"Science, Faculty of"@en . "Mathematics, Department of"@en . "DSpace"@en . "UBCV"@en . "Ree, Rimhak"@en . "2012-02-21T17:54:35Z"@en . "1955"@en . "Doctor of Philosophy - PhD"@en . "University of British Columbia"@en . "Let \u00C6\u00A3 be a commutative associative algebra over a field \u00CE\u00A6 of characteristic p > 0 , and \u00CF\u0091(\u00C6\u00A3) the derivation algebra of \u00C6\u00A3. A subalgebra [symbol omitted] of \u00CF\u0091(\u00C6\u00A3) is called regular if f D \u00C9\u009B for any f \u00C9\u009B \u00C6\u00A3 and D \u00C9\u009B [symbol omitted]. For a regular subalgebra [symbol omitted] of \u00CF\u0091(\u00C6\u00A3), if there exist D\u00E2\u0082\u0081,\u00E2\u0080\u00A6, D[subscript]m \u00C9\u009B [symbol omitted] such that every D \u00C9\u009B [symbol omitted] is expressed uniquely as D = f\u00E2\u0082\u0081D\u00E2\u0082\u0081+ ... + f[subscript]mD[subscript]m, where e f[subscript]i \u00C9\u009B \u00C6\u00A3, then [symbol omitted] is said to be defined by the system {D\u00E2\u0082\u0081, \u00E2\u0080\u00A6, D[subscript]m} and is denoted by the notation [symbol omitted] (\u00C6\u00A3 ; D\u00E2\u0082\u0081, ..., D[subscript]m). In this dissertation, the family of [symbol omitted] Lie algebras of the type [symbol omitted]( \u00C6\u00A3; D\u00E2\u0082\u0081, \u00E2\u0080\u00A6, D[subscript]m) is studied.\r\nIt is shown that if \u00C6\u00A3 is a field then all algebras in [symbol omitted] are simple except when p = 2, m = 1. It is also shown that if \u00CE\u00A6 is algebraically closed then every simple algebra in [symbol omitted] is a generalized Witt algebra of the type defined by I. Kaplansky [Bull. Amer. Math. Soc. vol. 10 (1943), pp. 107-121], and, conversely, that every generalized Witt algebra belongs to [symbol omitted]. A simpler form of the generalized Witt algebras is given. By using this form, the problem of whether every generalized Witt algebra can be defined over GF(p) is partly solved. It is shown also that a subfamily [symbol omitted] of [symbol omitted] , consisting for the most part of non-simple algebras, has a remarkable property: every algebra in [symbol omitted] has the same ideal theory as that of a commutative associative algebra. Jacobson's result on automorphisms of the derivation algebras of the group algebras of commutative groups of the type (p, ..., p) is extended to generalized Witt algebras, and, finally, it is shown that m is an invariant of the algebra\r\n[symbol omitted] = [symbol omitted] (\u00C6\u00A3; D\u00E2\u0082\u0081, ..., D[subscript]m) if [symbol omitted] is normal simple."@en . "https://circle.library.ubc.ca/rest/handle/2429/40825?expand=metadata"@en . "0 N \u00E2\u0080\u00A2 G E N E R A L I Z E D W I T T 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 in the Department of MATHEMATICS We accept this thesis as conforming to the standard required from candidates for the degree of Doctor of Philosophy. Members of the Department of Mathematics T E E U N I V E R S I T Y OF B R I T I S H C O L U M B I A April, 1955 Abstract Let 01 be a commutative associative algebra over a f i e l d $ of characteristic p > 0 , and &{0l) the derivation algebra of 01 . A subalgebra of friOl) i s called regular i f f D e iL for any f e 01 and D e ^ . For a regular subalgebra of ,$-( 0 have not been completely deter-mined. Four f a m i l i e s , which correspond to 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 family of simple L i e algebras, (the f i r s t examples were discovered by E. Witt) 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 i -see later, there are some families of inf i n i t e dimension-a l simple Lie algebras of characteristic zero which may be regarded as the zero characteristic counterparts of the above family. The Witt algebras are algebras over a f i e l d of characteristic p > 2 with basis e0> el\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB e p - l a n c* relations ej. o ej - (j - ije^+j , where i + j is to be calculated modulo p . H. Zassen-haus ([5], P\u00C2\u00BB 47) generalized the Witt algebras to alge-bras 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 a\u00C2\u00BB e^ \u00E2\u0080\u00A2 ( p - a)e a +|g \u00E2\u0080\u00A2 Another generaliza-tion of Witt Ts algebra was obtained by Jacobson[3j. In his investigations Witt [ 1 ] used implicitly the fact that the Witt algebras defined above are the derivation algebras of the group algebras of cyclic groups of order p \u00E2\u0080\u00A2 In the paper cited above, Jacobson proved that the derivation algebra of the group algebra of an elementary p-group, by which we shall mean throughout this disserta-tion an abelian group of the type (p, p, ... , p) , i s simple i f the order of the group i s > 2 . ingenious generalization of the Witt algebras, which i n -cludes the generalizations obtained by Zassenhaus and Recently, I. Kaplansky ([4], p. 471) gave an - I V -the Lie algebra^over # with basis { ( i , o\")} , where i e I, o\" e Ofy , and the multiplication (0.0.1) ( i , o > ( j , t ) = u ( i ) ( j , cr + 0 , and Cfy i s an elementary p-group. In this dissertation, 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 results i s that i f the ground f i e l d i s algebraically closed then any simple algebra i n our family.is a gener-alized Witt algebra. However, i f the ground f i e l d i s not algebraically closed then i n general our family con-tains many simple Lie algebras, i n particular many new. f i n i t e simple Lie algebras, which cannot be constructed by Kaplanskyfs method. Moreover, our formulation enables us to derive i n a natural fashion some of the remarkable properties of the generalized Witt algebras. The author would li k e to express his gratitude to Professor S. A, Jennings, under whose direction this dissertation was written, for his assistance and encouragement. - 1 -1. The algebra \u00C2\u00A3 (01; ... , Dm) . Throughout this paper, $ w i l l denote a f i e l d of characteristic p > 0 , and OL a commutative associa-tive algebra over $ , with a unit element. Suppose there exist derivations Dj_, ... , D m of OL such that m (1.0.1) D\u00C2\u00B1Dj - DjDi = Z I a i j k D k 0 for i , j = 1, ... , m , where ajjk e OL \u00E2\u0080\u00A2 Then the set (Ol; D^ , ... , Dm) of a l l derivations of Ol of the form fiD^ + ... + f m D m , where f^ e Oi , forms a subalgebra of (01) . More generally, the set of a l l derivations of Ol of the form f ] ^ + ... + f m D m , where fj[ runs over an ideal Q~ of Ol , forms a sub-algebra of ^ 9- (01) . For, m f i ^ i gjDj = f i ( D i g j ) D j - g ^ D j f i J D i + 2 1 f i g j a i j k D k , where a l l the coefficients of the right-hand side belong to 0\" . In the following we shall r e s t r i c t our Lie algebra \u00E2\u0080\u00A2\u00C2\u00A3, (01; D^ , ... , Dm) by imposing the following condition: (1.0.2) f1B1 + . . . + \u00C2\u00AB 0 implies f ] _ = . . . = f m = 0 . - 2 -The number m w i l l be called the D-dimension of \u00C2\u00A3 (OL ; D X , . . . , D M ) . Because of the condition (1.0.2) there exists a one-one correspondence f ^ + . . . + f M D M \u00C2\u00AB - \u00C2\u00BB (tlt . . . , fm) between the elements of J C (OL ; D ] _ , . . . , D M ) and the set of a l l vectors (f]_, . . . , fm) , where fj[ runs over OL, I f we identify f ]_D;L + . . . + f M D M with ( f l , . . . , fm) then o ( f l f . . . , fm) = (a f i , . . . , afm) for a e $ . (fl, . . . , f m ) + ( g l f . . . , gm) s ( f l + SI, , f m + Sm) , ( f l , . . . , f m ) \u00C2\u00B0 ( g i , . . . , gm) - ( h l r . . . , hm) , where (1.0.3) h i - 21 ( f s (D s gi ) - g s ( D s f i ) ) s + Z. \u00C2\u00A3 s S t a s t i \u00E2\u0080\u00A2 s , t Conversely, we may define a Lie algebra i \u00C2\u00A3 * by start-ing with derivations D]_, . . . , Dm satisfying (1.0.1) but not necessarily (1.0.2) and the set s\u00C2\u00A3 * of a l l vectors (f]_, . . . , fm) and defining scalar multipl ica-t i o n , addition, and multipl ication according to (1.0.3). In general, < C * i s different from \u00E2\u0080\u00A2\u00C2\u00A3 (OX ; D]_, ... , Dm) . It i s easily seen that the set 0 of a l l vectors ( f l , ... , f m ) satisfying f]_Di + ... + f m D m \u00E2\u0080\u00A2 0 forms an ideal of and that \u00E2\u0080\u00A2\u00C2\u00A3* /0 = H (01 ; D l f ... , Dm) Since we are mainly interested i n simple Lie algebras, we prefer to work with \u00E2\u0080\u00A2\u00C2\u00A3 (OX ; D]_, ... , Dm) rather than \"\u00C2\u00A3* . In what follows we study the properties of the alge bras \u00C2\u00A3(01*, D^ , ... , Dm) always assuming ( 1 . 0 . 2 ) . - 4 -2. Generalized Witt algebras* We show that any generalized Witt algebras \u00C2\u00A3. can be written i n the form aC (OL ; Di, ... , Dm) . Let it be defined with respect to the set I * { l , 2, ... , m j of indices and the total additive group (ty of functional s on I with values i n ft . Let ?5j = \u00E2\u0080\u00A2 [ U Q - , u^; ... ] be a multiplicative group isomorphic to Cfy via the corres-pondence U Q - \u00C2\u00AB \u00E2\u0080\u0094 * a \u00E2\u0080\u00A2 For each i e I we define the mapping O^ : \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 t \u00C2\u00AEm a r e homomorphisms' of ^ into the addi-tive group of ft , such that (2.0.1) eifuoO \u00E2\u0080\u00A2 ... \u00E2\u0080\u00A2 9 m(u f f) - 0 implies UQ- = 1 , (2.0.2) a^ei + ... + a m 9 m = 0 with e ft implies a]_ = ... = a m = 0 . Now let Ol be the group algebra of ^ over ft , and define the linear mapping D^ of Ol into i t s e l f by Di^o- s ei(uo-)u0- . Then Dj_ i s a derivation of Oi , since Di(uo-ut) - Di(uo-+t) - 6i(uo-+t)ucr+t - SitUo-Ju^UT; + eifutJUtfUfc \u00E2\u0080\u00A2 (DiUo-)ut + Uo-(DiUfc), It i s clear that (1.0.1) i s satisfied for D^ , ... , D m , since DiDj - DjDi = 0 for a l l i , j . We w i l l show that (1.0.2) i s also satisfied. Let fiD j + ... + fmDm - 0 , - 5 -with f\u00C2\u00B1E0l . Then we have X f i 6 i ( u c r ) - 0 for At all . Let f\u00C2\u00B1 \u00C2\u00AB* d ^ f t O u ^ . Then we have JEL a.^(X)Q^(ua) - 0 for al l t and cr . From ( 2 . 0 . 1 ) it it follows that a\u00C2\u00B1(X) \u00C2\u00BB 0 for al l i and t . Thus f]_ = . . . \u00C2\u00BB f m \u00E2\u0080\u00A2 0 . Therefore we can define the Lie algebra \u00E2\u0080\u00A2\u00C2\u00A3 (01 ; DT_, . . . , Dm) . The set { u^DiJ , where i \u00C2\u00A3 I , ff E ^ is a basis of this Lie algebra, and we have u cr D i - u t D j \u00E2\u0080\u00A2 Ucr(DiUt)Dj - ut^DjU^Di - U ( i ) u f f + c D j \" 0 < J ) u < r+t D i \u00E2\u0080\u00A2 Comparing the above with ( 0 . 0 . 1 ) , we see easily that the given generalized Witt algebra is isomorphic with H (OX ; D]_, . . . , Dm) . We note that ( 2 . 0 . 1 ) is equiva-lent to the following property of \u00C2\u00A3 (Ol\ D]_, . . . , Dm) : ( 2 . 0 . 3 ) D]_f - . . . = Pmf \u00E2\u0080\u00A2 0 implies f e * . Conversely, for any elementary p-group Cfr , i f there exist homomorphisms 8 i , ... , ^ of 0J into the additive group of $ such that ( 2 . 0 . 1 ) and ( 2 . 0 . 2 ) hold, then we can construct a generalized Witt algebra (01 ; D]_, . . . , Dm) by the above method. Suppose now that homomorphisms 0i , . . . , $m satisfy ( 2 . 0 . 1 ) and . ( 2 . 0 . 2 ) \u00E2\u0080\u00A2 Let the order of (ft be p n , and let xi , . . . , x n be a set of independent gen-erators of cj . We set 0j.(xj) = a j^ e $ . Then ( 2 . 0 . 1 ) \u00C2\u00AB- 6 \u00E2\u0080\u00A2\u00C2\u00BB and (2.0.2) are respectively equivalent to the following conditions: (2.0.4) i f k^, ... , k n are integers such that ^ a i j k j - 0 , i - l , ... ,m, then k i s ... s k n s. 0 (mod p ) , and (2.0.5) the rank pf the matrix ( a i j ) , i = l j \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 , m \u00C2\u00BB j = 1, ... , n , i s m . Thus a generalized Witt algebra whose dimension i s mpn i s completely characterized by mn elements a^j e ft satisfying (2.0.4) and (2.0.5). From (2.0.5) i t follows immediately that m ^ n . If m = 1 then (2.0.4) implies that ft i s of rank \u00C2\u00A3 n over GF(p) . Therefore i f m e 1 , and ft = GF(p), then n = 1 , so that the only generalized Witt algebra of D-dimension 1 over GF(p) i s the Witt algebra. o - 7 -3. Reduction of the algebras H((X\ D l , \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 , Dm) to orthogonal form. In this section, we show that any-simple algebra of the form = Dj, ... , Dm) can be written as ^ = i (Oi; Vj.t ... , Dffl) , where D^DJ - Dj-Dj = 0 for a l l i , j . An ordered set { Dl, ... , Dm J- of derivations of a commutative associative algebra Ot w i l l be called a system of derivations of OL or simply a system i f i t satisfies (1.0.1) and (1.0.2). We shall say that the Lie algebra {Oi; D^ , ... , Dm) is defined by the system { Dl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 i % ] \u00E2\u0080\u00A2 A system { D J L , ... , D m j - w i l l be called orthogonal i f DjDj - DjDj_ - 0 f o r . a l l i , j , that i s , i f i n (1.0.1) aijk - 0 for a l l i , . j , k, ort ho normal i f there exist m elements f ^ e OL such that D^fj o 8jj (Kronecker delta). An orthonormal system i s always orthogonal. Two. systems { B\u00C2\u00B1t ... D m j and. | D i , ... , Djjj } of Ot w i l l be called equivalent i f there exist c i j e Ot such that D i - TL cijDj ( i =1, ... , m) and such that det(cij) i s a unit of OL . { Dl\u00C2\u00BB > \u00C2\u00B0m } a n d { Di\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm } a r e equivalent i f and only i f \u00E2\u0080\u00A2\u00C2\u00A3 (01 ; Blt ... , Dm) = -\u00C2\u00A3 (OL; B{, ... , Dm) . Lemma 3.1. A system {^lt \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB ^m} of deriva-tions of OL i s equivalent to an orthonormal system i f - 8 -and only i f there exist \u00C2\u00A3]_, ... , f m e 01 such that det ( D i f j ) i s a unit i n Ol . Proof. Suppose that { DT_, ... , D m ]\u00E2\u0080\u00A2 i s equivalent to an orthonormal system { D^j ... , D m }\u00E2\u0080\u00A2 and l e t Dj[ = 2Z.CijDj , Djfj = 8^j , where det(cij) i s a unit i n Ol \u00E2\u0080\u00A2 Then we have Djf -j = C J . J . Thus det(Difj) i s a unit i n OL . Conversely, suppose that det(Difj) i s a unit i n Ol for some \u00C2\u00A3]_, ... , fm e Ol . Let ( c i j ) be the inverse matrix of the matrix (D-jfj) . We set D^ = \u00C2\u00ABZ1 CijDj . Then Dji, ... , D m i s equivalent c ^ t \"to 1^1\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm J a n d w e have \u00E2\u0080\u00A2 \u00C2\u00B0ij \u00C2\u00BB s o that ... , ^ i s orthonormal, which proves the lemma. For a given algebra - \u00C2\u00A3 (Ol 5 D l f ... , Dm) we denote by Ifl the set of a l l elements c E Ol such that Dc = 0 for a l l D e \u00C2\u00A3 \u00E2\u0080\u00A2 A i s a subalgebra of Ol \u00E2\u0080\u00A2 w i l l be called the algebra of constants of *C . Since OL i s always assumed to have a unity element, we have c e i f and only i f Die = ... = D mc \u00E2\u0080\u00A2 0 for some defining system {Di, ... , D m ^ of -s\u00C2\u00A3 \u00E2\u0080\u00A2 The following lemma i s useful. Lemma 3.2. If the algebra of constants has a divisor of zero, then \u00C2\u00A3.(011 D l , \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 , \u00C2\u00B0m) i s not simple. - 9 ~ Proof Let c e jpl be a divisor of zero. The set 3 of a l l cD , where D e , forms an ideal of s& . For, (cD)oD* - c(D\u00C2\u00ABD')e $ . If 3 m 0 then from (1.0.2) i t follows that c \u00E2\u0080\u00A2 0 , a contradiction. If $01 then = c(f i D i + ... + f mD m) for some f^, ... , f m e Ol . Then again from (1.0.2) i t follows that 1 \u00C2\u00AB= c f i , which i s impossible i f c divides 0 , and therefore s\u00C2\u00A3 i s not simple. A commutative associative algebra Ol i s completely primary i f the set of a l l non-units coincides with the radical of Ol . Lemma 3.3. If -JL (Ol \\u00E2\u0080\u00A2 D^, ... , Dm) i s simple then Ol i s completely primary. Proof Since (Ol; Di, ... , Dm) i s simple, from Lemma 3.2 i t follows that the algebra of con-stants has no divisor of zero. Since Ol i s commutative and i s f i n i t e dimensional over the ground f i e l d , i s a f i e l d . Let f e Ol be a non-unit. Since D \u00C2\u00B1fP - pfP-^D-jf - 0 for a l l i , we have fP e . If fP \u00C2\u00A3 0 then fP is a unit in 01 , and hence f i s also a unit. This i s a contradiction. Therefore f p \u00E2\u0080\u00A2 0 for a l l non-units f . Thus OL i s completely primary. Lemma 3.4. Let Ot be completely primary. - 10 If f i , ... , f n be such that f f i = ... = f f n \u00E2\u0080\u00A2 0 with f e Ol implies f - 0 , then at. least one f^ i s a unit i n Ol \u00E2\u0080\u00A2 Proof. Assume that a l l f i are non-units. Then there exists a positive integer k such that f l k * ... \u00E2\u0080\u00A2 f n k = 0 , and hence (3.4.1) f ^ l ... f n r n = 0 i f q + ... + r n a nk , where r i , ... , r n are non-negative integers. Suppose, therefore, that (3.4*1) holds whenever r i + ... + r n > r , a positive integer. Let r i + ... + r n \u00E2\u0080\u00A2 r , f - f i r l ... f n r n . Then ff1 \u00E2\u0080\u00A2 ... - f f n \u00E2\u0080\u00A2 0 , and hence f = 0 . Using complete o induction with respect to r , we can conclude that (3.4-1) holds whenever r i + ... + r n > 0 . In particu-l a r , f i = ... = f n \u00E2\u0080\u00A2 0 . Take a non-zero f e Ol . Then we have f f i = ... = f f n = 0 , a contradiction. Therefore at least one f i must be a unit. We are now i n a position to prove the follow-ing: Theorem 3.5. I f Ol i s completely primary, then any system Di, ... , D m of derivations of i s equivalent to an orthonormal system. In particular, any simple Lie algebra of the type ( 0l\ Di, ... D m^ i s defined by an orthonormal system. - 11 -Proof. Let u i , ... , u n be a basis of Ol over the ground f i e l d . We set (3.5.1) f . Dl*< D l u t . D r u i , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 D r u t r where 1 i> r \u00C2\u00A3 m . We shall prove by using Lemma 3*4 that, f < ' i s a unit for some choice of i l : Suppose, therefore, that f e Ol i s such that f f If \u00E2\u0080\u00A2m. = 0 for a l l . (3.5.2) f f . 0 i s true for some r , and a l l i]_, \u00C2\u00B12* by expanding the determinant f \u00E2\u0080\u00A2 column, we have . , i r , then along the r-th (3.5.3) ftj ir - U c i D i + ... + fe p D p J u ^ - - 0 , where c r \u00E2\u0080\u00A2 f / . V . Since (3\u00C2\u00AB5.3) i s true for a l l i r , we have f C T _ D I + '... + f c R D R \u00C2\u00AB 0 \u00E2\u0080\u00A2 Then from (1.0.2) we have f c i = ... = f c r = 0 , and i n particular f f ^ ... i 1 = 0 for a l l i]_, ... , i r\u00E2\u0080\u009Ei \u00E2\u0080\u00A2 Proceeding by induction with respect to r , we can conclude that (3.5.2) holds for a l l r \u00E2\u0080\u00A2 Taking the case r - 1 , we have f D ] U ^ - 0 for a l l i i . Therefore fDi = 0 . Hence from (1.0.2) we have f \u00E2\u0080\u00A2 0 . Therefore, by Lemma - 12 -3 \u00C2\u00BB4 f \u00E2\u0080\u00A2 \u00E2\u0080\u009E\u00E2\u0080\u00A2 i s a u n i t f o r some i i f ... i m . Then from Lemma 3.1 i t f o l l o w s t h a t \u00C2\u00A3 D i , ... , D m j- i s equiva-l e n t t o an orthonormal system. The second part of the theorem fo-llows immediately from the above r e s u l t and Lemma 3.3\u00E2\u0080\u00A2 - 13 -4 . Some lemmas. We establish here a number of results we w i l l need later. We assume throughout this section that { Di, ... , ]\u00E2\u0080\u00A2 i s orthonormal, that x i , ... , x m e Ol are such that % X J - $\u00C2\u00A3j t and that 3 i s an ideal of - (01 j Di, ... , Dm) . Lemma 4.1. If D = f^Di + ... + f m D m e 7^ > then f k D e for any k . Proof. Since Dxk = f k , we have Do(xkD) a f kD e J . Lemma 4.2. If D = f ^ + ... + f m D m e J and i f f k i s a unit i n Ol , then S1D1 + \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 + gm^ m e 1 t where g k = 1 and where g^ - 0 for any i such that f\u00C2\u00B1 = 0 . Proof. Consider the element U e 7 , where Since f k D e J by Lemma 4 . 1 , we have also V e $ , where * 4 % - 14 -Then we have V - 2U e Cf , where Setting V \u00C2\u00AB- 2U = glDi + ... + gmDm , we have g, - - 4 ^ ^ + , - I, 4 4 and for i / k , Therefore, i f f\u00C2\u00B1 = 0 then g^ * 0 , completing the proof. + Lemma 4.3. If f]_, ... , f m belong to the algebra of constants of and are such that 1 D1 + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + fmDm e 7^ > a n d i f some fk i s a unit, then Di e J for a l l i \u00E2\u0080\u00A2 1, ... , m . Proof. Suppose that f ^ i s a unit. Then (flDi + ... + f m D m ) \u00C2\u00BB ^ D*J - Di e ^ for a l l i s l , ... ,m. Lemma 4.4. Dj_ e implies $ = j\u00C2\u00A3 except when p \u00E2\u0080\u00A2 2 , m = 1 . - 15 -Proof. If D]_ e V then from Lemma 4.3 i t follows that Dj[ e 3 for i = 1, ... , m . Take an arbitrary element f e Ol . Then from Dji\u00C2\u00BB(fD i) = (D-jfjDi we have ( 4 . 4 . D (DjfjDi e ? for a l l i , j . First we consider the case p / 2 . Since D j j x i 2 ) = 2x^ , from (4.4.I) we have 2x iD i e t7 \u00E2\u0080\u00A2 Since p / 2 , we have xiDj[ e Cf> \u00E2\u0080\u00A2 Hence (4.4.2) (fDiJ^XiDi) = fD \u00C2\u00B1 - XitDifjDi e Cf . On the other hand., since Djjxif) \u00C2\u00AB f + Xj[(Dj[f) , from (4.4.1) we have (4.4.3) fDi + XifDifjDi e 3 . From (4.4.2) and (4-4.3) we have 2fDi e 3 . Since p / 2 we have fD^ e $ \u00E2\u0080\u00A2 Since f and i are arbitrary, we have $ - \u00C2\u00A3 \u00E2\u0080\u00A2 Now we consider the case p = 2 , m y 1 \u00E2\u0080\u00A2 For given i we may take j such that j ^ i . Since D J _ ( X J L X J ) = Xj , from (4.4.I) we have X J D ^ e -J \u00E2\u0080\u00A2 Then (fDj )o(xjDi) = fDi - Xj(Dif)Dj e ^ . However, we have xj(Dif)Dj - DitxjfjDj e jf from ( 4 . 4 . 1 ) . Therefore fD^ e jf . Since, f and i are arbitrary we have U - dL , completing.the proof. - 16 ^ 5. Derivations of a f i e l d . A subalgebra of the derivation algebra & (Ol) of Ol w i l l be called regular i f D e , f e Ol imply fD e -\u00C2\u00A3 . 3r(0l) i t s e l f i s a regular subalgebra of $-{01) . I f Ol i s i t s e l f a f i e l d , any regular subalgebra of &(Ol) may be considered as a vector space over the f i e l d Ol , since -JL i s closed under the scalar multi-plication by elements of Ol , that i s , i f D, D\u00C2\u00BB e , f-P+ f\u00C2\u00BBD\u00C2\u00BB e \u00C2\u00A3 , where f, f\u00C2\u00BB e Ol . If *L has a basis Dj, ... , D m over Ol , i t i s easily seen that Dl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm satisfy (1.0.1) a'nd (1.0.2). Therefore, i f Ol i s a f i e l d , any regular subalgebra of .\u00C2\u00A3-(01) i s of the type -\u00C2\u00A3_(0l\ D l , \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 , Dm) \u00C2\u00BB &nd we c a l l m the D-dimension of the regular subalgebra \u00E2\u0080\u00A2 Theorem 5.1. Let Oh be a f i e l d over \u00C2\u00A7 \u00E2\u0080\u00A2 Then any regular subalgebra -JL of the derivation alge-bra of OL over 3? is simple except when p = 2, m = 1 , where m i s the D-dimension of \u00E2\u0080\u00A2 Proof. \u00C2\u00A3 can be written i n the form ^ - (Ol; Di, ... , 1^) . By Theorem 3.5 we may assume that I D]_, ... , J i s orthonormal. 0 Let ^ be a non-zero ideal of and f^Di + ... + f m D m be a non-zero element i n such that the number of non-zero f ^ i s as small as possible. If fk \u00C2\u00A3 0 then by Lemma 4*2 ^ contains an element - 17 -S l D l + \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 + gmDm s u c n that g k = 1 and such that g\u00C2\u00B1 - 0 whenever f i = 0 , so we may assume at the out-set that fk = 1 for some k . Since J i s an ideal, we have B\u00C2\u00B1e(fiDi + ... + f mD m) = (D^f-j^D^ + ... + (L\jf m)D m e J for i = 1, ... , m . Since f k \u00E2\u0080\u00A2 1 , the number of non-zero coefficients i n (Difi)Di + ... + (Dif m)D m i s less than that of f ^ + ... + fjiPm . Therefore D-jfj - 0 for a l l i , j , and hence we have f i , ... , f m e $C , the algebra of constants of \u00C2\u00A3 . Since $C i s a sub-field of Qt , from Lemma 4.3 we have e $ for i = 1, ... , m , and = sC from Lemma 4*4. 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 character-i s t i c 0 , i f we start with an orthonormal system. For example, consider the f i e l d ft(x]_, ... , x m) of rational functions i n m variables x i , ... , x m over a f i e l d ft of characteristic 0 , and l e t Oi be a f i n i t e dimensional extension f i e l d of ft(xi, ... , x m) . Then Oi i s an in f i n i t e dimensional algebra over ft . It i s well known that there exist derivations r - , \u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00E2\u0080\u00A2. z~z of 01 over ft such that -x. a . \u00E2\u0080\u00A2 , and that every derivation D of OX. over ft i s written uniquely 2 o in the form VS!frz+---+fT? > where ~ f^, ... , f m e Ol \u00E2\u0080\u00A2 In other words, the derivation alge* bra \u00C2\u00A3 [Ol) of Ol over ft can be written as - lo -= -L(Ol\ ~ > \u00E2\u0080\u00A2 \u00E2\u0080\u00A2\u00E2\u0080\u00A2 j ^ ) . The above method enables us to prove that ^ ( t f t ) ; i s an i n f i n i t e dimen-sional simple Lie algebra of characteristic zero. If we consider the polynomial domain Ol = <&[xi, ... , x m J , instead of * ( x i , ... , x m ) , as an algebra over $ , 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 Lie algebras, together with i n f i n i t e dimensional algebras constructed by Kaplansky^ method, may be re-garded as analogues of the Witt algebra in the case of characteristic 0 \u00E2\u0080\u00A2 - 19 -6. Simple derivation algebras when $ i s algebraically closed. The main result of this section i s that i f Ol i s a commutative algebra over an alge-braically closed ground f i e l d <\u00C2\u00A3 , then any simple algebra of the type dL(Ol; D]_, ... , Dm) i s a gener-alized Witt algebra. Lemma 6.1. Suppose that \u00C2\u00A3 - \u00C2\u00A3, (Ol; DX, ... , Dm) is simple. If f e Ol i s such that Dif - X\u00C2\u00B1f ., X\u00C2\u00B1 e $ , for a l l i , then f = 0 or f is a unit i n . \u00E2\u0080\u00A2 Proof. If f i s as above, the set 0 of a l l elements of the form fD , where D e \u00C2\u00A3 , i s an ideal of \u00C2\u00A3, . For, i f X. S i D i e \u00C2\u00A3 then (fD). (Z g\u00C2\u00B1B\u00C2\u00B1) = L f(Dgi)Di - Z gi^ifD = f Z((.Dgi)Di - g\u00C2\u00B1XiD) e J . Since i s assumed to be simple, $ = 0 or 3 = 0 1 . If 3 - 0 then f - 0 by ( 1 . 0 . 2 ) . If 0 - 01 then again by (1.0.2) j i s a unit i n Ol , as required. By Theorem 3*5, any simple algebra of the form \u00C2\u00A3(01; D^ , ... , Dm) i s defined by an orthonormal system. Moreover, by Lemma 3\u00C2\u00AB2, the algebra fpt of constants for the simple algebra \u00C2\u00A3, [01; D]_, ... , Dm) is a f i e l d over $ , and i f $ is algebraically closed, we have $? \u00C2\u00BB $ . Since we are mainly interested i n this section i n simple algebras, we shall assume that the conditions (6.1.1) - (6.1 .1) , below hold. The last two of these are - 20 -necessary i f SL (Ol; D^, ... , Dm) i s simple, as i s seen from Lemma 6.1 and the above remark. The ground f i e l d .ft i s assumed algebraically closed. (6.1.1) The system {Di, ... , D m j i s orthogonal. (6.1.2) If f e Ol i s such that Dif - \$f with \\u00C2\u00B1 e ft for a l l i , then f = 0 or f i s a unit i n Ol \u00E2\u0080\u00A2 (6.1.3) Dif - ... = Dmf - 0 implies f e ft . These conditions and the fact that ft is algebraically closed w i l l enable us to prove that Ol i s the group algebra of an elementary p-group. We consider Ol as an ft -module, where the operator domain fi consists of multiplications by elements i n ft and the linear mappings Di, ... , D m (of Ol into i t s e l f ) . Since every two operators i n fL are commutative, and since ft i s algebraically 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 direct sum 2L fou of directly indecomposable fi -submodules. Then, since Dl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 i Dm are commutative, each Di has exactly one characteristic root X i v i n Oiv , when we consider Di as a linear mapping of Otv \u00C2\u00B0into i t s e l f , and there - 21 -exists a non-zero u v e 01 v such that D^uv = A . J [ V U V a l l i and v . By condition (6.1.2), u v i s a unit. Since u{ e 5 by (6.1.3), and since ft i s algebraically closed, we may assume (6.1.4) u\u00C2\u00A3 = 1 for a l l v . We shall prove that a l l the u v forms an elementary p-group with respect to the multiplication i n Ol \u00E2\u0080\u00A2 Lemma 6.2. If Djf = \ j f , \\u00C2\u00B1 e ft ,\u00E2\u0080\u00A2 for a l l i , and i f f ^ 0 , then there exists an Ol p such that f e Olp , \u00E2\u0080\u00A2 ^ip \u00E2\u0080\u00A2 Proof. Let f = J F f v , where f v e 0 l v . Then from Djf - \\u00C2\u00B1f i t follows that ZL D i * \" - H MA> \u00E2\u0080\u00A2 Since Djf v e v , we have Djf v \u00E2\u0080\u00A2 \ j f v for a l l i and v . Suppose that f v ^ 0 \u00C2\u00A3 f^ for two different indices v and u. \u00E2\u0080\u00A2 Then, by condition (6.1.2), f v and are units. By an easy calculation we obtain Djjfyf\"\" 1) = 0 for a l l i . Then by (6.1.3) we have f v ^ 1 e * \u00E2\u0080\u00A2 How-ever, this impossible since Oly C\ 01^ \u00E2\u0080\u00A2 0 , and there-fore a l l but one of the f v are zero. Thus there exists an Olp such that f e C%p . Since f ^ 0 i s assumed, and since V\u00C2\u00B1 has only one characteristic root K\u00C2\u00B1p i n <7Lp , we have \\u00C2\u00B1 = Xip \u00E2\u0080\u00A2 Now, for any two indices v and \i , we have Di(p-v^jx) \u00E2\u0080\u00A2 (^iv + Xiu.)uvU|j, for a l l i \u00E2\u0080\u00A2 Therefore, since - 22 -uvUu. ^ 0 by (6.1.4), i t f o l l o w s from (6.2) th a t t h e r e e x i s t s an OL p such t h a t u v u ^ e Olp and such th a t (6.2.1) \ - [ v + \u00E2\u0080\u00A2 \ \u00C2\u00B1 p f o r a l l i \u00E2\u0080\u00A2 From (6.2.1) i t f o l l o w s t h a t D^u^^u\" 1) = 0 f o r a l l i . Then by (6.1.3) we have u V U|j, - aup w i t h some a e ft , and t h e r e f o r e by (6.1.4) aP - 1 . Hence (a - 1)P = 0 , and we have a \u00E2\u0080\u00A2 1 . Thus U V U J J , \u00E2\u0080\u00A2 up . Therefore a l l the u v form a group O^. w i t h respect to the m u l t i p l i c a -t i o n i n Ol . i s an elementary p-group because of (6.1.4). We s h a l l show t h a t there e x i s t s o n l y one index v such t h a t \ i V \u00E2\u0080\u00A2 0 f o r a l l i \u00E2\u0080\u00A2 I f f - 1 i s the u n i t y element of Ol then D j f \u00E2\u0080\u00A2 0 f o r a l l i . There-f o r e by Lemma 6.2 t h e r e e x i s t s an index 0 such t h a t 1 e OIq . Suppose t h a t \u00E2\u0080\u00A2 0 f o r a l l i \u00E2\u0080\u00A2 Then D i ( u v ) - 0 f o r a l l i . By (6.1.3) we have u v e ft , and hence u v = 1 , v = 0 . We s h a l l prove t h a t (71 v = U v 0 L Q f o r a l l v . To do t h i s , we need some lemmas. Lemma 6.3. For any u- and v , u^Oly i s an fi -submodule of OL . On u^ v , each Dj[ has the one c h a r a c t e r i s t i c r o o t \ i p , where u^u v = Up . Proof. Since 0LV i s an fi-submodule i n which DJL has only the one c h a r a c t e r i s t i c r o o t \ j . v , t h e r e - 23 -exists a basis v i , ... , v n of OL v such that (6.3.1) Djivx = X i vvi , D i V k = X i v v k + ZL a i k s v s , (1 < k < n) , with ol\u00C2\u00B1\hs e $ \u00E2\u0080\u00A2 Then Di(unvi) - Uip, + \iv)u^vi , Di^ uH vk) 8 8 (^ijx + xiv) uM. vk + 2 L a i k s v s \u00C2\u00BB (1< k \u00C2\u00A3 n) . Therefore vi^Oly i s an SI -submodule of Ot , and has the only characteristic root + K\u00C2\u00B1v on v^OLv . By (6.2.1) we have + Xj_v = \\u00C2\u00B1p , completing the proof\u00E2\u0080\u00A2 Lemma 6.4. For any fixed n , Ol - TL u\u00E2\u0080\u009E0^ is a decomposition of OL into a direct sum of SI -submodules, and Ujj0l v i s fi -isomorphic to Ol p , where U^Uy \u00C2\u00AB Up . Proof. F i r s t , since u^ is a unit, any element f \u00C2\u00A3 Ol can be written i n the form f = u^g with g e Ol . Let g = Z. gv \u00C2\u00BB where g v e 0lv . Then f = Z. %gv and Ujj,gv e UpGlv . Moreover, i f Z_ u^gv \u00E2\u0080\u00A2 0 with g v e Olv then \u00C2\u00A3 g v = 0 \u00E2\u0080\u00A2 Since, however, Ol = \u00C2\u00A3 i s a direct sum, we have gv - 0 for a l l v . Hence Up,gv \u00E2\u0080\u00A2 0 . Thus we have proved that Ol \u00E2\u0080\u00A2 \u00C2\u00A3 u^C^v i s a direct decom-position of . By (6.3) above u^ v i s an 1^ -submodule, and the proof i s complete. - 24 -Since the number of direct components i n the decomposition Ol \u00C2\u00BB 21 u^tfly i s the same as that of Ol - 21 Ol v and since each component i n 21 0lv \u00C2\u00B1s directly indecomposable, by the Krull-Schmidt theorem we see that each component i n 21 Uu,Olv i s also directly indecomposable, and that u^^v i s fi -isomorphic to some Ol p . By comparing the characteristic roots, we see that p i s determined by the relation U|jUv \u00E2\u0080\u00A2 Up . (We have used Lemma 6.3 and the fact that \"k^ 5 8 Mp for a l l i i f and only i f cr \u00E2\u0080\u00A2 p) \u00E2\u0080\u00A2 Lemma 6.5. If f e Ol be such that Djf e OIq for a l l i , then f e OIq . Proof. Let f = Z f v , where f v e Olv . Then Djf = Z Djf v 6 OIq . Since Djf v e Olv , we have B\u00C2\u00B1fv - 0 for a l l i i f v ^ 0 . Then by (6.1.3) we have f v e ft . Since v ^ 0 , however, we have f v = 0 . Thus f = fo e OIq . Lemma 6.6. If -is i s an Jl -submodule of Ol such that every Di has only the characteristic root zero i n , then C OIq . Proof. Since 0 i s the only characteristic of every Di i n , there exists a basis wi, ... , wn of -ir such that, for a l l i , - 25 -(6.6.1) 0 ^ - 0 , Diwfc \u00E2\u0080\u00A2 21 <*iksws , (k > 1) , where aiks e \u00C2\u00AE \u00E2\u0080\u00A2 From (6.6.1) and ( 6 . 1 . 3 ) we have W]_ 6 ft . Hence w^ e Ol q . Suppose that wi, ... , Wk-i e Ol o . Then (6.6.1) yields DiWfc e 0 L Q for a l l i \u00E2\u0080\u00A2 Then by Lemma 6.5 we have w^ e Ol q . Pro-ceeding by induction with respect to k , we have Wfc e Qlq for a l l k . Therefore \"fs \u00C2\u00A3 OIq , as required. Now we are ready to prove Ol V \u00C2\u00AB U V ( 7 1 Q \u00E2\u0080\u00A2 By Lemma 6 . 4 , (71 v and U V 0 1 Q as vector spaces over ft have the same dimension. Let v i , ... , v n be a basis of 0 1 v such that ( 6 . 3.I) holds. We set w^ = \hr\u00C2\u00B1 for 1 = 1, ... , n . Then by an easy calculation we see that the basis wj, ... , wn of U y l ^ v satisfies ( 6 . 6 . 1 ) . Therefore by Lemma 6.6 we have v^^Oly C (%q . Hence 01m C u.v01q . Comparing the dimensions, we have (6.6.2) 0lv - U v 0 1 Q . Lemma 6 . 7 . OIq i s a subalgebra of Oh. Proof. Since 0 i s the only characteristic root of every Dj[ i n OIq , we may choose a basis w]_, ... , wn of Qlo such that (6.6.1) holds. Since wi e ft by ( 6 . 1 . 3 ) , we may assume that w^ = 1 . There-fore (6.7.1) wswt e QXq i f s + t S 3 \u00E2\u0080\u00A2 Suppose that (6.7.1) holds for a l l s and t such that s + t < r . Now let s + t = r , w s w t ~T. \u00C2\u00BB where f v e Olv . From (6.6.1) i t follows that Djjwswt) = (DiWs)w+; + ws(D.jwt) e OIq for a l l i . Therefore by Lemma (6.5) we have wsWfc 6 OIq . Proceed-ing by induction with respect to r , we see that (6.7.1) holds for a l l s and t . Therefore OIq i s a subalge-bra of Ol , as required. Since OIq depends on the system { D l f ... , D we may write OIq = ^C*(D1> \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 i Dm) \u00E2\u0080\u00A2 W e shall show that there exists an orthogonal system -^Ei, ... , E M J -equivalent to a given.orthogonal system {D^, Dm j-such that tfl0(Ei, ... , E m) = ft . To do this, i t w i l l b sufficient to show that we can always find an orthogonal, system \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm } equivalent to {^i, ... , Dmj such that the dimension of ^o^Dl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB \u00C2\u00AEm) * s less than that of ^o^Dl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > Dm) whenever the l a t t e r i s greater than one. Take a basis W]_ , ... , wn of OIq{Di, ... , D m), such that (6.6.1) holds. Since W]_ e $ by (6.1.3) we may assume W]_ = 1 . If w2 i s not a unit then by (6.1.3) we have w2P = 0 . Then 1 + w2 i s a unit. By replacing w2 by 1 + w2 i f w2 i s not a unit, we can always assume that w2 i s a unit. From (6.6.1) we have D J W 2 = e ft for a l l i . By (6.1.3) we see that not a l l ^ are zero, say j#i ^ 0 since w2 does not belong to ft . We set x = ^ \u00C2\u00A3 \" ' \" W 2 , - 27 -D l = D l \u00C2\u00BB D i = P l D i \" P i D l for i + 1 . Then 1^ 1> \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 > 1 i s an orthogonal system equivalent to {D1, ... , D m } such that DJX \u00E2\u0080\u00A2 1 , D\u00C2\u00B1x = 0 for i / 1 . We set Di = x D j , D| = Di for i \u00C2\u00A3 .1 . Then Di, ... , D^ | is. orthogonal and equivalent to {Di, ... , JJ^J and hence to {Di, ... , D mj , since x i s a unit. We have (6.6.1) DJx = x ^ 0 , where x e ^ o ( D l > \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > Dm) , (6 .8 .2) Di - 21 c i j D J \u00C2\u00BB where c i j e #?o( Dl, , Dm) \u00E2\u0080\u00A2 We shall show that ^o^ Dl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > Dm) * s pro-perly contained i n Qi Q ( D I , ... , Dm) . Take a,basis v i , ... , v r of ^ o ( D i \u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > Dm) such that (6 .8 .3) D^vi = 0 , D i v k = H a i k s v s , (k > l j , for a l l i , where c t i k s e f t . From,(6.8.3) and (6.1.3) we have vj_ e ft f and hence v i e #lo( Dl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > Dm) \u00E2\u0080\u00A2 Suppose that y i , ... , Vk\u00E2\u0080\u009Ei e ^ o( Dl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm) \u00E2\u0080\u00A2 Then from (6.8 .2) and (6.8 .3) we have (6 .8 .4) D i V k = 21 Z c i j a j k s v s \u00E2\u0080\u00A2 s< \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm) \u00E2\u0080\u00A2 Proceeding by induction with respect to k , we have 28 -vfc e ^ o ( D l > \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm) for a l l k . Therefore ^\u00E2\u0080\u00A2o( Dl> \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 * Dm) 9 ^o( Dl\u00C2\u00BB > Dm) i s proved. Suppose that Ol0(B{, ... , D^) - 0 l o ( D l > \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm) = <^0 \u00E2\u0080\u00A2 S i n c e f e OIq implies D-Jf e OIq , we can regard D{ as a linear mapping of OIq into i t s e l f . By the definition of 0 i s the only characteristic root of Dj[ i n O^q . However, this contradicts to ( 6 . 3 . 1 ) . Thus ^ 0 ( D i > \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm) i s properly contained i n 01q{Vi, \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 t Dm) , and hence the dimension of 0 t o ( D i f \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm) i s l e s s t h a n t h a t o f ^ 0 ^ D 1 \u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm) Repeating the above process, we obtain an orthogonal system f El> \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > Em)\" equivalent to the given system \"^ D^ , \u00E2\u0080\u00A2 \u00E2\u0080\u00A2\u00E2\u0080\u00A2 D m such that OtQiE^, ... , E m) i s one-dimensional. Since the algebras (01; D^ , ... , Dm) defined by equivalent systems are the same, we may suppose that Ol o ^ o ^ D l > \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm) i s one-dimensional. ,Then ^ Z Q = $ \u00C2\u00BB a n (* from (6.6.2) we have (6.8.5) 0? - YL\u00C2\u00AE uv \u00C2\u00BB D i u v e Mv uv , for a l l i and v . From (6.3.5) we see that Ol i s the group algebra of the elementary p-group <7J. formed by a l l u v . We shall show that i f (6.8.5) holds, then jl(0l; \u00C2\u00BB Dm) i s isomorphic to a generalized Witt algebra. We define mapping &\u00C2\u00B1 of Cfj. into ft by #i(u v) = \ i v . Then from (6.2.1) i t follows that 1^> \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB #m a r e ho.momorphisms of Ol into the - 29 -additive group of ft . We shall show that (2.0.1) and (2.0.2) are satisfied by ^1, \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB \u00E2\u0080\u00A2 Suppose ^l( ucr) ^m(ucr) \u00E2\u0080\u00A2 0 \u00E2\u0080\u00A2 T h e n M.cr \u00E2\u0080\u00A2 0 *\"\u00C2\u00B0 r a l l i , and hence cr \u00C2\u00BB 0 , u f f = 1 \u00E2\u0080\u00A2 Thus (2.0.1) i s satis-f i e d . Suppose now that + \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 + am^m s = 0 \u00E2\u0080\u00A2 Then J^LctiXiv = 0 for a l l v , and hence from (6.6.5) we i have aiDi + ... + a mD m = 0 . Then (1.0.2) yields a i = ...=.am = 0 . Thus (2.0.2) i s also proved to be satisfied. Therefore by the result i n section 2 &(0L\ D]_, ... , Dm) i s isomorphic to a generalized Witt algebra. Thus we have proved the following: Theorem 6 . 6 . Suppose that ft i s algebraically closed and that the system -[Di, ... , D m ]\u00E2\u0080\u00A2 i s orthogonal. Then the algebra \u00C2\u00A3, {Ol; Di, ... , Dm) 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) If f e Ol is such that Djf - X\u00C2\u00B1f , where Xi e ft , for a l l i , then f = 0 or f i s a unit in Ol . (6.1.3) D]f - ... \u00E2\u0080\u00A2 D mf m 0 implies f e ft . In particular, i f an algebra of the form \u00E2\u0080\u00A2\u00C2\u00A3{01; D]_, ... , Dm) , where { Di, ... , D m} i s not necessarily orthogonal, over an algebraically closed f i e l d ft i s simple, then -\u00C2\u00A3 i s isomorphic to a - 30 -generalized Witt algebra and Ol> to the group algebra of an elementary p-group. - 31 -7\u00C2\u00AB Nilpotent systems ( 1 ) . A system \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB ^mj w i l l be called nilpotent i f there exists a .positive integer k such that D^ k = ... = Dmk = 0 . If the ground f i e l d ft is algebraically closed then \u00E2\u0080\u00A2^Di, ... , D m | i s nilpotent i f and only i f OloiB]., \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 , Dm) = Ol . In the preceeding section we have proved that i f ft is algebraically closed then any simple algebra of the form (01; D^ , ... , Dm) can be defined by an orthogonal system for which OIQ = ft \u00E2\u0080\u00A2 The case OIQ \u00C2\u00BB Ot and the case Oto - ft are two extreme cases. Now we shall prove the following: Theorem 7.1. Suppose that ft i s algebraically closed. Then any orthogonal system \u00C2\u00A3^1, , D m} satis-fying (6.1.2) and (6.1.3) i s equivalent to a nilpotent orthogonal system. In particular, any generalized Witt algebra over ft can be written i n the form \"\u00C2\u00A3.(0i; Dj_, ... , Dm) , where Ol i s the group algebra of an elementary p-group and where {\u00C2\u00AElt \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > \u00C2\u00AEmj i s a nilpotent orthogonal system. Proof. We shall use the notations employed i n the preceeding section. Because of the remark i n the f i r s t paragraph of this section, i t i s sufficient to prove the following: If { ^ l , \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 , D m j- i s an orthogonal system satisfying (6.1.2) and (6.1.3) and i f OIQ = Ol Q( d1> \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 t \u00C2\u00AEm) ^ Ot then there exists an - 32 ~ orthogonal system {D-[, ... , j- equivalent to { D l i \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm}1 such that OIQ i s properly contained in OIQ - ^o^Di\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm) \u00E2\u0080\u00A2 B y (6.6.2) we have & ~ Z- u v '^o > D i u v 8 8 ^iv uv \u00C2\u00BB where e ft . There-fore, i f OIQ ^ Ol then there exists a v ^ 0 . Then Xiv t 0 for some i , say i \u00E2\u0080\u00A2 1 . We set = D]_ , n . - I D i ~ Hv D\u00C2\u00A3 ~ ^ i v D l ^ o r i r 1 i a n d x = \,\u00E2\u0080\u009Euv . Then x i s a unit and { D \u00C2\u00A3 , ... , i s an orthogonal system equivalent to {D'I, ... , D M J such that D^x = X , D J X - 0 for i ^ 1 . We set D{ - x - 1 ^ and D[ \u00E2\u0080\u00A2 Dj for i ^ 1 . Then {D{, ... , D m} i s an orthogonal system equivalent to ... , , and hence to |Di, ... , D m| , such that D^ x = 1 , D|X = 0 for i ^ 1 . Therefore x e Ol Q by Lemma 6 . 5 . On the other hand, since x = X'^ u v with v \u00E2\u0080\u00A2/> 0 , we have x i OIQ . Thus 0 l o ^ ^ 0 \u00E2\u0080\u00A2 Since x e OIQ , from the above con-struction we have (7.1.1) j)[ = I^eijDj- , C^. eOlQ . Using (7.1.1) and proceeding the same way as i n the pre-ceeding section, we see that OIQ i s properly contained in OIQ \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 Remark. A derivation D of Ol over ft w i l l be called normal i f Df = 0 implies f e ft . It i s clear from the above proof that i f Di i s normal then - 33 -Xlv ^ 0 f o r any v ^ 0 and hence =..x-lD]_ i s also normal. Therefore i f {D^, , D m 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 nor-mal then there exists a nilpotent orthogonal system {D-[, ... , } equivalent to ... , D m } such that D{ i s normal. This fact w i l l be used l a t e r i n section 9\u00C2\u00AB The above r e s u l t may be refined i f i t i s combined with the following: Theorem 7.2. I f a nilpotent orthogonal system {Dl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm } s a t i s f i e s (6.1.3) then there exist X ] _ , ... , x n e Ol such that the elements x\"' ... x^ , where 0 t> < p , xQ = 1 , xj? e ft , form a basis of Ol over ft and such that D-jxi e f t , DjXfc e ft(x]_, ... , xjj.i) , the subalgebra of Ol gen-erated by xj_,. ... , x k_]_ over ft , f o r a l l i and k > 1 . I f , i n p a r t i c u l a r , ft i s perfect i n the sense that every element i n ft i s a p-th power of an element i n ft , then x i , ... , x n may be taken such that either = ... = xP = 1 or = ... = = 0 . The proof follows e a s i l y from the following two lemmas. Lemma 7.3. Suppose that { D l f ... , D m } i s a nilpotent orthogonal system. I f V]_, ... , v r e Ol - 34 -are linearly independent over ft , i f D ^ V T _ = 0 , and i f DjV k i s a linear combination of v^, ... , v^ ...^ for a l l i and k 1 , then there exists an element v e 01 which i s not a linear combination of V T_, ... , v such that D^ v i s a linear combination of V ] _ , ... , v r for a l l i , provided that Ol i s not spanned by vl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 J v r \u00E2\u0080\u00A2 Proof. Denote by ^ k the fi -subspace of spanned v i , ... , v^ . Then 8?2 rO \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 <\u00C2\u00A3 # r and each factor space $ k/^k-1 * s one-dimensional. Since any increasing sequence of fi -subspaces of an (l-space (Jl can be refined into a composition series of Ol , there exists a composition series o \u00C2\u00B1 < p , x9 - 1 , are l i n e a r l y independent over ft and such that D i x k e $ ^ x l > \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > x k - l ) f o r a 1 1 1 a n d ^ ' I f x r + l & \u00C2\u00AE(xl> ... , x r) i s such that. \u00E2\u0080\u00A2 ^ i x r + l e ^ ( x l i \u00E2\u0080\u00A2\u00E2\u0080\u00A2.\u00E2\u0080\u00A2 \u00C2\u00BB x r ) f\u00C2\u00B0 r a l l i \u00C2\u00BB then the elements x u* ... x^*' , where 0 ^ v\u00C2\u00B1 < p , x? = 1 , are l i n e a r l y independent over ft . Proof. An element of the form where 0 < v-^ < p , w i l l be c a l l e d a monomialr and the number w \u00C2\u00AB= w(y) = + v 2p + ... \u00E2\u0080\u00A2 + v r p r \" \" l the weight of the monomial y \u00E2\u0080\u00A2 A monomial i s uniquely determined by its.weight. A monomial of weight w w i l l be denoted by y w . I f f - a 0yo + a i y i + ... + a w y w , where a \u00C2\u00B1 e ft , a^ .. / 0 , then the weight w(f) of f i s defined by w(f) = w . I t follows e a s i l y from our assumption that w(D if) < w(f) f o r a l l i i f 0 ^ f e ft(xx, ... , x r) . Any l i n e a r combination of the elements x ' ... x T can be w r i t t e n i n the form T +' 1-1 f o + f i x r + i + ... + f p - i x with fQ, ... , fp - 1 e ft(x].,-... , x r ) . We s h a l l prove by induction with respect to k that i f f o , ... , fk e ^(xl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > xr)\u00C2\u00BB 0 ^ k < p , then (7.4.1) f0 + f i x r + i + ... +-fk x r+ 1\"- 0 .. .implies f0 = ... = f k = 0 . - 36 -If k \u00E2\u0080\u00A2 0 then (7.4.1) i s clear. Suppose that (7.4.1) holds for a l l k < v but not for k = v . Let k *\u00C2\u00BB v , k / f 0 + f i x r + i + ... + fk x r + l \u00E2\u0080\u00A2 0 , fk r 0 , and. l e t f k be of minimal weight with respect to this property. For any 1 , we have. (7.4.2) ( D - j f ^ x ^ + ( ( k D i X r + 1 ) f k + Dif k - l ) * ^ , ' + ... - 0 . Since w(DjLfk) < w(fk) , we have D^k \u00E2\u0080\u00A2 0 for a l l i . Then (6.1.3) yields fk e ft . Since fk t 0 , we may assume f k = 1 . Then (7.4.2) yields ^(kXp+i + f ^ i ) = 0 for a l l i , and hence by (6.1.3) k x r + i + fk -1 e f t . Since a < k < p , this contradicts the assumption that x r + l i ^(xl\u00C2\u00BB \u00C2\u00AB.. \u00C2\u00BB. x r ) \u00C2\u00BB Thus (7*4.1) i s proved f o r a l l k , completing the proof of the Lemma. . An algebra s\u00C2\u00A3 over ft i s called normal simple i f -JLYL * s simple for any extension K of ft . . i s normal simple i f -S\u00C2\u00A3K i s simple for any algebraically closed extension K of ft . It i s known [ 4 j that the generalized Witt algebras are normal simple i f p > 2 or i f p = 2 , m > 1 . Theorem 7.5. Suppose that p > 2 or that p \u00E2\u0080\u00A2 2 , m > 1 . If J % , ... , D m | i s a nilpotent orthogonal system then -\u00C2\u00A3 = (01 ; D]_, ... , Dm) i s simple i f and only i f the algebra <\u00C2\u00A3T' of constants of \u00E2\u0080\u00A2jC i s a f i e l d , while -\u00C2\u00A3 i s normal simple i f and only i f \u00C2\u00A3T= ft . We need a general remark. Let s\u00C2\u00A3 be an algebra over ft , and ftT a subfield of ft \u00E2\u0080\u00A2 Since ^ i s a vector space over ft , s\u00C2\u00A3 can be regarded as a vector space -\u00C2\u00A3J over ft* . The multiplication xy in i\u00C2\u00A3 is bilinear as a multiplication i n T , and we have (ax)y = x(ay) = a(xy) for a e f t 1 , x , y e f t . Therefore s\u00C2\u00A3 T i s an algebra over ft1 , although not necessarily f i n i t e dimensional. If { u^ J i s a basis of -s\u00C2\u00A3 over ft , and i f | aj J i s a basis of ft over ft1 , then the set {aju^} i s a basis of -\u00C2\u00A3 f over ft1 \u00E2\u0080\u00A2 We refer to the algebra -\u00C2\u00A3J as n \u00C2\u00A3 regarded as an algebra over ftT n . Lemma 7*6 below i s probably well known, and i n any event the proof may be readily supplied by the reader. Lemma 7\u00C2\u00AB6. \u00C2\u00A3 i s simple i f and only i f \u00C2\u00A3 i s simple. Lemma 7\u00C2\u00BB7* If ft has a f i n i t e degree > 1 over ft1 , then \u00E2\u0080\u00A2\u00C2\u00A3} i s not normal simple. Proof. Since ft is algebraic over ft1 , there exists an extension K of ft1 such that ftg has a zero divisor a . The set $ of a l l elements of the form af , where f e \u00C2\u00A3fc i s an ideal of , since (af)g = a(fg) for a l l f, ge s C j . J i s different from .- 38 -zero, since a ^ 0 . We shall show that 7^ ^ S^ K \u00E2\u0080\u00A2 The set of a l l x e ftj^ such that ax = 0 i s a sub-algebra of ftg of dimension 1 , so l e t a i , ..* a r be a basis of this subalgebra over K .. Take ar+l\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > a s 6 % such that a^, ... , a s i s a basis of % over K . Since a ^ 0 , we have r < s . Let ul\u00C2\u00BB > u n be a basis of \u00E2\u0080\u00A2\u00C2\u00A3 over ft .. Then a J U J L , j \u00C2\u00AB 1, ... , s, i = 1, ... , n , form a basis of over K . Then {aajU^j i s a basis of CJ over K , and aa^ = ... = aa r = 0 , so that jf \u00C2\u00A3 \u00E2\u0080\u00A2 There-fore -sCg i s not simple, and therefore i s not simple, as required. Consider the algebra -\u00C2\u00A3.(01 J ^ l , .... , Dm) whose.algebra & of constants i s a f i e l d . Since $C i s a subfield of the algebra Ot , we may consider Ol as an algebra Ol over fif . Since D^c = 0 for a l l c e jfC , Di defines a derivation \"D i of Ol . It i s easily seen that it (01; ^ l , \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 , Dm) i s t h e algebra \"3^ (01 ; Di, ... , D m) regarded as an algebra over ft . Therefore by Lemma 7\u00C2\u00AB6 -\u00C2\u00A3,(Ol] D]_, ... , Dm) i s simple i f and only i f \u00C2\u00A3(01 ; D^ , ... , Dm) i s simple, provided that \u00C2\u00A3C i s a f i e l d . Lemma 7 .8. Let $C be the algebra of constants of -\u00C2\u00A3_(0l\ Di, ... , Dm) , and K an extension of ft . Then the algebra of constants o f ^ \u00C2\u00A3 (Otj^; Di, ... , D m) - 3 9 -i s \u00E2\u0080\u00A2 Proof. Let uj_, ... , u r be a basis of and U]_, ... , u r, ... , u n a basis of Ot . Suppose f = 21 a i u i \u00C2\u00BB where e K ., belongs to the algebra of constants of (#K> d 1 \u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > Dm) \u00E2\u0080\u00A2 W e shall show that c t r + i = ... = a n = 0 . For any i , we have a r +iDjU r +]_ + ... + ct nDiU n = 0 \u00E2\u0080\u00A2 If ar+l> \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB a n were not a l l zero, then there would exist | 3 r + i , ... , j 3 n e ft , not a l l zero, such that P r + l D i u r + l + + P n D i u n \" 0 f o r a l l i \u00C2\u00BB since D i u j E Ol \u00E2\u0080\u00A2 Then we have j3 r + i u r + i + ... + |3 n u n e , a contradiction. Thus ct r +]_ = ... = ctn = 0 . \u00E2\u0080\u00A2 Therefore the algebra of constants for -\u00C2\u00A3{0l%; D]_, ... , Dm) i s K \u00E2\u0080\u00A2 Proof of 7.5. Suppose that \u00E2\u0080\u00A2\u00C2\u00A3 is simple. Then, by Lemma 3.2, <\u00C2\u00A3T i s a f i e l d . Suppose that i s normal simple. Let K be an algebraically closed extension of ft . By Lemma 7.6 the algebra of constants of is K \u00E2\u0080\u00A2 Since C^ K i s a f i e l d , 8l- $ \u00E2\u0080\u00A2 Conversely suppose that i s a f i e l d . F i r s t consider the case ft , and l e t K be an algebrai-cally closed extension of ft . Then by Lemma 7.8 the algebra of constants of i s K . Since K i s algebraically closed, and since j ^ i , ... , D m f i s nilpotent and orthogonal, by Theorem 6.8 K i s a generalized Witt algebra. Hence is simple. There-fore i s normal simple. Since the algebra of con-stants of -L(0\,\ Di, ... , D m ) i s always \u00E2\u0080\u00A2\u00C2\u00A3.(01', D]_, . . . , D m ) i s normal simple, and hence \u00E2\u0080\u00A2\u00C2\u00A3.{01; Di, ... , D m ) i s simple. Corollary 7 . 9 . The derivation algebra of the group algebra OL over ft of an abelian group 0\u00C2\u00A3 whose order i s divisible by p i s simple i f and only i f 0^ i s an elementary abelian group, provided that the order of i s > 2 . Proof. Suppose that Cfy i s an elementary p-group with independent generators x i , . . . , x m . Then Ol = ft(xi, ... , x m ) and i t i s easily seen ( [ 2 ] , p. 2 1 7 ) that 3r(0l) - -\u00C2\u00A3(01; \u00C2\u00A3 ? , J * ' v f e ^ ^ ^ be the algebra of constants for , and l e t f e Then _9_f - 0 for a l l i clearly implies that f e ft . Hence where a^k e ft(xi, ... , xk-i) , o^k e \u00C2\u00AE f \u00C2\u00B0 r a l l i and k . Unless p = 2, m = 1, \u00E2\u0080\u00A2\u00C2\u00A3 (01 ; D]_, ... , DM) i s normal simple i f and only i f the following two conditions are satisfied: (7.10.1) For any k , there does not exist f e ft(xi, ... , xk-l) such that aik = Dif . for a l l i . (7.10.2) If integers p.^ , ... , u.g are such that s 1^ ^ikM-k = 0 f \u00C2\u00B0 r a l l i , then m = . . . = n s \u00C2\u00AB o (mod p ). - 42 -Proof. We may assume at the outset that ft is algebraically closed. By Theorem 6.S, \u00C2\u00A3 = H(Ol\ J ) L F ... , Dm) . i s simple i f and only i f (6.1.2) and (6.1.3) are satisfied. Suppose that i s simple. Since DjXk = aik , (7.10.1) follows from s (6.1.3) . Suppose now that 0 ?> < p, ^LLaikM-k = 0 for a l l i . We set f = y^* ... y r* . Then D^f - 0 for a l l i . Therefore (6.1.3) gives f e ft . Hence M-i \u00C2\u00BB \u00E2\u0080\u00A2.. = u-s = 0 \u00E2\u0080\u00A2 Thus we have (7 .10.2) . Conversely, suppose that (7.10.1) and' (7*10.2) are satisfied. F i r s t we shall prove (6.1.3) for the case when f e ft(x]_, ... , x r ) . If r = 1 then this i s clear, since DjX]_ = a i i e ft and not a l l a^i are zero. We shall proceed by induction with respect to r . Suppose that r > 1 and that (6.1.3) i s true i f f e ft(x]_, ... , x r _ i ) . Suppose now that f = bo + b]_x r + \u00C2\u00AB\u00C2\u00BB\u00E2\u0080\u00A2 + bk x\u00C2\u00A3 \u00C2\u00BB where b o , . . . , bjj E ft(xi, .... , x r\u00C2\u00ABi), bk ^ 0 , and that D\u00C2\u00B1f - 0 for a l l i . Then (7.10.3) Djf - (Djb 0 + bT_air) + ... + (Djb k_i + kbkairix^\" 1 + (DjbjJxJ; = 0 . Therefore D^bk - 0 for a l l i . Then the induction assumption gives bk E ft . From (7.10.3) we have D i D k - l + k b k a i r = 0 f o r 9 1 1 1 \u00E2\u0080\u00A2 I f k ^ 0 we set h = ( k b k ) \" ^ ^ ! \u00C2\u00AB then we have h s ft(xi, ... , x r - i ) - 43 -and a i r + Dih = 0 for a l l i , a contradiction. There-fore k = 0 . Thus f e ft(xi, ... , x r _ i ) and the i n -duction assumption gives f s ft . Thus (6.1.3) i s proved for f e ft(xi, ... , x r) . To prove (6.1.3) for the general case, suppose that .f *2If u 7^ ... 7 ^ where f \" e $ ( x i . . . . , Xy.) , and that Dn-f \u00C2\u00AB 0 for a l l f-t / * s \u00C2\u00B1 1 i . Then, since Dif = JT (D \u00C2\u00B1f . . . yus + f aivuujyA 1' ... y/4* = 0 , we have D i f - u M + f M T a i k ^ k - 0 for a l l Hi, ... , u.s and i . Since 0 i s the only character-i s t i c root of Di in ftfx^, ... , x r) , we have X. ctikM-k ~ 0 f \u00C2\u00B0 r a l - \" - 1 a n <* f^ l> > l^s s u c n \"that f tt f 0 . Then (7.10.2) yields u-n - ... - n_ \u00C2\u00AB 0 . Therefore f e ft(xi, ... , x r) and hence f e ft . Thus (6.1.3) i s proved. We shall prove next that (6.1.2) holds. Suppose that g = T~ g y^' ... y^ s , where g e ft(xi, ... , x r) , and that Dig = \ i g with \,- e <\u00C2\u00A3 for a l l i . Then we have D,-g + < I a i t f k - , \u00E2\u0080\u00A2 0 f o r a 1 1 M-l, ... , H s and i . Therefore, as before, g eft and ' >/\u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 /*\u00E2\u0080\u00A2 s (7.10.4) aikl J'k - x i \u00E2\u0080\u00A2 0 for a l l i i f g f 0 . If g / 0 / g , then from (7.10..4) we have - 44 -Z - a i k ^ k - n\u00C2\u00A3) - 0 f o r a l l i . Then (7.10.2) gives u.^ & ^ ( mod P ) f o r a l l k . Since 0 => 1% < p , 0 S < p , we have \u00C2\u00AB . Therefore g \u00E2\u0080\u00A2\u00C2\u00BB a y ' * ' . . . y ^ s with a e ft and some u^, ... , u-s . Then g = 0 or g i s a unit according as a =\u00C2\u00BB 0 or a ^ 0 , since y]_, . . . , y s are u n i t s . Thus (6.1.2) i s also proved. - 45 -8. Nilpotent systems (2). The case m = 1 . If the D-dimension m = 1 , then we can s t i l l further sharpen the results obtained i n the preceding section. In particular, i t w i l l be proved that any generalized Witt algebra of the form \u00E2\u0080\u00A2\u00C2\u00A3 ( O l ; D) over an algebrai-cally closed f i e l d i s uniquely determined by i t s dimen-sion. The result obtained here w i l l be the basis of the argument in the next section. Consider the group algebra Ol = ft(xi, ... , x r) of an elementary p-group with independent generators X]_, ... , x r and the derivation D of Ol defined by (8.0.1) D = \u00C2\u00A3 + - <., 4 \u00E2\u0080\u00A2 Then D i s nilpotent. Let y w = x^' ... x^r be a monomial of weight w = v-j_ + v\u00C2\u00A3p + ... + v r p r ~ l . Then Dy w i s easily seen to be a linear combination of monomials of weight 2 then the algebra -\u00C2\u00A3_(0\; D) i s normal simple. Remark. Jacobson ( [ 3 j , Theorem 4- ) proved the existence of a derivation D of Ol satisfying (6.0.1) - 46 - x under the condition that ft is i n f i n i t e . However, the above argument shows that such a derivation exists for any f i e l d ft . This fact w i l l be further generalized in the next section. Lemma 8.1. If f e OL i s of weight w \u00C2\u00A3 1 then Df is of weight w - 1 . Proof. We may assume that f = y w i s a mono-mail of weight w . Suppose that' Dy w i s of weight < w - 1 . Then Dy^, ... , Dy w are linear combinations of YQt \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > yw-2 \u00C2\u00BB a n <* bence there exist a l > \u00C2\u00BB aw e ^ > which are not a l l zero, such that X- a i D 7 i = 0 \u00E2\u0080\u00A2 Hence we have D( Ji^\u00C2\u00B1Y\u00C2\u00B1) \" 0 , u\u00C2\u00B1Y\u00C2\u00B1 e ft , and a-^ \u00C2\u00BB ... = a w = 0 , a contradiction. Therefore Dy w i s of weight w - 1. As an immediate consequence of Lemma (8.1) we have: Lemma 8.2. If 0 ^ w < p r - 1 then there exists an element f e OL such that Df = y w . Now we consider an arbitrary algebra ;\u00C2\u00A3 (01; D) of D-dimension m = 1 , where D is a nilpotent deriva-tion such that Df = 0 implies f e ft . We shall assume that ft is perfect. If (% i s of dimension > 1 then we can easily find an element x e OL such that Dx = 1 , xP = 1 . Then 1, x, ... , xP\"-1- are linearly independent. - 47 -Suppose we have already found x^, ... , x k e Ot satis-fying (8.3.I) - (8.3.3) below: (8.3.1) x? = 1 for a l l i = 1, ... , k ; (8.3.2) The elements x ... x * , where 0 ^ < p , x^ = 1 , are linearly independent over ft ; h-\ f-l j>-l (8.3.3) Dx^ = 1, Dx2 = x^ , ... , Dxk = ... x If Ot i s not spanned by the elements x ... x !^ , then by Lemma 7\u00C2\u00BB3 there exists v e Ol such that Dv e ft(xi, ... , x k) while v i ft(xi, ... , x k) . We set Dv = ax ... x + g , where a e ft and where g i s a linear combination of monomials of weight < p r - 1 . By Lemma 8.2 there exists f e ft(xi, ... , x k) such that Df = g . Then D(v - f) \u00E2\u0080\u00A2 ax^\"'... x^\"' . Hence a ^ 0 , otherwise D(v - f) - 0, v - f E ft , and v e ft( xl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > xk) \u00E2\u0080\u00A2 Since ft i s perfect, there exists y3 e ft such that x^+i = a~l(v - f) + y3 satisfies x * - 1 . satisfying x = 1 . Thus we have proved the existence of x k + i (8.3.4) Dx k + 1 = x r ... x ^ , x ^ = 1 , xk+l t $ ( x l \u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB *k> \u00E2\u0080\u00A2 Then by Lemma 7.4 the elements x ' ... x are linearly independent over ft . Repeating the above \u00E2\u0080\u0094 46 \u00E2\u0080\u0094 process we obtain x j , ... , x r e Ol such that the elements x u> ... x \" r , where 0 S VA < p . form a i r ' x * basis of Ol and such that (6.3.4) holds for a l l k . Let OJ. be the multiplicative, group generated by x^, ... , Xj, . Then Ol - ^(xl\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > x r ) i s t n e group algebra of 0^ over ft , and D can be written in the form ( 6 . 0 . 1 ) . By a similar argument we may choose x^, ... , x r satisfying xP \u00C2\u00BB ... xP = 0 instead of p r x = ... = x p = 1 . Thus we have proved 1 r Theorem 6.3\u00C2\u00BB Suppose'that .ft i s a perfect f i e l d . Then any algebra \u00C2\u00A3 (01 ; D) defined by a n i l -potent derivation D such that Df = 0 implies f e ft i s isomorphic to an algebra (6.3.5) \u00E2\u0080\u00A2\u00C2\u00A3 (ft(xi, ... , x r) ^ + x + \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 + x*\" ... x>\"' A ') , l ' r-i 7xr where X;L, ... , x r are such that the elements x^ ' ... x r , 0 > < p , form a basis of ft(xi, ... , x r) and such that ^ (6.3.6) x? = ... = xP = 1 . 1 r We may take 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 in the form ( 8 . 3 . 5 ) . If Ol \u00E2\u0080\u00A2\u00C2\u00BB ft(xi, ... , x r) then any generalized Witt algebra \u00E2\u0080\u00A2\u00C2\u00A3 (Ol ; DQ_, ... , D R) of D-dimension r i s isomorphic to the algebra \u00C2\u00A3 (OX ; , . . . , ^ ) . The proof of the second part of Corollary 6.4 is as follows: Let D-^ = 2_ jXj ^ , where a^j.e.ft . Then the matrix (a\u00C2\u00B1jl i s non-singular. Therefore { D l , \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > Dm} i s equivalent to | x x ^ , ... , x r ^ ] and hence to \u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 , } * Therefore \u00C2\u00A3{0LI D L ... , D R) - , . . . , X ) . Thus the problem of c l a s s i f i c a t i o n of the gen-eralized Witt algebras i s completely solved for the two extreme cases: m = 1 and m = r . The author has been unable to solve this problem in general. - 50 -9- Normal systems* We recall that a deriva-tion . D of Ol i s called normal i f Df = 0 implies f e ft , where ft is the ground f i e l d . A system [D^, ... , D m | w i l l be called normal i f some D^ i s normal. A system { D]^ .... , D m J w i l l be called principal i f Ol = ^ ( ^ i , ... , x n) i s the group algebra of an elementary p-group with independent generators x 1 ? ... , x n , and i f DjXj = a i j x j w i t n a i j e * f o r a l l i and j , where the matrix ( a i j ) satisfies (9.0.1) and (9.0.2) below: (9.0.1) i f integers k^,.... , k n are such that Principal systems, which were used in section 2 to define generalized Witt algebras, are always orthogonal and the matrix ( T^j) is non-singular. Theorem 9.1. Suppose that ft i s an i n f i n i t e a-ij-kj = 0 for a l l i , then, k^ s. ... =. k n = 0 (mod p ) ; (9.0.2) the rank of ( a 1 s ) i s m . f i e l d . Then for any principal system , Dm ) \u00C2\u00BB m. J there exists a normal scalar-equivalent to Proof. Let ... , $ m be indeterminates over ft , and consider the p n linear forms ^ ( k i , ... , & n) a l i a i j k j , where the kj are integers and 0 S kj < p . Since these p n forms are distinct because of (9.0.1), and since ft i s i n f i n i t e , by the theory of specialization there exist #1> , 7m e $ such that the p n values atk^, ... , k n) = 2Z. ^ i a i j k j a r e distinct. We set D = + ... + VMDM . Then D(x*\u00C2\u00BB ... xf\") = a(ki, ... , kn)x*^' ... x^ n , and hence D i s normal. If * i / 0 we set . D ^ = D , v[ = D A for i > 1 . The system {D ^ , . . . , D M j i s normal and principal, as 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. Suppose that ft i s algebraically closed. Then any orthogonal system -[D ^ , . . . , D M J satisfying (6.1.2) and (6.1.3) i s equivalent to a normal nilpotent orthogonal system. The characterization of the generalized Witt algebras given in the following theorem contains consider-ably fewer parameters than that given by Kaplansky. Theorem 9.3. Suppose that ft i s algebraically - 52 -closed. Then any generalized Witt algebra over ft can be written in the form -at (01 ; D]_, ... , Dm) , where Ot = ft(x]_, ... , x n) is the group algebra of an elementary p-group with independent generators x j , ... , x n , and where ( 9 . 3 . D V, = \u00C2\u00A3 + + \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 + *T~ ' \"^4 (9.3-2) \u00E2\u0080\u00A2 o - i . A - * Tin, ^X-n) with \u00E2\u0080\u00A2 pj_ ^ e ft \u00E2\u0080\u00A2 (i>0. Proof. By Theorem 9 . 2 , a generalized Witt algebra can be written i n the form \u00C2\u00A3(0t; D -L, ... , Dm) , where [Di, ... , D m } i s a nor-mal nilpotent orthogonal system. We shall assume that i s normal. Then by Theorem 8.3 there exist x-i, ... , x n e Ot such that x p = ... = x p = 1 , such 1 n that the monomials x ' ... x^*1 , 0 ^ VA < p form a basis of Ol over ft , and such that D]_ takes the form <9.3\u00C2\u00AB1)\u00C2\u00AB Suppose that D is an arbitrary deriva- 0 tion of Ol commutative with D]_ . From D^Dx^) = D(Dixi) = 0 , we have Dxi = fii e ft . For any k > 0 , we have Di(Dx k + 1) = D(D l X k + 1) = DUD^x*\" 1 J = (DD^x*\"' - (D-,xk)(Dxk)x*'a = D 1((Dx k)x*\"' ) . jb-l Therefore we have ^(Dx^+i - (Dx k)x^ ) = 0 , and hence D x k + i - (Dxk)x\u00C2\u00A3 ' = e \u00C2\u00AE > ^ r o m which we see easily that ( 9 . 3 . 3 ) D-r,(&\u00E2\u0080\u0094\u00E2\u0080\u00A2 Since every commutes with D-^ , i t has the form (9 \u00C2\u00AB3 \u00C2\u00BB3 ) \u00E2\u0080\u00A2 Then by taking a suitable scalar-equivalent system we obtain { D^, ... , D m | of the form (9.3.2). Remark. If we take 1 + x^, ... , 1 + x n as independent generators of Ot instead of X=L, .... , x n , 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 this case, i t i s easily seen that - 54 -10. The case ft - GF(p) . Let ^\u00C2\u00A3 be an algebra over ft, u^, ... , u n be a basis of over ft . Then u^ u-j = 2L ^ i j k ^ k \u00C2\u00BB where ^ i j k e $ \u00E2\u0080\u00A2 If we can choose a basis u^, ... , u n of g\u00C2\u00A3 over ft such that a l l the #ijk belong to a subfield ft1 of ft , then we shall say that the algebra i s defineable over ft* . In other words, an algebra over ft i s defineable over ftT i f and only i f there exists an alge-bra \u00C2\u00A3J over ft* such that \u00C2\u00A32\u00C2\u00A7 8 8 j\u00C2\u00A3 . Theorem 6.3 shows that any generalized Witt algebra of D-dimension m = 1 over an algebraically closed f i e l d ft is defineable over GF(p) , which may naturally be regarded as a subfield of ft . Whether this i s true for an arbitrary D-dimension m i s not known. By an application of Theorem 9\u00C2\u00AB3, we shall show that i f Ol i s the group algebra of an elementary p-group of order p3 then any generalized Witt algebra of the form (01 ; D]_, D2) over an algebraically closed ft i s defineable over GF(p) . Let ^x^y^z^ } be a basis of Ol , where xP = yP \u00E2\u0080\u00A2 zP \u00E2\u0080\u00A2 0 . 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 . If, furthermore, 7 = 0 , then our assertion i s proved. Suppose y ^ 0 . Taking a non-zero element X e ft , we set x 1 \u00C2\u00BB Xx, y' = XPy, z T = X^ z . Then the set { x'iy'Jz'k J forms a basis of c% , and we have Di = XD^ , D2 = XPD2 , where D l = dx' * X If + x $ W y Therefore i f we determine X by the equation 2 X p ~ p = 1 , then we see that i s defineable over GF(p) . I f $ \u00E2\u0080\u00A2 0 then we may take y = 1 , and hence our assertion i s also clear. In section 2, we have remarked that the only algebras constructed by Kaplansky's method for the case when D-dimension m = 1 and ft = GF(p) are the original Witt algebras. Theorem 7.10, 8 . 3 , or 9.3 shows that we can construct normal simple algebras of the type \"\u00C2\u00A3(01) D i , ... \u00C2\u00BB 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 algebraically closed f i e l d . Therefore we see that we have constructed some new f i n i t e simple Lie algebras. Remark. If we construct a generalized Witt - 56 -algebra \u00C2\u00A3 over ft and regard i t as an algebra over GF(p) , as i s done in 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 alg e b r a s . Let j\u00C2\u00A3 be a L i e algebra over ft w i t h the m u l t i p l i c a t i o n * * . For any two i d e a l s . ^ i , J2 o f \u00C2\u00A3 w e s h a l l denote by 0 72 the i d e a l of \u00C2\u00A3, generated by a l l X}ax 2 , where x i 6 ^ i * ^et & be a commutative a s s o c i a t i v e a l g e -bra over ft , and denote by A [ST) and the l a t t i c e s (defined by i n c l u s i o n ) of a l l i d e a l s of 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 isomor-phism cr: A(<\u00C2\u00A3~) \u00E2\u0080\u00A2* A ( \u00C2\u00A3 ) such t h a t ( ^ i ^ ) 0 - &f\u00C2\u00BB b\u00C2\u00A3 h \u00C2\u00B0 l d s f o r any two i d e a l s 0^, d~2 e A(< )^ , then we s h a l l say t h a t (x) be the algebra over ^ w i t h the b a s i s 1, x, x 2 , ... , x n P \" l , where xP s a t i s f i e s the equation cp(xP) = 0 , and l e t Ol be the algebra ^ ( x ) regarded as an algebra over ft . C l e a r l y t h e r e e x i s t s a d e r i v a t i o n D of Ol such - 58 -that Dx = 1 and such that Da = 0 for a l l a e -3> . Then the algebra \u00E2\u0080\u00A2\u00C2\u00A3> - \u00E2\u0080\u00A2\u00C2\u00A3. (01 > D) i s uniquely determined by the polynomial 9 , provided that ft and ^ are fixed, so that ((% ; D) may be denoted by - \u00C2\u00A3 . ( y ) with-out ambiguity. It i s easily seen that the algebra $C of constants for (01 ; D) i s generated by xP over , and that jfc \u00C2\u00AB \"$CxJ/(9(X)) as algebras over ft . Hence dT' i s a principal ideal ring. Every ideal of &C can be written as 0~s <\u00C2\u00A3\"a = (a). , where , and i t i s always possible to choose a monic factor a(X) , i.e. a factor whose leading coefficient i s 1 , of cp(X) such that 0~- (a(xP)) , since 2 . Then the algebra (01 ; D) defined above has no nilpotent ideal except the zero ideal. The algebra \u00C2\u00A3 (01; D) and i t s algebra ST of constants have the same ideal theory. Proof. We shall prove f i r s t that and have the same ideal theory. For any ideal & of <\u00C2\u00A3\" we define 0r(x2D) = 4ac p_ 1xP~ 1D , we have 4aCp_]xP~-1-D e 0 , and hence aCp.^ xP'-'-D e \"3 . Thus afD e CJ for any f e OX . Now, for any h(X) e '$\u00C2\u00A3\]j such that h(x)D e 0 , we set h(\) = a(X)j(\) + r(\) , where ^ ( \ ) , f { \ ) e J and where deg r(\) < deg a(\) . Since h(x)D, a(x)^(x)D e \"J t we have r(x)D e 0 . Then the minimality of the degree of a(X) yields r(\) = 0 . Thus we have proved that every element in ^ i s of the form afD , where f e Ol . Hence fr*''* \"7 i f we denote by & the ideal of generated by . Let & 2 be ideals of <$7 such that i t : i s sufficient to show that a]^ 0*\"*\" e ^ \u00E2\u0080\u00A2 6-f, and hence aj^cx^D e &fo 6-f* Since aixDoa2Cxp~^D = -2aia2CxP~lD , we have a ^ c x P ^ D e flf . Thus ( ^ i ^ ) 0 \" 9 \u00C2\u00A3 - 0-\u00C2\u00B0~o ( O c T ) * ~ 0* - J \u00E2\u0080\u00A2 Therefore J i s not milpotent unless 7^ = 0 . Thus Theorem 11.1 is completely proved. Lemma 11.2. With the notation as in the proof of (11.1), i f & i s an ideal of e\u00C2\u00A3\" and i f a(\) i s a divisor of cp(\) such that Ct = (a(xP)) , then ^ / 0 \" a = -d\u00C2\u00A3(a(\)) as Lie algebras over ft . Proof. We define a mapping 0 : d\u00C2\u00A3(?(X)) - \u00C2\u00A3(a(X)) by 0(f(x)D) = f(x)D . If f(x)D = g(x)D in -\u00C2\u00A3iq>M) then f (\) s g(X)(mod 2 then any semi-simple algebra of the type -\u00C2\u00A3 (cp) can be decomposed into a direct sum of simple algebras of the same type. Proof. By Theorem 11.1, is semi-simple i f and only i f \u00C2\u00A3L i s semi-simple, and so, i f and only i f cp can be expressed as a product cp = q>i ... cpr of distinct irreducible polynomials i n ^ [\J . Suppose then that -s\u00C2\u00A3(cp) is semi-simple and that q> = cpi ... cpr \u00E2\u0080\u00A2 - 62 -We set ^ = cp/cpi , 0\u00C2\u00B1 = (xi(xP)) . Then \u00C2\u00B0~\u00C2\u00A9 . . . \u00C2\u00A9 From the definition of fr\u00C2\u00B1 i t follows easily that (5*2 + ... + 0~r = (q>i(xP)) . Hence by Lemma 11.2 we have -\u00C2\u00A3( ... \u00C2\u00AE #y~) = -sd^) . Then from (II . 3.I) we have &f \u00C2\u00A3 \u00C2\u00A3(i) for a l l i . 0 Since q>i is irreducible, \u00E2\u0080\u00A2\u00C2\u00A3(\u00C2\u00B1) i s simple. 0 - 63 -12. Automorphisms of -\u00C2\u00A3(0l', D]_, \u00E2\u0080\u00A2 \u00E2\u0080\u00A2\u00C2\u00BB , Dm) \u00E2\u0080\u00A2 By an automorphism of an algebra \u00C2\u00A3^ over ft we mean a 1-1 mapping o* of onto i t s e l f such that (x + y)* = x\u00C2\u00B0* + yff,. (xy)0\" - x V , (ax)0\" = ax0\" for a l l x, y e -3^. and a e ft . Because.of the last property, any automorphism over ft i s completely determined by i t s effect on a basis of \u00E2\u0080\u00A2\u00C2\u00A3 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 this section f i r s t we discuss certain relationships between automorphisms of Ot and \"\u00C2\u00A3,(01 ; D^, ... , Dm) . Let o* be an automorphism of Ot and D a derivation of Ot . The mapping D0\" which i s defined by Dafa = (Df) a is easily seen to be a derivation of Ot . For two derivations D]_, D2 of Ot we have (D-L + D2)V = Dj + D*, (DX Da)* = Dj D* , and . (fD) f f = f 0!) 0* for any f e 01 . Let ^ be a subalge-bra of the derivation algebra of Ol \u00E2\u0080\u00A2 An automorphism o- of Ol w i l l be called admissible to -s\u00C2\u00A3 i f D\u00C2\u00B0* e \u00C2\u00A3 for any B e . If cr i s admissible to then the mapping D -\u00C2\u00BB\u00E2\u0080\u00A2 DC i s an automorphism of , which w i l l be said to be induced by 0* . If an automorphism ff of ^ i s admissible to s\u00C2\u00A3 (01 J Di, ... , Dm) then from - 64 -( f l D l + ... + fmDm)<^ = f j D j + ... + f j D j i t follows that -\u00C2\u00A3D 0, ... , | is a system equivalent to -[Di, ... , D m j . Thus we have proved the \"only i f \" part of the following Theorem 12.1. Suppose that p = 5 and that ... , Dm J is orthonormal. Then an automorphism c of \u00C2\u00A3 = -\u00C2\u00A3_{oi ; ... , D m) i s induced by an auto-morphism of Ol i f and only i f { Dj, ... , J i s a system equivalent to j D j , ... , D m | . To complete our proof, suppose that o* i s an automorphism of such that {D J, ... , Dj* j i s equiva-lent to {^l, \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00BB Dm } \u00E2\u0080\u00A2 Then we may define linear map-pings c*ij of Ol into i t s e l f such that (12.1.1) (fL\u00C2\u00B1)\u00C2\u00B0 = \u00C2\u00A3 f ^ S D 0 s = / for a l l f e 01 and i = 1, ... , m . Setting f = 1 i n (12.1.1) yields cr.. (12.1.2) 1 <0 = S i : } (Kronecker delta) . From ( f D ^ M g D j ) * = (fD^D.^ )* = (f (D i g)D.j) f f - (g(Djf )D\u00C2\u00B1)<* and (12.1.1) we have (fDi) 0\" (gDj)0\" = 2 Z [ ( f D i S ) ^ - ( S D j f ) q : * ] D k ' -ft On the other hand, from DfoD^ '= 0 and (12.1.1) we have - 65 -(fDi) 0 , (gDj)0\" -\u00C2\u00A3.[\" D g g l * - g ^ D j f ^ j D j . sA Therefore we have (12.1.3) ( f D j g ) * * - ( g D j f ) * * = X f ^ ' D j g ' S * X D*f . 7 s Setting f - 1 in (12.1.3) yields (D i g ) H DJg 5* . Substituting this i n (12.1.3) yields (12.1.4) ( f D i g ) ^ - ( g D j f r 4 = X . f * $ ( D s g ) ^ err or. 5 - L f 3 J ( D g f ) * * s We shall use the fact that -\u00C2\u00A3l>i, ... , D m J i s orthonormal. Let x^, ... , x m e OL be such that D J ^ C J = S^j . Setting i = j = k, g = X i in (12.1.4) yields (12.1.5) ( x ^ f ) ** = X \u00E2\u0080\u00A2x.* r(D rf) . r * Setting f = X j , where j \u00C2\u00A3 i , in (12.1.5) yields or-(12.1.6) 0 = x y ( i ^ j) . Substituting (12.1.6) in (12.1.5), we have (12.1.7) (xiPitfi = x?* (D \u00C2\u00B1f . Setting j = i ^ k , g = x^ i n (12.1.4) and using (12.1.6), we have f - ( x ^ f ) - - x . ^ ( D i f ) ^ . - 66 -Setting f = X J , where j \u00C2\u00A3 i , i n the above, we have x.* = 0 for j \u00C2\u00A3 i ^ k . Combining this result with (12.1.6), we conclude that i f i ^ j then e r f . (12.1.8) x / 4 = o for a l l k . Setting k = i ^ j , g^x.^ i n (12.1.4) and using (12.1.8\"), we have (12.1.9) f '* - ( x ^ f ) ^ = - xj* ( D j f ) ^ - , (j / i ) . Setting f = Xj in (12.1.9) and using (12.1.8), we have (12.1.10) ' - xT J \u00E2\u0080\u00A2 Ai 4. n Setting f \u00E2\u0080\u00A2 Xj,Xj , where j \u00E2\u0080\u00A2\u00C2\u00A3 i , i n (12.1.7), we obtai (12.1.11) ( x i X i ) ^ - x \"x.^' . Setting f = x2. in (12.1.9), we have ix 2.) ^ 3 cr . o r - J - 2(x ix i)** - - 2x.**x.*x . Therefore, using (12.1.10) and (12.1.11), we have or. (12.1.12) , ( x j } ^ - \" 0 \u00C2\u00BB ( i / J) \u00E2\u0080\u00A2 Setting i = j = k, f = x 2 i n (12.1.4) and using (12.1.12) we have (12.1.13) (x 2D i g r \" - 2 ( g X i r * = ( x 2 ) ^ ( D i g ) ^ 1 or- c- \u00E2\u0080\u00A2 - 2g \u00E2\u0080\u00A2* x .** . Setting f = gxi i n (12.1.7), we have - 67 -(x.D^g + x^g) * := x .** (xiD^g) + x .** g ** . Therefore, 1 >C A* by (12.1.7), we have (12.1..14) ( x ? D i g ) ^ + (gXir* - (x.^' ) 2 ( D 1 g ) < & 1 cr; . \u00C2\u00ABJT , * + g * * X * * 2 Setting f - x? i n (12.1.7) yields 2(x?) ** = 2(x.**) and hence (x?) **\u00E2\u0080\u00A2 = (x.** ) 2 , since p + 2 . Then 1 * or- en \u00E2\u0080\u00A2 err-(12.1.13) and (12.1.14) yield 3(gxj[)** = 3g **x.** and hence (12.1.15) (g X i) = g x. t for a l l g , since p ^ 3 . By using (12.1.15) and (12.1.10) in (12.1.9), we have for i ^ j ' and f e \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > fm a r e m indeterminates over P , and let Ol *= ft(x]_, , x m) , where x? = \u00C2\u00A3\"i \u00E2\u0080\u00A2 W e s e t n ^ . l>-i d b-i 4>-> 2 Then the algebra -3^(01; D) over ft - has the desired property. In the course of the proof of Theorem 12.1, only the fact that p ^ 2, 3 was used. Therefore Theorem 12.1 holds even when p <= 0 \u00E2\u0080\u00A2 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 characteristic 0 is induced by an automorphism of Ol over ft . Now we shall consider automorphisms of the generalized Witt algebras. In the following, 01- ft(x^, \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 , x n) w i l l denote the group algebra of - 70 -an elementary p-group with independent generators x i , ... , x n . A polynomial f(X) e ft[X3 i s called a p-polynomial i f f(X) is of the form f (X) \u00C2\u00BB a 0X + a^X\" + ... + ajj-X , where 0.^ e ft . Lemmas 12.3 and 12.4 are proved i n f 3 j , P\u00C2\u00BB 110. Lemma 12.3. If 1, u^, u 2, ... , , where N = p n , is a basis of Ol over ft , then there exist n distinct 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. The characteristic polynomial of any derivation i n (JL i s a p-polynomial. Lemma 12\u00C2\u00B0. 5. I f a l l the roots of the minimum polynomial of a derivation D i n Ol are i n ft and distinct, and i f D does not satisfy any non-zero p-polynomial of degree less than p n , then a l l the charac-t e r i s t i c roots of D are in ft and distinct. Proof. Since a l l the roots of the minimal polynomial of D are i n ft and distinct, D can be diagonalized, that i s , there exists a basis 1, U]_, u 2, < of Ol such that Dui = XjUi, \\u00C2\u00B1 e ft , for a l l i . By Lemma 12.3 we may assume that the elements u ' ... u ^ f - 71 -form a basis of Ol over ft . Since D(u^' ... uf~) = (T \ n - k 4)u 1 ... u , i t i s sufficient to show that X ^ i ^ i - 0 with 0 < p implies k-^ = ... = k n = 0 . Suppose that there exist . (k^, ... , k n) \u00C2\u00A3 (0, ... , 0), 0 = k i < p , such that X ^ i k i = 0 . Since s k (mod p ) we have X \^ kj. = 0 for j \u00C2\u00BB 0, 1, 2, ... . Then the matrix' [\T ) , where At l ^ i ^ n , O ^ j ^ n - 1 , is \u00C2\u00AB0\u00C2\u00AB\u00C2\u00BBsingular. Therefore there exists f^li \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 i Pn-1 e ^ \u00C2\u00BB n o t z e r o \u00C2\u00BB such that 1 8 . / = 0 for a l l i . Since u-- XT Ui we have (XI )ui - 0 for a l l i . Then the derivation X_ = 0 > since U]_, ... , u n generate tfl . This contradicts tm our assumption. Therefore X ^ i k i ~ 0 implies k]_ = ... s k n = 0 mod p , as required. The following two lemmas are easily verified. . Lemma 12.6. Suppose that p 2 5 \u00E2\u0080\u00A2 If a0\u00C2\u00BB a l > \u00E2\u0080\u00A2\u00E2\u0080\u00A2* \u00C2\u00BB a p - l 6 \u00C2\u00AE a r e s u c h that a i a j ~ a i + j \u00C2\u00BB where i + j i s calculated ...mod p , for a l l i / j , and i f CLQ \u00C2\u00A3 0 , then ctQ = a]_ = ... = a p _ i - 1 \u00E2\u0080\u00A2 Lemma 12.7. Suppose p 2* 5 \u00E2\u0080\u00A2 If a o = 0 \u00C2\u00BB a l > \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 J a p - l 6 ^ are such that jctj - io^ - (j - i ) a ^ + j . , where i + j is calculated mod p $ for a l l \u00E2\u0080\u00A2 i and j , then = ia-^ for a l l i \u00E2\u0080\u00A2 ^ 72 -Let \u00C2\u00A3 - \u00C2\u00A3 (01; D l f ... , Dm) be a general-ized Witt algebra defined by a principal system {^1, ... , D m } . We shall assume that ft is a perfect i n f i n i t e f i e l d and that p 2 5 \u00E2\u0080\u00A2 Let cr be an automor-phism of -j\u00C2\u00A3 . By Theorem 9.1 , there exist Kl> \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 > tfm 6 \u00C2\u00AE s u c h t h a t D = ^1D1 + \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 + ^mDm is normal. By Lemma 12.4 the characteristic polynomial %(\) of D i s a p-polynomial of degree p n . A l l the roots of are i n ft. and distinct. We shall show that the characteristic polynomial of D0\" is also %{X) . Since (12.8.1) Do(D\u00C2\u00AB ... (DoX) ... ) = D r cX for any i and X e u2\u00C2\u00BB \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00C2\u00B0f ^ over ft such that .D^ Ujj = \-jUi , \ i e ft , for a l l i . By Lemma 12.3 we o 2 may assume that the elements u*' ... u * form a basis of Ot . Then the p n . elements 21 ^ i k i > 0 5 kj[ < p , are precisely the (distinct) characteristic roots of \u00E2\u0080\u00A2 %{\"K) . On the other hand, since \\u00C2\u00B1 i s also a characteristic root of D , there exists a non-zero element x\u00C2\u00B1 e Oi such that Dxi = ^iX^ . Then 1, x^, ... , x ^ i , where N = p n , form a basis of Ot . Since D]_, ... , D m are commutative with D , we have D(DjXi) ~ ^ i D j x i > a n d hence D^x^ = a j i x i with e ft for a l l i and j . Since ; L^, ... , Dm) is simple and since x^ ^ 0 , by Lemma 3.2 we see that x^ i s a unit i n 01 . Therefore we may assume x? = 1 for a l l i . The elements x^' ... x ^ , 0 ^ k,- < p , form a basis of Ot over ft . We note that the matrix (ctij) i s of rank m . Similarly, D ? U J = J3J [ JU J for 0 i = 1, ... , m and j = 1, ... , n . The matrix ( jS-jj) i s also of rank m . We consider the subspace ^/T(ki, ... , k n) of , which w i l l also be denoted by flfly. , spanned by X E \u00E2\u0080\u00A2\u00C2\u00A3 such that D\u00C2\u00BBX = (\]ki + ... + ^ nk n)X . It is easily seen that 001^ consists of elements of the form x^l ... x^-ii^Pl + \u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 + tfnrV > where <#\u00C2\u00B1 e ft , - 74 - \u00E2\u0080\u00A2 so that '2ftTk i s of dimension m . The image fift^ of / ki < p - 1 for a l l i , then ^ k consists of elements of the form ufi . . . U ^ ^ D - L + . . . + ^ M D M ) , where fa e ft.. We are now ready to prove u? ^ 0 for a l l i . Suppose u^ = 0 . We shall denote flT7((p--- 2)A.i, 0, ... , 0 ) , W(P - 3 ) ^ i , 0, ... , 0) simply by ^ ( p - 2), fl)7(p - 3) respectively. Then u^ = 0 implies Y Y1 = 0 for any Y e fl7(p - 2)a and Y\u00C2\u00AB e flfl(p - 3) f f . Hence X-X' = 0 for any X e | f | ( p - 2 ) and X\u00C2\u00BB e ?#(p - 3) \u00E2\u0080\u00A2 This i s a contradiction, since (12.8.2) xf~Z D X * x^*3 D X = - ^IX^ D-L / 0 . Therefore u^ ^ 0 , and similarly u? ^ 0 for a l l i . Hence we may assume u? = 1 for a l l i . - 75 -Now that we have shown that u? = 1 for a l l i , i t i s easily seen that \u00C2\u00A3 consists of elements of the form uf* ... i^Pl + ... + tfnJDJj) , where ^ e $ , without any rest r i c t i o n on . Since a\u00C2\u00A3 i s the sum of a l l ?)Tk , ^ is also the sum of a l l 1#7k \u00E2\u0080\u00A2 Therefore every element i n can be written i n the form giD^ + ... + grrPm \u00C2\u00BB where gi e Ol . Thus we have proved that {D\u00C2\u00A3, ... , DjJ J- i s a system equivalent to |D\"L, ... , Dmj- . By taking a suitable scalar-equivalent system i f necessary, we may assume without loss of genera-l i t y that D J_XJ = ^ l j x j > where 8ij- i s a Kronecker delta, for i , j = 1, ... , m . Note that m ^ n . Similarly, there exists a system { E^, ... , E m J scalar-equivalent to ^D*, ... , D^ ]\u00E2\u0080\u00A2 such that E i U j = ^ i j u j f \u00C2\u00B0 r i> j = 1, ... , m . We set , (12.8.3) ( x ^ ) * = u j ( p i l E l + + pim Em } \u00C2\u00BB where p^j e f t . We also set (xkDfe.)* - u kF for any fixed k > 1 . Since F commutes with every Ej , Di (x^D^) 0 - 0 yields Pokuk^ = ^ ' a n d hence we have (12.8.4) P Q 1 ^ 0 , p Q k = 0 , (k > 1) . Now (12.8.3) yields easily P u P j l * P\u00C2\u00B1+j i f \u00C2\u00B0 r i \u00C2\u00A3 3 \u00E2\u0080\u00A2 Hence by Lemma 12.6 and (12.8.4) we have p^ ]_ = 1 for a l l i . Hence (12.8.4) yields Dj = E x . Similarly Df = E i for a l l i . Again (12.8.3) yields, for any - 76 -k > 1, j p j k - i p i k - (j - i)Pi+j }k \u00E2\u0080\u00A2 H e n c e fey ( 1 2-7) we have = iPlk f \u00C2\u00B0 r a H * \u00E2\u0080\u00A2 W e shall write p k for pjk \u00E2\u0080\u00A2 Then (12.8.3) can be written as (12.8.5) ( x ^ ) 0 \" = uj(E! + i ( p 2 E 2 + ... + P mE m)) \u00E2\u0080\u00A2 As before, we set (xk^k) 0 = u k F , F u l = 9^1 f \u00C2\u00B0 r k > l . Then [x^)0e{xkDk)a - 0 and (12.8.5) imply, for i # 0 (mod p ) , (12.8.6) p kF = 7 k ( E l + i (P2 E2 + .\u00C2\u00AB. + PmEm)) \u00E2\u0080\u00A2 By changing i i n (12.8.6), we obtain p kF - 7 y^i and tfk(P2E2 +'\u00E2\u0080\u00A2\u00E2\u0080\u00A2\u00E2\u0080\u00A2 + PmEm) \" 0 \u00E2\u0080\u00A2 Therefore i f p k 0 then 7k ^ 0 > a n d hence we have p 2E 2 + ... + p m E m = 0 , a contradiction. Hence Pk - 0 f \u00C2\u00B0 r a H k > 1 . Since E l = D l \u00C2\u00BB (12.6.5) yields ( x ^ ) 0 \" = u^D0 . Similarly we have (x^D^) 0 - u^ D 0 for a l l i and j . We set Dj = x~} Dj . Then \u00C2\u00AB... , ^ m} *-s a n orthonormal system equivalent to \u00C2\u00A3 Dl> \u00E2\u0080\u0094 \u00E2\u0080\u00A2 \u00C2\u00BB Dm^ \u00E2\u0080\u00A2 Since (Dj) 0 = u\".1 Dj, {(D^) 0, ... , {Vm)GJ i s equivalent to { D 0 , ... , D 0 | which i s equivalent to { D]_, ... , Dm^ . Hence. ^(D^) 0, ... , (D^) 0J is equivalent to {^i> ... \u00C2\u00BB ^ m} * B y Theorem 12.1, cr is induced by an automorphism c of Ol \u00E2\u0080\u00A2 Suppose that D? = D\u00C2\u00B1 for a l l i . Then D\u00C2\u00B0\" = D .. We set y = x^' ... x ^ . Then we have - 77 -DyC = Dcyr = ( D y)ff , ( X l k l + ... + XnknJy0* . Hence y\u00C2\u00B0\" = ay with a e ft . Since (yff)P \u00C2\u00AB (yP)0\" - 1 , we have aP = 1 , a <= 1 . Thus y* = y . Since x, ' ... x n form a basis of Ol , the automorphism o* of Ol i s the identity. Since ifD\u00C2\u00B1)a - fffD? - fD \u00C2\u00B1 for a l l i and f 6 Ol , the automorphism a of \u00E2\u0080\u00A2\u00C2\u00A3 i s also the identity. Thus we have proved the following Theorem 12.8. Suppose that ft i s an in f i n i t e perfect f i e l d and that p = 5 \u00E2\u0080\u00A2 Then any automorphism cr of a generalized Witt algebra ^(tfT; ... , Dm) i s induced by an automorphism of Ol \u00E2\u0080\u00A2 If = D-^ for a l l i , then o* i s the identity. , * Corollary 12.9. Let Di, ... , D m) be a generalized Witt algebra, and assume that there exist non-zero elements x^, ... , x m e Ol such that DjXj = $ i j X j for i , j = 1, ... , m . If an automorphism \u00C2\u00B0* of Ol admissible to -g\u00C2\u00A3 leaves every Xj invariant, then o\" i s the identity. Proof. Since \u00E2\u0080\u00A2\u00C2\u00A3 D^, ... , j- is equivalent to ... , D m j. ,' we may set D?.= c i j D j \u00E2\u0080\u00A2 Then D\u00C2\u00B0fxa = SJ.-X? = c,* ,-X.? . Since x,- i s a unit, we have S J L J = c^j , and hence D?/ = for a l l i . Therefore by Theorem 12.8 cr is the identity. What automorphisms of Ol are admissible to - 78 -\"e\u00C2\u00A3 (01 l Di\u00C2\u00BB \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 > Dm) ? \u00E2\u0080\u00A2\"\u00E2\u0080\u00A2n t n e following we shall consider only the case m = 1 . If ft i s algebraically closed, then any generalized Witt algebras of D-dimension 1 can be written in the form \u00C2\u00A3 (Ol ; D) , where 01= ft(xj, ... , x n) i s the group algebra of an elemen-tary p-group with independent generators 1 + X]_, ... , 1 + x n , and where (Once is formulated i n this form, we may prove, with-out any condition on ft , that any automorphism of i s induced by an automorphism of 01 ) . Denote by y w the monomial x ... x ^ of weight / 71 w = v x + v 2p + ... + v n p n _ 1 . If f = a w y w + a w + 1 y w + 1 + .. where a w , c t w + 1 , ... e ft, a w ^ 0 , then we define the weight w(f) of f to be w . Lemmas 12.10 and 12.11 are easily verified. Lemma 12.10. If f s ^ is of weight w > 0 , then Df i s of weight w - 1 . Lemma.12.11. Let 9fl be the radical of Ol . If f 6 <$f- then w(f) i s not a power of p . Lemma 12.12. Let 1 then from (12.12.1) we have (x ... x ) b = a ^ T + a^ox + ... \u00E2\u0080\u00A2 + ct i nx ... x (mod ffl ) . Therefore CLAJI - 0 for i > 1 . We set (12.12.2) x? \u00E2\u0080\u00A2 a ^ X i + ... + c t i n x n + f i f Take a fixed i > 1 and assume that (12.12.3) cV.rs = 0 for s < r , and that w(f r) > p ^ 1 whenever r < i . Suppose that = ... = c^ijk-l = 0 > a i k ^ ^ *\"or s o m e k such that 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 easily that w((xj ... xP\"1)^ ' b) = p1\"\"1 - 1 > p ^ 1 - 1 . Therefore (12.12.4) yields w(Df\u00C2\u00B1) = p k - 1 - 1 . Then from Lemma 12.10 we have w(f \u00C2\u00B1) \u00C2\u00BB p k _ 1 , a contradiction by Lemma 12.11 . Hence ct^. = 0 for - 80 -j < i . Then (12.12.4) yields w(f \u00C2\u00B1) Z p i _ 1 - 1 . Hence w(fi) p^r^ . Then by Lemma 12.11 we have w(fi) > p 1 \" 1 . 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 auto-morphisms of Ol . Then the mapping cr -*\u00E2\u0080\u00A2 (ct^) defined by (12.12.1) is a homomorphism of n* . By Theorem 9.1 there exists D e ^ whose characteristic roots (as an operator on Oi ) are distinct. Let Df e \u00C2\u00AB^\u00C2\u00A3T be the element corres-ponding to D, % x (\) the characteristic polynomial of D' .. M i s a p-polynbmial by Lemma 12.4, and of degree p n t . From %* (D1) - 0 and (12.8.1) i t follows easily that %X(B) - 0 , since no non-zero derivation of Ot commutes w i l l a l l elements i n s\u00C2\u00A3 . This is a contra-diction, since D does not satisfy any non-zero poly-.nomial of degree less than p n . Therefore m = m* must hold. - 82 -References Ho-Jui Chang, \"Ueber Wittsche Lie-Ringe\", Abhandlungen aus dem Mathematischen Seminar der Hansischen Universitat, vol. 14 (1941), pp. 151-164. N. Jacobson, \"Abstract derivation and Lie algebras\", Transactions of American Mathematical Society, vol. 42 (1937), pp. 206-224. N. Jacobson, \"Classes of restricted Lie algebras of characteristic p, II\", Duke Mathematical Journal, vol. 10 (1943), pp. 107-121. I. Kaplansky, \"Seminar on simple Lie-algebras\", Bulletin of American Mathematical Society, vol. 60 (1954), pp. 470-471. H. Zassenhaus, \"Ueber Lie Tsche Ringe mit Primzahl-characteristik\", Abhandlungen aus dem Mathema-tischen Seminar der Hansischen Universitat, vol. 13 (1940), pp. 1-100. "@en . "Thesis/Dissertation"@en . "10.14288/1.0080649"@en . "eng"@en . "Mathematics"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "On generalized Witt algebras"@en . "Text"@en . "http://hdl.handle.net/2429/40825"@en .