  Then y x f ~ D x = 0 1  p:  x& = 1  and hence  , where  Dx = 0  for  - 41 any  D s  .  J - {(x - 1)D|D and that  0 /  It i s e a s i l y seen that the set  e &(Ol)j  ?  ^  MOl)  forms an i d e a l of .  ^-(01)  J5H0L)  Hence  i s not  simple. The following theorem w i l l be used l a t e r . Theorem 7*10, Ol a  ... , x ,  Let  y i , ... , y )  r  be the group algebra  s  of an elementary p-group with independent generators x i , ... , x ,  y i , ... , y  r  .  s  Suppose that  D^,  ... ,  D  M  i s an orthogonal system such that D  i  - i:L%. a  - uyi-^ a  where k .  +  ...  •••  +  +  a  + a  isy .^ s  i r - ^  s  >  a^k e ft(xi, ... , xk-i) , o^k  e  ®  f°  all i  r  p = 2, m = 1, •£ (01 ; D]_, ... , D )  Unless  M  simple i f and only i f the following two  and  i s normal  conditions are  satisfied: (7.10.1)  For any  k , there does not  f e ft(xi, ... , xk-l)  exist  such that  aik = D i f  . for a l l i . (7.10.2)  I f integers  p.^, ... , u. are such that g  s  ^1 ^ikM-k (mod  p ).  =  0  f°  r  a l l i , then  m = ...  = n  s  « o  - 42 -  Proof.  We may assume at the outset that ft By Theorem 6.S,  i s a l g e b r a i c a l l y closed. £ = H(Ol\ J )  L F  ... , D ) . i s simple i f and only i f m  (6.1.2) and ( 6 . 1 . 3 ) are s a t i s f i e d . i s simple.  Suppose that  DjXk = aik , (7.10.1) follows from  Since  s  (6.1.3) .  for a l l i .  We set  for a l l i .  Therefore  M-i » •.. = u-s = 0 •  r  ( 6 . 1 . 3 ) gives  Then  D^f - 0  f e ft .  Hence  Thus we have ( 7 . 1 0 . 2 ) . suppose that (7.10.1) and' (7*10.2)  f e ft(x]_, ... , x )  .  r  If  r = 1  then t h i s i s  DjX]_ = a i i e ft and not a l l  a ^ i are zero.  We s h a l l proceed by induction with respect to r >1  Suppose that  0  =  F i r s t we s h a l l prove ( 6 . 1 . 3 ) f o r the case  are s a t i s f i e d .  c l e a r , since  < p, ^LLaikM-k  f = y^* ... y * .  Conversely,  when  0 ?>  Suppose now that  r .  and that (6.1.3) i s true i f  f e ft(x]_, ... , x _ i ) .  Suppose now that  r  f = b o + b]_x + «»• + b k £ » where x  r  bo,  ...  , bjj E ft(xi, .... , x « i ) , bk ^ 0 , and that r  D±f - 0  for a l l  (7.10.3)  i .  Djf - (Djb  Then + bT_ai ) + ... + ( D j b _ i  0  r  k  + kbkairix^" + (DjbjJxJ; = 0 . 1  D^bk - 0  Therefore  bk E ft .  assumption gives D  i k-l D  +  k b  k ir a  =  h = (kbk)"^^!  0  for a l l  f  o  r  9  1  1  1  i .  Then the induction  From (7.10.3) we have •  I  « then we have  f  k  ^  0  we set  h s ft(xi, ... , x  r -  i)  - 43 and  a i + Dih = 0  f o r a l l i , a contradiction. There-  r  fore  k = 0 .  Thus  f e ft(xi, ... , x _ i ) and the i n r  duction assumption gives  f s ft .  Thus (6.1.3) i s  proved f o r f e ft(xi, ... , x ) .  To prove (6.1.3) f o r  r  the general case, suppose that .f f  *2If "  f-t  i  .  7^  u  D -f « 0  for a l l  n  ±  1  Then, since  Dif = JT (D f  . . . yu  ±  s  aivuujyA ' ... y/ * = 0 , we have 1  -u  Hi,  where  , Xy.) , and that  e $ ( x i . ...  /*s  + f D i f  ... 7 ^  + M  f  ... , u.  i s t i c root of X. ctikM-k ~ 0 f  Therefore  -  a  i .  Di f°  in  r  f 0 .  tt  T ik^k  M  and  s  4  a  Since ftfx^,  l-"-  1  a n <  0  for a l l  0  i s the only character-  ... , x ) , we have r  *  f^l>  > l^s  Then (7.10.2) y i e l d s  f e ft(xi, ... , x )  u  c  n  "that  u- - ... - n_ « 0 . n  and hence  r  s  f e ft .  Thus  (6.1.3) i s proved. We s h a l l prove next that (6.1.2) holds. g = T~ g  that g  y ^ ' ... y^ s  +  r  Then we have  <I itfk -  and  ,  a  i .  •  0  (7.10.4) for a l l i  f o r  Therefore, as before, ' s  g  , where  e ft(xi, ... , x ) , and that  \,- e <£ f o r a l l i .  a  ikl 'k J  if g / 0 / g  x  i•  Suppose  Dig = \ i g with  D,-g a 1 1  M-l, ... , H  s  g e f t and >/••• /*•  0  f  0 . If  , then from (7.10..4) we have  - 44 - n£)  Z- ik^k a  u.^ & ^ 0 S  - 0  ( od P )  < p , we have  g = 0  a ^ 0 , since also  proved.  or  i .  for a l l k .  m  g •» a y ' * ' . . . y ^ Then  for a l l  «  with  s  g  Then (7.10.2) g i v e s  Since .  0 => 1% < p ,  Therefore  a e ft and some  u^, ... , u- . s  i s a u n i t a c c o r d i n g as  y]_, . . .  , y  s  are u n i t s .  a =» 0  or  Thus (6.1.2) i s  - 45 8.  Nilpotent systems  I f the D-dimension  (2). The case  m = 1 .  m = 1 , then we can s t i l l further  sharpen the r e s u l t s obtained i n the preceding section. In p a r t i c u l a r , i t w i l l be proved that any generalized •£ ( O l ; D)  Witt algebra of the form  over an algebrai-  c a l l y closed f i e l d i s uniquely determined by i t s dimension. the  The r e s u l t obtained here w i l l be the basis of  argument i n the next section. Consider the group algebra Ol = ft(xi, ... , x ) r  of an elementary p-group with independent generators X]_,  ... , x  and the derivation  r  (8.0.1) Then  D = £  D  of  Ol  defined by  - <.,  +  i s nilpotent.  of weight  D  Let  y  4  •  = x^' ... x^ be a monomial r  w  w = v-j_ + v£p + ... + v p ~ l r  r  .  Then  Dy  w  is  e a s i l y seen to be a l i n e a r combination of monomials of weight  <w  .  Since  maximal weight i n  x  ftfx^,  f e ft(x-L, ... , x ) k  ... x  i s the monomial of  ... , x^) , there does not exist  such that  Df = x  Jb-I  ... x  Jb-I  Therefore from Theorem 7.10 i t follows that (6.0.2)  Df - 0  Hence i f p > 2  implies  f e ft .  then the algebra  -£_(0\; D)  i s normal  simple. Remark.  Jacobson ( [ 3 j , Theorem 4- ) proved the  existence of a d e r i v a t i o n  D  of  Ol  s a t i s f y i n g (6.0.1)  - 46 -  x  under the condition that ft i s i n f i n i t e .  However, the  above argument shows that such a d e r i v a t i o n exists f o r any f i e l d ft . This f a c t w i l l be further generalized i n the next section. Lemma 8.1. then  Df  i s of weight Proof.  mail of weight  w £1  w - 1 .  We may assume that  f = y  w . Suppose that' Dy  < w - 1 . Then  Dy^, ... , Dy  i s a mono-  w  i s of weight  w  are l i n e a r combinations  w  YQt ••• > w-2 » * bence there exist  of a  I f f e OL i s of weight  y  l>  »w a  X- i 7 i a  D  u±Y±  =  0  e  an<  ^ > which are not a l l zero, such that  • Hence we have  D( Ji^±Y±)  e ft , and a-^ » ... = a  Therefore  Dy  w  i s of weight  w  " 0 ,  = 0 , a contradiction.  w - 1.  As an immediate consequence of Lemma (8.1) we have: Lemma 8.2. I f 0 ^ w < p exists an element  r  f e OL such that  - 1  then there  Df = y  w  .  Now we consider an a r b i t r a r y algebra of D-dimension  m = 1 , where  t i o n such that  Df = 0  that ft i s perfect.  D  implies  I f (%  ;£ (01; D)  i s a nilpotent derivaf e ft . We s h a l l assume  i s of dimension > 1  we can e a s i l y f i n d an element  x e OL such that  xP = 1 . Then  1  1, x, ... , xP"--  then  Dx = 1 ,  are l i n e a r l y independent.  -  47 -  Suppose we have already found  x^, ... , x  e Ot  k  satis-  f y i n g (8.3.I) - (8.3.3) below: (8.3.1)  x? = 1  (8.3.2)  The elements 0 ^  i = 1,  for a l l  x  ... , k ;  ... x * , where  < p , x^ = 1 , are l i n e a r l y independent  over ft ; (8.3.3)  Dx^ = 1, Dx  Ot  If  2  h-\ = x^ , ... , Dx  i s not spanned by the elements  then by Lemma 7»3 there exists Dv e ft(xi, ... , x ) while k  set g  Dv = ax  ... x  f-l k  j>-l ... x  =  x  ... x !^ ,  v e Ol such that  v i ft(xi, ... , x ) . We k  + g , where  a e ft and where  i s a l i n e a r combination of monomials of weight  < p  r  - 1 . By Lemma 8.2 there exists  such that  Df = g . Then  a ^ 0 , otherwise  Hence  f e ft(xi, ... , x ) k  D(v - f ) • ax^"'... x^"' . D(v - f ) - 0 , v - f E ft , and  v e ft( x  l»  ••• > k ) • Since ft i s perfect, there exists x  y3 e ft that wex^+i a~l(v f )existence + y3 s a tof i s f i exs i x = 1 such . Thus have=proved the x* -1 . satisfying k +  (8.3.4)  Dx x  k + 1  = x  k+l t  ... x ^  r  $ ( x  l»  ,x ^  = 1 ,  ••• » *k> •  Then by Lemma 7.4 the elements  x ' ... x  are  l i n e a r l y independent over ft . Repeating the above  — 46 —  process we obtain  x j , ... , x  elements  x > ... x "  basis of  Ol  Let  i  such that the  0 S VA < p . form a x  r ,'where  u  e Ol  r  r  *  and such that ( 6 . 3 . 4 ) holds f o r a l l  k .  OJ. be the multiplicative, group generated by  x^, ... , Xj, .  Then  group algebra of  Ol - ^( l»  ••• > r )  x  x  0^ over ft , and  D  i s  t  n  e  can be written  i n the form ( 6 . 0 . 1 ) . By a s i m i l a r argument we may choose  x^, ... , x  r  s a t i s f y i n g xP » ... xP = 0 instead of p x = ... = x p = 1 . Thus we have proved r  1  r  Theorem 6.3» field.  Suppose'that  Then any algebra  potent d e r i v a t i o n  D  .ft i s a perfect  £ (01 ; D)  such that  defined by a n i l -  Df = 0  implies  f e ft  i s isomorphic to an algebra (6.3.5)  r  +  where  ^  •£ (ft(xi, ... , x )  x*" '  X;L, ... , x  x^ ' ... x  r  ... x>"'  r-i  r  , 0 >  A  ') ,  7xr  •••  l  < p , form a basis of  ft(xi,  ... , x ) r  ^ x? = ... = xP = 1 1 r  We may take  +  are such that the elements  and such that (6.3.6)  + x  x^, ... , x  r  .  satisfying  xP = ... = xP = 0  instead of ( 6 . 3 . 6 ) . Corollary 6.4^.  Suppose that  ft  i s algebraically  - 49  closed.  Then any generalized Witt algebra of  D-dimension  1  i s completely determined by i t s dimen-  sion and can be written i n the form ( 8 . 3 . 5 ) . Ol  •» ft(xi, ... , x ) r  •£ (Ol ; DQ_, ... , D ) R  to the algebra  If  then any generalized Witt algebra of D-dimension  £ (OX ;  , ...  r  , ^  i s isomorphic ) .  The proof of the second part of C o r o l l a r y 6.4 i s as follows: Then the matrix { l, D  ••• > m} D  Let ( ±jl a  jXj ^  , where  i s non-singular.  i s equivalent to | x  and hence to  £{0LI  D-^ = 2_  » ••• ,  }  *  x  ^  R  Therefore , ... , x  r  ^  ]  Therefore  , ...  D L ... , D ) -  a^j.e.ft .  , X  ) .  Thus the problem of c l a s s i f i c a t i o n of the generalized Witt algebras i s completely solved f o r the two extreme cases:  m = 1  and  m = r .  The author has been  unable to solve t h i s problem i n general.  -  9tion . D  50 -  Normal systems* Ol  of  We r e c a l l that a deriva-  i s c a l l e d normal i f Df = 0  f e ft , where ft i s the ground f i e l d . [D^,  ... , D  normal.  m  implies  A system  | w i l l be c a l l e d normal i f some  D^  is  A system { D]^ .... , D J w i l l be c a l l e d m  p r i n c i p a l i f Ol = ^ ( ^ i ,  ... , x )  i s the group algebra  n  of an elementary p-group with independent generators x  1 ?  all  ... , x i  and  , and i f DjXj = i j j a  n  x  j , where the matrix  w  i  t  n  a  i j  e  *  f  o  r  ( i j ) satisfies a  (9.0.1) and (9.0.2) below: (9.0.1)  i f integers  k^,.... , k  a-ij-kj = 0 k^ s. ... =. k (9.0.2)  for a l l n  are such that  i , then,  = 0 (mod p ) ; (a )  the rank of  n  1 s  is m .  P r i n c i p a l systems, which were used i n section 2 to define generalized Witt algebras, are always orthogonal and  the matrix  ( T^j)  i s non-singular.  Theorem 9.1. field.  Suppose that  Then f o r any p r i n c i p a l system  there exists a normal scalar-equivalent to  ft  i s an i n f i n i t e , m D  )  »  m. J  Proof.  Let  ... , $  over ft , and consider the ^(ki,  ... , & )  , where the  a  0 S kj < p .  indeterminates  l i n e a r forms  n  li ijkj  a  n  integers and  p  be  m  Since these  d i s t i n c t because of ( 9 . 0 . 1 ) , and since  p  kj  forms are  n  ft  are  is infinite,  by the theory of s p e c i a l i z a t i o n there exist #1>  , 7m  $  e  such that the  atk^, ... , k ) = 2Z. ^ i i j j a  k  + ... + V D  a r e  n  D  =  M  M  .  = a ( k i , ... , k )x*^' ... x ^  n  n  If  * i  / 0  we set  system { D ^ , . . .  .D^  =  D,  Then  p  values  n  distinct. D(x*» ...  , and hence  v[  = D  D  for  A  We set  xf")  i s normal. i > 1 .  The  , D j i s normal and p r i n c i p a l , as M  required. From Theorem 9.1 and the remark following the proof of Theorem 7.1, we obtain the following refinement of Theorem 7.1. Theorem 9.2. closed.  Suppose that  ft  Then any orthogonal system -[D^,  s a t i s f y i n g (6.1.2) and  (6.1.3)  i s algebraically ...  , D  M  J  i s equivalent to a normal  nilpotent orthogonal system. The c h a r a c t e r i z a t i o n of the generalized Witt algebras given i n the following theorem contains considerably fewer parameters than that given by Kaplansky. Theorem 9.3.  Suppose that  ft  i s algebraically  -  closed.  52 -  Then any generalized Witt algebra over ft  can be written i n the form  -at (01 ; D]_, ... , D ) , m  Ot = ft(x]_, ... , x )  where  n  i s the group algebra of  an elementary p-group with independent generators x j , ... , x (9.3.D  , and where  n  V, = £  + • • • + *T~ '  +  ^"4  (9.3-2)  A  •o-i.  with  * Tin,  -  ^ -n X  (i>0.  )  • pj_ ^ eft• Proof.  algebra  By Theorem 9 . 2 , a generalized Witt  can be written i n the form  £ ( 0 t ; D-L, ... , D ) , where m  [ D i , ... , D } i s a norm  mal nilpotent orthogonal system. i s normal. x - i , ... , x  n  Then by Theorem 8.3 there exist  e Ot such that  that the monomials basis of  = ... = x = 1 , such 1 n  x  p  p  x ' ... x^* , 0 ^ VA < p 1  Ol over ft , and such that  form <9.3«1)« t i o n of  We s h a l l assume that  Suppose that  Ol commutative with  = D(Dixi) = 0 , we have  D  D]_ takes the  i s an a r b i t r a r y derivaD]_ . From  k + 1  ) = D(D  D^Dx^)  Dxi = fii e ft . For any  k > 0 , we have Di(Dx  form a  ) = DUD^x*" J 1  l X k + 1  0  = ( D D ^ x * " ' - (D-,x )(Dx )x*' k  a  k  = D ((Dx )x*"' ) . 1  k  jb-l  Therefore we have hence  ^(Dx^+i - (Dx )x^ k  D x i - (Dx )x£ ' = k +  e  k  ) = 0 , and  ® >^  r o m  which we  see e a s i l y that (9.3.3)  D-r,(&—•  Since every (9 «3 »3 ) •  commutes with  D-^ , i t has the form  Then by taking a suitable scalar-equivalent  system we obtain { D^, ... , D | m  Remark.  I f we take  as independent generators of X=L, .... , x  n  of the form ( 9 . 3 . 2 ) .  1 + x^, ... , 1 + x  Ot  n  instead of  , then the form (9*3.1) and (9.3.2) can  s t i l l be preserved, and we. have  x^ = ... = x^ = 0 .  In t h i s case, i t i s e a s i l y seen that  - 54 -  10.  The case ft - GF(p) .  algebra over ft, u^, ... , u ft .  Then  n  Let  such that a l l the  u^, ... , u  #ijk  be an  be a basis of  u^u-j = 2L ^ i j k ^ k » where  we can choose a basis  ^£  n  over  ^ijk $ •  If  e  of  g£  over ft  belong to a subfield  ft  of  1  ft , then we s h a l l say that the algebra  i s defineable  over ft* .  over ft i s  In other words, an algebra  defineable over £J  bra  over  ft  i f and only i f there e x i s t s an alge-  T  ft*  such that  £2§  88  j£ .  Theorem 6.3 shows that any generalized Witt algebra of D-dimension closed f i e l d  ft  m = 1  over an a l g e b r a i c a l l y  i s defineable over  GF(p)  , which may  naturally be regarded as a subfield of ft . t h i s i s true f o r an a r b i t r a r y D-dimension  m  Whether i s not  known. By an a p p l i c a t i o n of Theorem 9«3, we s h a l l show that i f Ol p3  of order form  i s the group algebra of an elementary p-group then any generalized Witt algebra of the  (01 ; D]_, D2)  i s defineable over of  Ol  over an a l g e b r a i c a l l y closed ft  GF(p) .  Let  ^x^y^z^ } be a basis  xP = yP • zP • 0 .  , where  By Theorem 9 . 3 , we  may assume that  where  0,  e ft .  Suppose f i r s t that  0 .  Then  we may assume  /3 = 1 .  then our assertion i s proved. a non-zero element z  Suppose  X e ft , we set x  1  y ^ 0 .  Taking  » Xx, y' = XPy,  = X^ z . Then the set { x'iy'Jz'k J forms a basis  T  of  c% , and we have  D  l  =  dx' *  X  If  +  Di = XD^ , D $  x  Therefore i f we determine 2  X  7 = 0 ,  I f , furthermore,  p  ~  . I f  = XPD  2  , where  y  X  by the equation  = 1 , then we see that  p  GF(p)  W  2  i s defineable over  $ • 0 then we may take  y = 1 , and  hence our assertion i s also c l e a r . In section 2, we have remarked that the only algebras constructed by Kaplansky's method f o r the case m = 1  when D-dimension Witt algebras.  and ft = GF(p) are the o r i g i n a l  Theorem 7.10, 8 . 3 , or 9.3 shows that we  can construct normal simple algebras of the type "£(01)  D  i,  ... » ) D  m  over  GF(p) , which cannot be con-  structed by Kaplansky's method, and which, however, reduce to the generalized Witt algebras i f ft = GF(p) i s extended to an a l g e b r a i c a l l y closed f i e l d . we see that we have constructed some new f i n i t e Lie  Therefore simple  algebras. Remark.  I f we construct a generalized Witt  - 56 -  algebra GF(p)  £  over ft and regard i t as an algebra over  , as i s done i n section 7, then we can obtain  simple algebras over  GF(p) . However, Lemma 7.7  shows that such algebras are not normal simple.  11.  Non-simple a l g e b r a s .  L e t j£  be a L i e  a l g e b r a o v e r ft w i t h t h e m u l t i p l i c a t i o n * * . F o r any two . ^ i , J2  ideals  o  t h e i d e a l o f £, x  i  ^ i  6  *  f  £  w  e  generated  ^et &  s h a l l denote by by a l l X}ax  ,  2  7  0  2  where  be a commutative a s s o c i a t i v e a l g e A [ST)  b r a over ft , and denote by  and  the  l a t t i c e s ( d e f i n e d by i n c l u s i o n ) o f a l l i d e a l s o f and  r e s p e c t i v e l y . I f there e x i s t s a l a t t i c e  cr: A(<£~) •* A ( £ )  phism  - &f» b£  h  °l  <JR~ and  I n t h i s c a s e , i f 3?  then  s t r u c t L i e algebras  { £  •  £  n  I n t h i s s e c t i o n we s h a l l con}• f o r w h i c h t h e r e  ft .  exist  <6TJ- such t h a t  xP  ft  •£  ^  with the basis  .  cp(M  of  L e t -$>(x) be t h e  1, x , x , ... , x P " l ,  s a t i s f i e s the equation  be t h e a l g e b r a  extension  and a p l o y n o m i a l  w i t h c o e f f i c i e n t s i n ~£  algebra over  Ol  8C  have t h e same i d e a l t h e o r y .  o f t h e ground f i e l d  where  ,  $C = ft have  Consider a f i n i t e dimensional  degree  A(<^)  Note t h a t any s i m p l e  commutative a s s o c i a t i v e a l g e b r a s and  e  have t h e same i d e a l  o v e r ft and t h e f i e l d  t h e same i d e a l t h e o r y .  d~2  0  i s the radical of  i s the radical of  L i e algebra  0^,  f o r any two i d e a l s  d s  t h e n we s h a l l s a y t h a t theory.  (^i^)  such t h a t  isomor-  ^ ( x ) regarded  2  n  cp(xP) = 0 , and l e t as an a l g e b r a o v e r  C l e a r l y there exists a d e r i v a t i o n  D  of  Ol  such  -  Dx = 1  that  and such that  f i x e d , so that  ((% ; D)  of  i s uniquely determined ^  may be denoted by  are with-  -£.(y)  $C  It i s e a s i l y seen that the algebra (01 ; D)  of constants f o r  jfc «  , and that  f o r a l l a e -3> .  9 , provided that ft and  by the polynomial  out ambiguity.  Da = 0  •£> - •£. (01 > D)  Then the algebra  Hence  58 -  i s generated by  "$CxJ/(9(X))  0~  s  over  as algebras over ft .  dT' i s a p r i n c i p a l ideal r i n g .  &C can be written as  xP  Every ideal  <£"a = (a). , where  , and i t i s always possible to choose a monic factor is  a(X) , i . e . a f a c t o r whose leading c o e f f i c i e n t  1 , of  cp(X) such that  Thus there exists a (1:1) correspondence  <p(xP) e 0* .  between ideals of  and monic factors of  Theorem 11.1.  (01 ; D)  algebra  ST  The algebra  0  r<r  a e  i s an i d e a l of -£  £  A (<6T) and l e t  For any ideal  Qr and  , since  We s h a l l show that  of  £ (01; D)  and i t s theory. and  &  of <£"  to be the set of a l l elements of the  afD , where  between  Then the  We s h a l l prove f i r s t that  have the same i d e a l theory.  form  p > 2 .  Suppose that  of constants have the same ideal  Proof.  we define  <p(X) .  defined above has no nilpotent i d e a l  except the zero i d e a l . algebra  0~- (a(xP)) , since  f e  Ot  .  Then  Qr^  afD gD = a(fDg - gDf)D e fr* .  0" , i s the desired l a t t i c e isomorphism Let  ^ £ 0  and  A(£) .  be an i d e a l  a(X)  have the minimal p o s i t i v e degree  - 59 -  a(X) £ -£ix]  among polynomials Then  D»a(x)D - (Da(x))D e J  the degree of  a(X)  a = a(x) e iT . f e Ol . where  yields  such that  , and the minimality of Da(x) = 0 , i . e . ,  We s h a l l show that  Express  f  c± e <fl .  7  afD e  0 £ i < p - 1 , then  = ( i + l)aCiX^D e jf  aD  , and hence a C i x D e Since  CiX  i  +  1  1  ,  D  0 for  i  i = 0, ... , p - 2 .  f o r any  f = co + c i x + ... + Cp_ixP"  as  If  0 .  a(x)D e  aCp_ixP"" D e  ^  2  a n (  j since  (ac xP- D)*>(x D) = 4ac _ xP~ D , 2  2  1  p-1  we have Thus  p  0 , and hence  4aCp_]xP~- -D e 1  afD e  CJ f o r any  f e OX .  h(\) = a(X)j(\) + r ( \ ) , where deg r ( \ ) < deg a(\) .  we have of  r(x)D e  a(X)  yields  0 .  fr*''*  Hence  Since  .  a i e 0\  t  Thus we have proved that  i s of the form  Let  0* , we have  .  &  2  &  ajD = a f D . 2  Hence  f e Ol .  afD , where the i d e a l of <$7  be ideals of &  ±  0~ •  C  2  such Sup-  Then, by the d e f i n i t i o n of the map-  a]_D e  Therefore, there exist  J and  h(x)D, a(x)^(x)D e "J  <S£"C QT . We s h a l l show that  pose that ping  ^(\), f{\) e  "7 i f we denote by  generated by that  ^  , we set  Then the minimality of the degree  r(\) = 0 .  every element i n  aCp.^xP'-'-D e "3 .  Now, f o r any  h(x)D e 0  h(X) e '$£\]j such that  where  1  a  02  cr f  , and hence  £ $"  2  ai = a f • 2  and  a^D e  f e 0t  Express  f  . co-  &  z  .  such that  i n the form  - 60 f = CQ l  a  C]_x + ... + c_]_xP~-'- , where  +  c± s $C .  p  = a2Co + a2Cix + ... + a c _ i x P " 2  .  1  p  Since  Cj_ are polynomials i n xP , we have  and Hence Of  e (^2  &f  =  fore  Qf C  then  O'l ^ #2  a  n  &\ £ $"2  and  QT  &2 ^  d  •  i  QTc  a n d  H  e  If  frf  i m p  ^ 1 = @2  n c e  a i , a2  a^ = a2CQ .  proved.  s  Then  a  iy n  d  there-  0 £ /\ {<ft ) -»• '/\ (s£, ) i s a l a t t i c e isomorphism.  ( &i t\^  s h a l l prove  =  a  We  &z  6  ' f o r any two i d e a l s 0~2  $C •  of  i - 1, 2 .  Take  Then  a^ e ^  ^  and  elements of the form  aifD  respectively, since a  l l * 2 2 f  D  &e  a  f  6 <-  r  D  @~±@2  " ia2(fiDf .  a  2  0  0  a i a f D , where  f e  2  D  2  are the sets of a l l  a  - (aia2) .  i t :  i s  f o r any  <r  0 ^ i < p - 1 , then e Of t> 6-f,  and  $\ = ( a i ) ,  D  Ol,  From  we have  In order to prove  ( ^ 1 ^ 2 ^ £ tf/" &f > a]^ *"*" e ^• 0  ( C^i L%)  - f2 *i)  a  ( (9"i 0- )  r  such that  s u f f i c i e n t to show that c e l$C and  a]D a 2 C x  and hence  i+1  0 ^ i < p . If  D - (i+  Da^cx^  a j ^ c x ^ D e &fo 6-f*  Since  aixDoa2Cx ~^D = -2aia2CxP~lD , we have p  a^cxP^D e proved. -JL  Hence  flf  .  Thus  (^ i ^ )  0  <T  is  " 9  ( 0 i 0 £ ) O " « (9-^ A * ' .  Therefore  a  and  have the same i d e a l theory. To complete the proof, l e t 0 .  Then there exists an i d e a l  -jf = (J' 0'* Since  = t^""", we have  O  D e  of  a  n  i d e a l of such that  - 61 a  J>£  - 0-°~o  ( O c T ) * ~ 0* - J ^7 = 0 .  i s not milpotent unless completely  of (11.1), i f & a d i v i s o r of a  Thus Theorem 11.1 i s  i s an ideal of  e£" and i f a(\) i s  Ct = (a(xP)) , then  = -d£(a(\)) as L i e algebras over ft .  d£(?(X)) -  f(x)D = g(x)D and hence  We define a mapping  £(a(X))  0 ( f ( x ) D ) = f(x)D . I f  by  -£iq>M) then  in  f (\) s g(X)(mod <p(\)) ,  f ( \ ) =. g(\)(mod a(\)) .  f(x)D = g(x)D  i n -£{a{\)) .  It i s e a s i l y seen that bra  J  With the notation as i n the proof  cp(\) such that  Proof. 0:  Therefore  proved. Lemma 11.2.  ^/0"  •  -s£ (<p)  &  onto the algebra  a  .  algebra of the type  i s well defined.  i s a homomorphism of the alge-s£(a) .  Now  0(f(x)D) =° 0  , i . e . i f and only i f  Q - = -£_(a) as required.  Therefore  Theorem 11.3«  B  Thus  i f and only i f f (\) = 0 (mod a ( M ) fD e Q -  Therefore  a  If p > 2  -£ (cp)  then any semi-simple  can be decomposed into a  d i r e c t sum of simple algebras of the same type. Proof. i f and only i f £L if  By Theorem 11.1,  i s semi-simple  i s semi-simple, and so, i f and only  cp can be expressed as a product  cp = q>i ... cp  r  d i s t i n c t i r r e d u c i b l e polynomials i n ^ [\J . then that  -s£(cp)  i s semi-simple  and that  of  Suppose  q> = cpi ... cp • r  - 62 We set ^  = cp/cpi , 0± = (xi(xP)) .  composed into the d i r e c t sum: Hence, by Theorem 1 1 . 1 ,  Then  <ft i s de-  © ...  ®  frj*  .  we have  «  My)  (11.3.1)  = G>°~© . . . ©  From the d e f i n i t i o n of (5*2 + ... + 0~  r  have  -£(<p)/( &^G>  •£(<f>±)  ^(q>i)  i t follows e a s i l y that  = (q>i(xP)) .  ( I I . 3 . I ) we have 0?=  fr±  Hence by Lemma 11.2  ... ® #y~) = - s d ^ ) . &f  £  £(<p±)  for a l l i .  0  i s simple.  0  we  Then from  , and s i m i l a r l y Since  q>i  i s irreducible,  12.  63 -  -£(0l',  Automorphisms of  By an automorphism of an algebra a  1-1  mapping  o* of  D]_, • •» , Dm) •  ^£ over ft we mean  onto i t s e l f such that  (x + y ) * = x°* + y ,. (xy) " - x V , (ax)0" = ax0" f o r a l l ff  x, y e -3^. and  0  a e ft .  Because.of the l a s t property,  any automorphism over ft i s completely determined by i t s effect on a basis of •£  over ft . The automor-  phism group of the Witt algebra was determined by Ho-Jui Chang [.lj, and that of a generalization of the group algebra of an elementary p-group by Jacobson 3 . In t h i s section f i r s t we discuss c e r t a i n relationships Ot  between automorphisms of Let  Ot  Df a  a  +  (fD)  ff  D )V 2  = Dj + D*, (D  = f !) * 0  0  D  a  D" which i s defined 0  X  D]_, D  Da)*  2  of  Ot we have  = Dj D* , and  f o r any f e 01 . Let ^  .  be a subalge-  of the derivation algebra of Ol • An automorphism  o- of Ol  w i l l be c a l l e d admissible to  for any  B e  mapping  D -»• DC  .  -s£ i f D°* e £  I f cr i s admissible to i s an automorphism of  be said t o be induced by  s£ (01 J Di, ... , D ) m  then the , which w i l l  0* .  I f an automorphism ff of to  Ot and  i s e a s i l y seen to be a derivation of  . For two derivations  (D-L  bra  = (Df)  a  m  o* be an automorphism of  derivation of Ot . The mapping by  "£,(01 ; D^, ... , D ) .  and  then from  ^  i s admissible  -  (f  l D l  ...  +  f D )<^ = f j D j  +  m  i t follows that to  64 -  m  -£D , ... ,  fjDj  +  | i s a system equivalent  0  -[Di, ... , D j .  ...  +  Thus we have proved the "only i f "  m  part of the following Theorem 12.1. ... , D c  £  of  m  Suppose that  J i s orthonormal.  = -£_{oi ;  p = 5  and that  Then an automorphism  ... , D )  i s induced by an auto-  m  Ol i f and only i f { Dj, ... ,  morphism of  J is a  system equivalent to j D j , ... , D | . m  To complete our proof, suppose that automorphism of lent to pings  such that  { ^ l , ••• » m } • D  c*ij  of  (12.1.1)  Ol  (fL )°  { D J , ... , Dj* j  i s equiva-  Then we may define l i n e a r map-  into i t s e l f such that  = £  ±  o* i s an  f ^ D S  0  s=/  for a l l f e  01 and  i = 1, ... , m .  Setting  f = 1  i n (12.1.1) y i e l d s cr.. 1 = S  (12.1.2) From  <0  (fD^MgDj)*  i:}  (Kronecker delta) .  = (fD^D.^ )* = (f (D )D.j)  ff  ig  - (g(Djf )D )<* and (12.1.1) we have ±  (fDi) " (gDj) " = 2 Z [ ( 0  0  f D  iS)^  - S j (  D  f ) q :  *] k ' D  -ft  On the other hand, from  DfoD^ '= 0  and (12.1.1) we have  - 65 (fDi)  (gDj) " - £ . [ "  0,  Dggl* - g ^ D j f ^ j D j .  0  sA  Therefore we have (12.1.3)  ( f D j g ) * *  X  -  ( g D j f ) * *  D*  =  X  .  f  f^'Djg'S*  7  s  Setting  f - 1  ( D ) H DJg 5* .  i n (12.1.3) y i e l d s  i g  Substituting t h i s i n (12.1.3) y i e l d s (12.1.4)  (fDig)^ -  L f s  3  err J  x^, ... , x  or.  i = j = k, g = X i (12.1.5)  s  5  -£l>i, ... , D  be such that  J i s orthonormal.  m  DJ^CJ  = S^j .  Setting  i n (12.1.4) y i e l d s  ( x ^ f ) ** = X  •x.* (D f) r  r  r f = X j , where  Setting  (D g)^  $  g  e OL  m  = X. f*  4  (D f)**  We s h a l l use the fact that Let  r  (gDjf  .  *  j £ i , i n (12.1.5) y i e l d s  or(12.1.6)  0 =  x  y  (i ^ j) .  Substituting (12.1.6) i n (12.1.5), we have (12.1.7) (xiPitfi = x ? * (D f . ±  Setting  j = i ^ k , g = x^  i n (12.1.4) and using  we have f  - (x^f)  - - x .^(Dif)^  .  (12.1.6),  - 66 f = X J , where  Setting =0  x.*  j £ i , i n the above, we have  for j £ i ^ k .  Combining t h i s r e s u l t with  ( 1 2 . 1 . 6 ) , we conclude that i f i ^ j  then  erf.  (12.1.8) for a l l  x/4 = o k .  Setting  k = i ^ j ,  (12.1.8"), we have  using  (12.1.9) Setting  '  - xT  Setting  i  X  i  2  )^  and  i  i  - x "x.^' .  cr.  3  2(x x )**  •  J  ix 2.)  i n ( 1 2 . 1 . 9 ) , we have  f = x.  (12.1.8), we have  j •£ i , i n ( 1 2 . 1 . 7 ) , we obtai n  f • Xj,Xj , where (x  (j / i ) .  4.  Ai  (12.1.11)  (Djf)^- ,  i n (12.1.9) and using  f = Xj  Setting  = - xj*  f '* - ( x ^ f ) ^  (12.1.10)  -  i n (12.1.4) and  g^x.^  or-  - - 2x.**x.*  x  ^  J  .  (12.1.10)  Therefore, using  (12.1.11), we have  or. (12.1.12) Setting  ,  (  x  j ^ }  i = j = k, f = x  "  2  0  »  ( i / J) •  i n (12.1.4) and using  we have (12.1.13)  (x D r" 2  ig  1  Setting  f = gxi  -2(g  X i  r*  =  (x )^(D  or- c- • 2g •* x .** .  i n ( 1 2 . 1 . 7 ) , we have  2  i g  )^  (12.1.12)  - 67 (x.D^g + x^g) * := x .** (xiD^g) 1  + x .** g **  >C  .  Therefore,  A*  by (12.1.7), we have (12.1..14)  (x?D )^  + (gXir*  i g  1  +  Setting  f - x?  g  2  *  2(x?) ** = 2(x.**)2  i n (12.1.7) y i e l d s , since  2  *  1  p + 2 . or-  3(gxj[)**  (12.1.13) and (12.1.14) y i e l d  < &  1  cr;. «JT , ** X **  (x?) **• = (x.** )  and hence  - (x.^' ) ( D g )  Then en • err-  = 3g **x.**  and hence (12.1.15)  (g ) X i  = g  x.  t f o r a l l g , since  p ^ 3 .  By using (12.1.15) and  (12.1.10) i n (12.1.9), we have f o r i ^ j ' and  f e  <X •  (12.1.16)  f  » 0 .  Setting  k » j•/*i , g = x* orcr.-  we have  f  - f **  3 3  fore we may set  i n (12.1.4) and using (12.1.16)  for. any  f e 01 , i  ting  c  i s an automorphism of  i = j = k  j . .There-  C T Q = ... = o*^ = cr , using the same  l e t t e r as the given automorphism ff of prove that  and  ?t .  We s h a l l  01 over ft .  i n (12.1.4) y i e l d s  (12.1.17)  ( f D ) f f - (gDjf )cr = fff(D )ff - g (D.f)c .  Replacing  g  (12.1.18)  (fg +  ff  ig  ig  i n (12.1.17) by  X i  fD  X j g , we have  - x ^ f ) ° = f (g + ff  i g  x±B±g)V  Set-  - 68 (x-jg) " = x?g f o r any g e Ot .  (12.1.5) y i e l d s  There-  0  f o r e , by (12.1.17) and (12.1.18), we have  fg  (fg) " = 0  a  G  f o r a l l f , g e Ol . We s h a l l show that every element h e Ol can be written i n the form (12.1.1) we have [fD ) ±  {fD )t  • (gD )  f  a  = h .  If  f  a  = g°" then  (fDi) " 0  ±  , f e 0? .  (fD) " = J . ( f f i ^ i ) * =  f i D i , and  . From  Therefore i f  i s an automorphism of 01 . Let D e D =X  a  f D » gDi, f = g . Therefore cr  and hence  cy  i  = f°D? .  ±  - hD* then  a  h • f  Then  HCffi^D*  0  • 21 f J i • f ^ • Therefore the given automorphism f C T  c  D  0  0  of ;£ i s induced by the automorphism  a  of  Ot .  Thus Theorem 12.1 i s proved. Corollary 12.2.  Suppose that  p _ 5  and that  Ol  i s a f i e l d over ft . Then any automorphism of an alge-  bra  of the form -JL (Ol ; D)  of  Ot . The automorphism group of  i s induced by an automorphism  (OX ; D)  over ft  i s isomorphic to a subgroup of the automorphism group of <£T over ft , where  i s the algebra of constants of  (01 ; D) . In p a r t i c u l a r , i f <£f = ft then  •£ (01 ; D)  has no automorphism except the i d e n t i t y . Proof. Then lent. of  B  a  Let cr be an automorphism of J C (Ol ; D) .  = aD with  a ^ 0 . Hence  By Theorem 12.1,  OL • If f e ^  Df* = 0, f  a  D" and D 0  are equiva-  cr i s induced by an automorphism  then  D°"fO- = (Df) " - 0 .  e £ T . Therefore  0  Hence  cr induces an automorphism  - 69 of  £C .  If  o* induces the i d e n t i t y automorphism on  fif  , then we have  f  e <£~.  p  hence  Therefore,  o* = 1 .  •jL (Ol\ )  (f^JP *= fP  f e 01 , since  f o r any  (f * - f )P = 0 , f " = f , and 0  0  Hence the automorphism group of  over ft i s isomorphic to a subgroup of the  D  $C  automorphism group of  over ft , as required.  By the above r e s u l t , we can construct e a s i l y simple. L i e algebras which have no automorphism except the  identity.  where fl>  P  For example, l e t ft • P(  ... ,  i s a f i e l d of c h a r a c t e r i s t i c  ••• > fm  let  Ol *=  n  ^  .  a  r  e  m  ft(x]_, l>-i  , x ) , where m  P , and  £"i •  W  e  s e t  4>-> 2  b-i -3^(01;  x? =  m  and where  indeterminates over  d  Then the algebra  p  £ )  over ft - has the desired  D)  property. In the course of the proof of Theorem 12.1, only the fact that  p ^ 2, 3  was used.  Theorem 12.1 holds even when  p <= 0 •  Therefore  Thus any automor-  phism of the derivation algebra of the function f i e l d  0\, of one variable over a f i e l d of c h a r a c t e r i s t i c i s induced by an automorphism of  0  Ol over ft .  Now we s h a l l consider automorphisms of the generalized Witt algebras. 01- ft(x^, ••• , x ) n  In the following,  w i l l denote the group algebra of  - 70 an elementary p-group with independent generators xi,  ... , x  n  . A polynomial  f ( X ) e ft[X3 i s c a l l e d  a p-polynomial i f f ( X ) i s of the form + a^X"  + ... + ajj-X , where  f (X) » a X 0  0.^ eft.  Lemmas 12.3 and 12.4 are proved i n f 3 j , P» 110. Lemma 12.3. N = p  n  I f 1, u^, u , ... ,  , where  2  , i s a basis of  Ol over ft , then there exist  n  d i s t i n c t indices, say, 1, 2, ... , n , such that the A Jk 0 elements u ' ... u ^ , where 0 <. k< < p , u. - 1 ,  form a basis of  (71 over ft .  Lemma 12.4. any derivation i n  The c h a r a c t e r i s t i c polynomial o f  (JL i s a p-polynomial.  Lemma 12°. 5.  I f a l l the roots o f the minimum  polynomial of a derivation d i s t i n c t , and i f D  D  does not s a t i s f y any non-zero p-  polynomial of degree l e s s than t e r i s t i c roots of D Proof. polynomial of D  i n Ol are i n ft and  p  n  , then a l l the charac-  are i n ft and d i s t i n c t .  Since a l l the roots of the minimal are i n ft and d i s t i n c t ,  diagonalized, that i s , there e x i s t s a basis of  Ol  such that  can be  1, U]_, u , 2  Dui = XjUi, \± e ft , f o r a l l i . By  Lemma 12.3 we may assume that the elements  f  D  u ' ... u ^  <  - 71 form a basis of Ol over ft . =  ... u  1  (T\ -k4)u n  X ^i^i - 0  with  Since  D(u^' ... u f ~ )  , i t i s s u f f i c i e n t to show that  0  < p  implies  k-^ = ... = k  n  = 0.  Suppose that there exist . (k^, ... , k ) £ (0, ... , 0), n  X ^i i  0 = k i < p , such that  = 0 .  k  Since  s k (mod p ) we have X \^ kj. = 0 f o r j » 0, 1, 2, ... . Then the matrix' [\T ) , where At  l ^ i ^ n ,  O ^ j ^ n - 1  f^li ••• i Pn-1  there exists such that  , i s «0«»singular. ^ »  e  18./  = 0  - XT U i we have  n  for a l l i .  (XI  )ui - 0  the d e r i v a t i o n X_  t  z e r o  Since  »  u-  f o r a l l i . Then  0 > since  =  o  Therefore  U]_, ... , u  n  generate tfl . This contradicts tm our assumption. Therefore  X ^i i  ~  k  implies  0  k]_ = ... s k =  0 mod p ,  n  as required. The following two lemmas are e a s i l y v e r i f i e d . . Lemma 12.6. a  0»  a  l > ••* » p - l a  where  i + j  6  ®  a  Lemma 12.7. l>  ••• J p - l a  6  ^  for  all • i  e  s  u  c  h  that  a  i j  ~ i+j »  a  ctQ = a]_ = ... = Suppose  p 2* 5 •  are such that  - ( j - i ) a ^ j . , where +  r  p 2 5 • If a  i s calculated ...mod p , f o r a l l i / j ,  and i f CLQ £ 0 , then  a  Suppose that  i + j  and j , then  a p  _ i  - 1 •  If o a  =  0»  jctj - i o ^  i s calculated  mod p $  = ia-^ f o r a l l i •  ^ 72 -  Let  £ - £ (01; D  ... , D )  lf  be a general-  m  ized Witt algebra defined by a p r i n c i p a l system {^1,  ... , D } .  We s h a l l assume that ft i s a perfect  m  p 2 5 • Let cr be an automor-  i n f i n i t e f i e l d and that  phism of -j£ . By Theorem 9 . 1 , there Kl>  ••• > tfm ® 6  i s normal. %(\)  s  u  c  h  t  h  a  t  D  ^1 1  =  D  exist •••  +  of D  i s a p-polynomial of degree  p  are i n ft. and d i s t i n c t .  n  . A l l the  We s h a l l show  that the c h a r a c t e r i s t i c polynomial of D" 0  .  D  By Lemma 12.4 the c h a r a c t e r i s t i c polynomial  roots of  %{X)  ^m m  +  i s also  Since  (12.8.1)  Do(D« ... (DoX) ... ) = D  f o r any i  and X e <g£, , and since no non-zero d e r i v a t i o n  of  Ol  cX  r  commutes with a l l elements i n ^  /£(D ) = 0 and that ff  , we see that  D" does not s a t i s f y any non-zero 0  p-polynomial of degree l e s s than  p  .  n  %(D ) = 0 a  implies that the minimum polynomial of B roots contained  a  has d i s t i n c t  i n ft . Therefore by Lemma 12.5 - a l l the  characteristic roots of D" 0  are d i s t i n c t , and hence the  minimal polynomial of TP coincides with the characteri s t i c polynomial of D" . Therefore  #(\) i s the  0  c h a r a c t e r i s t i c polynomial of TP . In p a r t i c u l a r D°f - 0  implies  f e ft , that i s , D  ff  the c h a r a c t e r i s t i c roots of D* 0  i s normal.  Since  are i n ft and d i s t i n c t ,  - 73 D*  can be diagonalized, so that there e x i s t s a basis  1» l > 2 » ••• u  °f ^  u  over ft such that  .D^Ujj = \-jUi , \ i e ft , f o r a l l i . By Lemma 12.3 we  o may assume that the elements basis of Ot  . Then the  p  2  u*' ... u * form a  n  . elements  21 ^ i i > k  0 5 kj[ < p , are p r e c i s e l y the ( d i s t i n c t ) c h a r a c t e r i s t i c roots of • %{"K) . On the other hand, since  \±  i s also  a c h a r a c t e r i s t i c root of D , there exists a non-zero x± e Oi such that  element  1, x^, ... , x ^ i , where Since  D]_, ... , D  D(DjXi) ~ ^ i j i D  x  a  e ft f o r a l l is  n  hence  d  i  n  a  x  D , we have with  Since  ; L^, ... , D ) m  x^ ^ 0 , by Lemma 3.2 we see that  i s a unit i n 01 . Therefore we may assume  for a l l i . The elements form a basis of matrix  (ctij) for  J3J[JUJ  The matrix  Ot over ft . We note that the  i s of rank 0  x? = 1  x ^ ' ... x ^ , 0 ^ k,- < p ,  m . Similarly,  i = 1, ... , m  ( jS-jj)  D?UJ  and j = 1, ... , n .  i s also of rank  We consider the subspace  m . ^/T(ki, ... , k ) n  , which w i l l also be denoted by flfly. , spanned  of  X E •£ such that  by is  , form a basis of Ot .  D^x^ = j i i  and j .  simple and since  x^  =  N = p  are commutative with  m  >  Dxi = ^ i X ^ . Then  e a s i l y seen that  form  D»X = (\]ki + ... + ^ k ) X . I t n  n  001^ consists of elements of the  x^l ... x^-ii^Pl  +  ••• tfnrV > where +  <# eft, ±  - 74 - • so that '2ftTk of <fl7k /  i s of dimension  m .  under the isomorphism  o*  The image  i s also of dimension  m , and can be characterized as the set of such that  B 'Y - [K^l  + ... + X k )Y  a  n  u ^ ' ... uf~( ftD* + . . . $  ±  e ft .  If  Y e  . Therefore  n  + ^ M D S ) efl?£  0 ^ - k i < p - l  fift^  f o r any  f o r a l l i , then  u'*' ... u ^ D ^ , i = 1, ... , m , are l i n e a r l y independent. For, i f u^« ... u* ( y u  (IT  7 i j 5 i j ) u j U ~ « ... u ^ =  for a l l j .  7l all  ufi  OfeD?)  = ... =  Since  <tf  m  i , then ^ k . . .  0  then  i s of rank  Therefore i f  +  ...  +  ^MDM)  u^ = 0 .  Then  Y e fl7(p - 2)  and  a  for any  = 0  0 l> k i < p - 1  for  fa  u? ^ 0  e ft.. for a l l  i .  We s h a l l denote flT7((p--- 2)A.i, 0, ... , 0 ) ,  - 3 ) ^ i , 0, ... , 0)  respectively.  fa  m , we have  , where  We are now ready to prove  W(P  #±  consists of elements of the form  U ^ ^ D - L  Suppose  X  0 , and hence  ( j^ij)  - 0 .  =  u^ = 0  simply by implies  Y« e flfl(p - 3 )  Xe|f|(p-2)  and  f f  ^ ( p - 2 ) , fl)7(p - 3) Y Y .  1  = 0  Hence  X» e ?#(p - 3) •  f o r any X-X' = 0 This i s a  contradiction, since (12.8.2) Therefore  xf~  Z  D X * x^* D X = - ^IX^D-L / 3  u^ ^ 0 , and s i m i l a r l y  Hence we may assume  u? = 1  u? ^ 0  for a l l  i .  0 . for a l l  i .  - 75 -  Now that we have shown that i  , i t i s e a s i l y seen that  uf* ...  the form ^  £  i^Pl  u? = 1  consists of elements of  + ... + tfnJDJj) , where  e $ , without any r e s t r i c t i o n on  the sum of a l l ?)T  k  a£ i s  can be w r i t t e n i n the g i e Ol . Thus we have  { D £ , ... , DjJ J- i s a system equivalent to  ... , D j- .  |D"L,  Since  k  giD^ + ... + grrPm » where  proved that  .  i s also the sum of a l l 1#7 •  , ^  Therefore every element i n form  for a l l  m  By taking a suitable scalar-equivalent  system i f necessary, we may assume without l o s s of general i t y that for  DJ_XJ  = ^lj j  8-  > where  x  i s a Kronecker d e l t a ,  ij  i , j = 1, ... , m . Note that  m ^ n . Similarly,  there exists a system { E^, ... , E J  scalar-equivalent  m  to  ^D*, ... , D^ ]• such that  E  iUj = ^ i j j  f ° i>  u  r  j = 1, ... , m . We set , (12.8.3)  (x^)*= uj  ( p  il l E  where  p^j e f t . We also set  fixed  k> 1 .  Since  F  Di (x^D^) - 0  yields  (12.8.4)  ^ 0 ,p  0  P  Q 1  u  Now (12.8.3) y i e l d s e a s i l y  +  im m »  p  E  (xkDfe.)* - u F  =  = 0 ,  ^  '  a  n  d  Ej ,  hence we have  (k > 1) .  P u P j l * P±+j i  i . Hence (12.8.4) y i e l d s  Df = E i  f o r any  k  Hence by Lemma 12.6 and (12.8.4) we have all  }  commutes with every  Pok k^ Q k  +  Dj = E  x  f°  r  i £ 3•  p^]_ = 1 f o r . Similarly  f o r a l l i . Again (12.8.3) y i e l d s , f o r any  - 76 k > 1, j p j  - ip  k  we have for  - (j - i)Pi+j k •  i k  = iPlk  f°  k>l  * •  W  2  (xk^k)  = k  0  u  [x^) e{x D ) 0  y ( -7)  e  fe  12  p  k  + ... + P E ) ) •  ,  F  2  F  m  l  u  - 0  a  k  k  p F = 7 ( l E  k  By changing and  = 9^1  m  f°  r  and (12.8.5) imply,  i(  7k ^ 0 >  a  n  +  E  .«.  +  (x^D^)  p F - 7 y^i  0  - u^D  p E 2  f°  ra  for a l l i «... , ^m}  D  0  + ... + p E  H  k > 1 .  m  0  0  D  ... , {V ) J m  0  k  2  ( x ^ ) " = u^D  0  .  and *-  s a  system equivalent to £ l > — • » m^ • = u". Dj, {(D^) ,  k  • Therefore i f p  0  Pk - 0  Dj = x~} Dj . Then  1  PmEm)) •  +  hence we have  d  l » (12.6.5) y i e l d s  we have  E  Pm m) "  a contradiction. Hence D  P2 2  i n (12.8.6), we obtain  +'•••  E  k  +  k  i  tf (P2 2  then  0  c  i # 0 (mod p ) ,  (12.8.6)  {D ,  n  s h a l l write  e  0  . Then  for  =  H  ra  ( x ^ ) " = uj(E! + i ( p E  As before, we set  l  e  pjk • Then (12.8.3) can be written as  (12.8.5)  E  H  }  n  = 0 ,  m  Since  Similarly j .  We set  orthonormal  Since ( D j )  0  i s equivalent to  G  ... , D | which i s equivalent t o { D]_, ... , D ^ . 0  m  Hence. ^ ( D ^ ) , ... , (D^) J 0  0  {^i> ... » ^m} * automorphism  c  B  y  of  i s equivalent to  Theorem 12.1,  cr i s induced by an  Ol •  Suppose that  D? = D  ±  fora l l i .  D°" = D .. We set y = x^' ... x ^ .  Then  Then we have  - 77 DyC Hence  =  D  cyr  (Dy  )ff , (  y°" = ay with  we have  Ol  + ... + XnknJy * . 0  Xlkl  a e ft .  aP = 1 , a <= 1 .  x, ' ... x of  =  Since  Thus  (y )P « (yP) " - 1 , ff  0  y* = y .  Since  form a basis of Ol , the automorphism o*  n  i s the i d e n t i t y .  Since  ifD )  - f D? - f D  a  ff  ±  and f 6 Ol , the automorphism  for a l l i  i s also the i d e n t i t y .  a  of •£  Thus we have proved the following  Theorem 12.8.  Suppose that ft i s an i n f i n i t e  perfect f i e l d and that  p = 5 • Then any automorphism  cr of a generalized Witt algebra  ^(tfT;  i s induced by an automorphism of  Ol • I f  f o r a l l i , then  ±  ... , D ) m  o* i s the i d e n t i t y .  Corollary 12.9.  = D-^  ,  *  Let  D i , ... , D ) m  be a generalized Witt algebra, and assume that there exist non-zero elements DjXj = $ i j X j  °* of Ol then  for  x^, ... , x  e Ol such that  i , j = 1, ... , m .  admissible t o  I f an automorphism  -g£ leaves every  Xj  invariant,  o" i s the i d e n t i t y . Proof.  Since •£ D^, ... ,  ... , D j. ,' we may set m  D°fx = SJ.-X? = c,* ,-X.? . a  SJLJ  m  = c ^ j , and hence  by Theorem 12.8  D?.=  Since D?/ =  j- i s equivalent to c  ij j D  •  Then  x,- i s a u n i t , we have for a l l i .  Therefore  cr i s the i d e n t i t y .  What automorphisms of  Ol are admissible to  - 78 "e£ (01 l i» • • • > m) D  D  only the case  •"•  ?  t n  m = 1 .  n  following we s h a l l consider  e  I f ft i s a l g e b r a i c a l l y  closed,  then any generalized Witt algebras of D-dimension  £ (Ol ; D) , where  be written i n the form 01= ft(xj, ... , x )  1 can  i s the group algebra of an elemen-  n  tary p-group with independent generators 1 + X]_, ... , 1 + x  (Once  n  , and where  i s formulated i n t h i s form, we may prove, with-  out any condition on ft , that any automorphism of i s induced by an automorphism of 01 ) . Denote by the monomial  x  x  where weight  71  + v p + ... + v p 2  w  n _ 1  n  a , ct  w + 1  w  ... x ^ of weight  / w = v  y  , ... eft,a  w(f) o f f  . If f = a y w  w  w  +a  w + 1  y  w + 1  + ..  ^ 0 , then we define the  to be w . Lemmas 12.10 and 12.11  are e a s i l y v e r i f i e d .  then  If  Df  Lemma 12.10.  If f s ^  i s of weight  w - 1 .  Lemma.12.11.  Let 9fl be the r a d i c a l of  f 6 <$f- then  w > 0,  Ol .  w(f) i s not a power of p .  Lemma 12.12. o* an automorphism of (12.12.1)  i s of weight  Let <TZ be the r a d i c a l o f Ol ,  Ol admissible to £  x? - ctiixi + ... + a  i n  x  n  (mod  , and l e t  <tfl ) 2  - 79 -  for  i = 1, ... , n , where  for  3 <  i> 1 (x  then from (12.12.1) we have  ... x  )b =  + a^ox  a ^ T  CLAJI  f o r i > 1 . We set  0  n  Take a fixed  i> 1  cV.  = 0  rs  ik ^ ^  *"  o rs  o  m  > p^  i f  1  Suppose that k  e  + f  f o r s < r , and that r  r < i .  n  and assume that  w(f ) whenever  (mod ffl ) .  x? • a ^ X i + ... + c t i x  (12.12.2)  (12.12.3)  -  + ... •  ... x  in  Therefore  b e Ot .  Let bD°" = D , where  + ct x  a  a^j = 0  Then  i. Proof of 12.12:  If  a ^ je f t .  such that  = ... = c^ijk-l  =  0>  1 < k < i . From  (12.12.2) we have (12.12.4)  (x? ... x ^  JP-^b -' a-soc*"' ... x?"' + ...  + OHnxf"' ... x*~' + Dfj . From (12.12.3) i t follows e a s i l y that = p "" - 1 > p ^ 1  = p  1  k - 1  1  w((xj ... xP" )^ ' b) 1  - 1 . Therefore (12.12.4) y i e l d s  - 1 . Then from Lemma 12.10 we have  a contradiction by Lemma 12.11 . Hence  w(Df )  w(f ) » p ±  ct^. = 0 f o r  ±  k _ 1  ,  - 80 j < i . Then (12.12.4) y i e l d s Hence  w(fi)  w(fi) > p " 1  1  w(f ) Z p ±  i _ 1  - 1 .  p^r^ . Then by Lemma 12.11 we have . Thus (12.12.3) holds f o r a l l r , com-  pleting the proof. Denote by Itf the group of a l l admissible morphisms of Ol . Then the mapping  cr -*• (ct^)  auto-  defined  by (12.12.1) i s a homomorphism of <Vl onto a group of nxn  matrices, which i s solvable by Lemma 12.12. Let  /y£  be the kernel of the homomorphism.  group of  The automorphism  Ol over ft i s e s s e n t i a l l y the same as that o f  i t s r a d i c a l ^fl , since Ol /tfl = $ • Therefore  can  be regarded as a subgroup of the group s\Q of a l l automorphisms of  1{X  which induce the i d e n t i t y on  Since  ^ff i s nilpotent, ^  Hence  rQ\J i s solvable.  tfl/Ifl  2  .  i s solvable (See Q3j, P« 117). Therefore <Vl i s also solvable.  Thus we have proved the following Theorem 12.13. phism group of the algebra  Suppose  -£,(01  p 2 5 . The automorD)  given i n Theorem  8.3 i s solvable. F i n a l l y we s h a l l prove the following Theorem 12.14. -L  D  lf  ... , D ) m  I f two normal simple and  -£{Ol'l  algebras  Dj, ... , D^)  over the same ground f i e l d ft are isomorphic, then t h e i r D-dimensions coincide:  m = m' .  - 81 Proof. we may  Since  and  -g^'  assume without l o s s of generality that  a l g e b r a i c a l l y closed, and that ized Witt algebras.  Ql^^t  t~^£ T  D e ^  Oi )  , and hence  n  n > n* .  are d i s t i n c t . D,  p  n t  e a s i l y that  Let  % (\) x  M  D' .. degree  p , p  are  general-  be the dimensions of  n t  mp  n  = m'p  .  nT  Suppose  By Theorem 9.1 there exists  D  f  e «^£  T  be the element corres-  i s a p-polynbmial by Lemma 12.4, .  From % (B) X  d i c t i o n , since  D  on  the c h a r a c t e r i s t i c polynomial of  % * (D )  - 0  1  and  and of  (12.8.1) i t follows  - 0 , since no non-zero d e r i v a t i o n of  commutes w i l l a l l elements i n  s£  .  This i s a contra-  does not s a t i s f y any non-zero poly-  .nomial of degree l e s s than hold.  is  whose c h a r a c t e r i s t i c roots (as an operator  ponding to  Ot  Let  ft  and  respectively, so that  T  m < m  are normal simple,  p  n  .  Therefore  m = m*  must  - 82 -  References  Ho-Jui Chang, "Ueber Wittsche Lie-Ringe", Abhandlungen aus dem Mathematischen  Seminar  der Hansischen Universitat, v o l . 14  (1941),  pp. 151-164.  N. Jacobson, "Abstract derivation and L i e algebras", Transactions of American Mathematical Society, v o l . 42 (1937), pp. 206-224. N. Jacobson, "Classes of r e s t r i c t e d L i e algebras of c h a r a c t e r i s t i c p, I I " , Duke Mathematical Journal, v o l . 10  (1943), pp.  107-121.  I. Kaplansky, "Seminar on simple Lie-algebras", B u l l e t i n of American Mathematical Society, v o l . 60 (1954), pp. 470-471.  H. Zassenhaus, "Ueber L i e s c h e Ringe mit PrimzahlT  c h a r a c t e r i s t i k " , Abhandlungen aus dem Mathematischen Seminar der Hansischen Universitat, v o l . 13  (1940), pp. 1-100.  

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