UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

A general Cartan theory Foster, David Merriall 1969

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-UBC_1969_A1 F68.pdf [ 4.92MB ]
Metadata
JSON: 831-1.0080531.json
JSON-LD: 831-1.0080531-ld.json
RDF/XML (Pretty): 831-1.0080531-rdf.xml
RDF/JSON: 831-1.0080531-rdf.json
Turtle: 831-1.0080531-turtle.txt
N-Triples: 831-1.0080531-rdf-ntriples.txt
Original Record: 831-1.0080531-source.json
Full Text
831-1.0080531-fulltext.txt
Citation
831-1.0080531.ris

Full Text

A GENERAL CARTAN THEORY by DAVID MERRILL POSTER . . B.A..,. Northwestern University,, 1962. M;Sc.s San Diego State College, 1965 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 required standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1969 In p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h Columbia, I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and Study. I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department or by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g or p u b l i c a t i o n of t h i s thes,is f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada i i . ABSTRACT Recent results of Jacobson and Barnes indicate that Lie, Jordan and alternative algebras may have a common Cartan theory. In this thesis, we show this i s indeed the case. We also show that for certain classes of non-associative algebras,\called E-classes, that possess an Engel function, a general Cartan theory i s possible. In\Chapter One, a generalization of nilpotence and:: solvability i s introduced that permits our Cartan theory for E-classes. In Chapter Two, we construct Cartan subalgebras for alternative algebras based on a given Engel'function. Jacobson's Cartan theory for Jordan algebras is given i n Chapter Three along with our extensions of his results. We point out that the Engel function for alternative algebras and Jordan algebras coincides, and may be used to give the classical Cartan theory for Lie algebras Commutative power associative algebras" are. discussed in Chapter. Four, and some results, are obtained. i i i . . TABLE OP CONTENTS Page INTRODUCTION 1 PRELIMINARIES 4 ' CHAPTER 1 GENERAL CARTAN THEORY 1.1 Generalized Solvable Radical Properties . 6 1 .2 Properties of f-solvability 11 1 .3 f - n i l Algebras 14 1.4 Cartan Subalgebras 20 1 .5 Construction of Cartan Subalgebras 2 3 1.6 The. Inner •Automorphism Group of an Algebra 29 CHAPTER 2 ALTERNATIVE ALGEBRAS 2 . 1 Introduction 4 l 2 . 2 The Universal Multiplication Envelope of an Alternative Algebra - _ 47 2 . 3 Existence of an Engel Function for Alternative' Algebras 53 2 . 4 Cartan Subalgebras of-Alternative Algebras 63 2 . 5 Properties of Cartan Subalgebras 65 . CHAPTER 3 JORDAN ALGEBRAS 3 .1 . Introduction ' • - 69 3 . 2 \ The Universal Multiplication Envelope of a~ Jordan Algebra 72 3 . 3 Cartan Subalgebras of Jordan Algebras 76 3 . 4 A-solvable Jordan Algebras ~ 8 l CHAPTER 4 . COMMUTATIVE POWER ASSOCIATIVE ALGEBRAS 4.1 Introduction 4.2 Cartan Theory of Commutative Power Associative Algebras BIBLIOGRAPHY \ / ACKNOWLEDGEMENTS I wish to express my appreciation to my research supervisor, Dr. C. T. Anderson, for the encouragement and advice given by him during the preparation of this thesis. I wish to thank Dr. P. Lemire for his many helpful suggestions. The financial support of. the University of Br i t i s h Columbia.and the National Research Council of Canada i s grate-f u l l y acknowledged. •., INTRODUCTION The concept of a Car tan subalgebra p l a y s a key r o l e i n the s t r u c t u r e theory of L i e a lgeb ra s . In 1°>66, Jacobson [16] i n t roduced the n o t i o n of a Car tan subalgebra f o r Jordan a lgeb ra s , and showed that an ana logous Car tan theory i s v a l i d f o r Jordan a lgeb ra s . His main r e s u l t s showed that i n any f i n i t e d imens iona l Jordan a lgeb ra J over an i n f i n i t e f i e l d F there do e x i s t Car tan subalgebras . He a l s o proved a conjugacy theorem f o r Car tan subalgebras of J when F i s a l g e b r a i c a l l y c l o s e d and of c h a r a c t e r i s t i c ze ro . Because of the c l o s e r e l a t i o n s h i p o f Jordan , L i e and a l t e r n a t i v e a l g e b r a s , the f o l l o w i n g ques t ion a r i s e s : does there e x i s t a common Cartan theory f o r L i e , Jordan , and a l t e r n a t i v e a lgebras ? In t h i s t h e s i s , we g ive an a f f i r m a t i v e answer to t h i s ques t i on . In Chapter One, we i n t roduce a g e n e r a l i z a t i o n of n i l -potence and s o l v a b i l i t y . We f i n d tha t f o r f i n i t e d imens ional a l g e b r a s , the g e n e r a l i z e d s o l v a b i l i t y i s a r a d i c a l p r o p e r t y . We then def ine an Engel f u n c t i o n . U s i n g these func t ions and some l i n e a r a l geb ra arguments o f Barnes [ 8 ] , we o b t a i n a genera l Car tan theory . F i n a l l y , we c o l l e c t those r e s u l t s of Cheva l l ey [11] tha t are necessary f o r the conjugacy theorems. In Chapter Two, we develop the Car tan theory f o r a l t e r -n a t i v e a l g e b r a s , and show that i t p a r a l l e l s the theory f o r L i e algebras. Our Cartan subalgebras are characterized as minimal Engel subalgebras. As a r e s u l t , we show that i f the ground f i e l d F has "enough" elements and c h a r a c t e r i s t i c d i f f e r e n t than 2 , then a l t e r n a t i v e algebras w i l l always have Cartan subalgebras. Furthermore, i f F i s a l g e b r a i c a l l y closed and of c h a r a c t e r i s t i c zero, any p a i r of Cartan subalgebras i s conjugate. Chapter Three contains a sketch of Jacobson's Cartan theory f o r Jordan algebras. We add to his theory our character-i z a t i o n of Cartan subalgebras as minimal Engel subalgebras, thus extending his \existence theorem for Cartan subalgebras to Jordan algebras over f i n i t e f i e l d s having "enough" elements, We close Chapter Three with a discussion of associator solvable Jordan algebras, and introduce a class of nilpotent derivations. Because of two recent results [ 6 ] and [ 2 2 ] , we try i n Chapter Four to extend the Cartan theory to commutative power associative algebras. We prove that i f X i s a commutative power associative algebra with unity and stable i n the sense of Albert [ 4 ] , and R i s an A-nilpotent subalgebra containing 1 then R can be used to generate a nilpotent L i e algebra of l i n e a r transformation of X . I f X i s not stable, then we can only prove that this L i e algebra i s solvable. An example shows that this r e s u l t cannot be improved. There i s also trouble i n the existence of Cartan subalgebras. For to prove t h i s , we need to know that n i l algebras are nilpotent. This i s a d i f f i c u l t ximsolved problem. We remark that no study was made into forms of uniqu ness of Engel functions, ot i f d i f f e r e n t Engel functions could resu l t i n d i f f e r e n t Cartan theory. \ PRELIMINARIES Suppose U Is a vector space over a f i e l d F . We w i l l say that U i s an algebra i f and only i f there i s a multi-p l i c a t i o n defined on U , denoted by ab f o r a,b e U , such that t a(b+c) = ab + ac (a+b)c = ac +• be a,b,c e U a(ab) = (aa)b = a(ab) a e F , a,b e U The commutator [x,y] i s defined [x,y] = xy - yx . I f [a,b] = 0 f o r a l l a,b e U , then U i s commutative. The associator (x,y,z) i s defined (x,y,z) = x y z - x-yz . I f (a,b,c) = 0 f o r a l l a,b,c e U , then U i s associative. We note that i n any algebra U we have the following i d e n t i t i e s : (P-l) (wx,y,z) - (w,xy,z) + (w,x,yz) - w(x,y,z) - (w,x,y)z = 0 (P-2) [xy,z] - x[y,z] - [x,z]y - (x,y,z) - (z,x,y) + (x,z,y) = 0 . We also note that the standard concepts of subalgebra, i d e a l , homomorphism, isomorphism, and anti-isomorphism carry over from associative, algebras to non-associative algebras. I f A i s an i d e a l of U , we w i l l write A<J U . Furthermore, the fundamental isomorphism theorems are v a l i d . When we speak of a class IX of algebras, we w i l l assume that U i s homomorphically closed. I f K i s an extension of F , by U K we mean U ® F K . For a general introduction to non-associative algebras, the reader i s referred to Schafer [ 2 5 ] . An algebra L i s a L i e algebra i f and only i f [x,x] = 0 arid [[x,y],z] + [[y,z],x] + [[z,x],y] = 0 where [x,y] denotes m u l t i p l i c a t i o n i n L . For a theory of L i e algebras, the reader i s referred to Jacobson [ 18 ] . A l i n e a r transformation D on U i s c a l l e d a derivation i f and .only i f f o r a l l x,y e U , xyD = xD-y + x.yD . The set of derivations of U forms a L i e algebra, denoted by D(u) , and D(U) i s c a l l e d the derivation algebra of U . Most of the notation i n the thesis i s standard. Def i n i t i o n s are indicated by underlining the term being defined. Theorems, lemmas,and c o r o l l a r i e s are numbered with three integers to denote the chapter, section and order i n which they appear. CHAPTER ONE GENERAL CARTAN THEORY 1.1 Gene ra l i zed So lvab l e R a d i c a l P r o p e r t i e s Suppose P i s a p r o p e r t y that an a lgeb ra may possess . We say an a lgeb ra A i s a T?-.algebra i f i t .possesses the p rope r ty P . An i d e a l o f A i s c a l l e d a P - i d e a l i f , as an a l g e b r a , i t i s a P - a l g e b r a . . We say that P. i s a r a d i c a l p r o p e r t y i f the f o l l o w i n g three c o n d i t i o n s are s a t i s f i e d : (A) Any homomorphic image of a' P - a l g e b r a i s a P - a l g e b r a . (B) Every a lgeb ra A con ta ins a P - i d e a l which con ta ins every o ther P - i d e a l of A . We denote t h i s i d e a l by P(A) , and c a l l P(A) the P - r a d i c a l o f A . (C) For every a lgeb ra A , P ( A / P ( A ) ) = 0 . I f P i s a r a d i c a l p r o p e r t y and A i s an a lgeb ra such that P(A) = 0 , then A i s c a l l P semi s imp le , whereas i f P(A) = A, A i s c a l l e d a P r a d i c a l a l g e b r a . We r e c a l l tha t f o r f i n i t e d imensional a l g e b r a s , s o l v a -b i l i t y i s a r a d i c a l p r o p e r t y . We w i l l r e f e r to t h i s as the c l a s s i c a l r a d i c a l . We b e g i n our study by i n t r o d u c i n g s e v e r a l new p r o p e r t i e s f o r f i n i t e d imens iona l a lgebras tha t g e n e r a l i z e the concepts of s o l v a b l e and n i l p o t e n c e . 7 . Throughout the r e s t of Chapter One, U w i l l denote a f i n i t e d imens iona l a lgebra oyer an a r b i t r a r y f i e l d F . Suppose f ( x 1 , . . . , x n ) i s a l i n e a r homogeneous element of the f ree n o n - a s s o c i a t i v e a lgeb ra on the n generators Xn , . • . , x over F . Then f o r n elements u _ , . . . , " u of U , 1 , n i n f ( u - ^ , . . - . , u ) i s an element of U . ¥ e set f 1 ^ , . . . , u n ) = f ( u 1 , . . . , u n ) f o r any set {ti , . . . , u n ) of n elements from U , and f o r k > 1 and k ( n - l ) + l elements u i > • • • . > u k ( n _ i ) + i i n U , • f k ( u l ^ * ^ u k ( n - l ) + l ) =  f l ( f k - 1 ( u l > * * * j U ( k - l ) ( n - l ) + l ^ u ( k - l ) (n- l )+2>' * * >"\(n-±)+l^ We w i l l say U i s f - n i l p o t e n t , i f there i s a k > 0 such that f o r a l l se ts • * ' ^ ^ ( n - l j + l ^ o f e l e m e n t s f r o m u > w e h a v e f k ( u 1 , . . . , u k ( n _ l ) + 1 ) = 0 . Next we set f ^ (u^,... , u n ) = f ( u ^ , . . . ,\x^) f o r any set { u j , . . . , ^ } o f n elements from U , and f o r - . k > 1 and k n elements u - ^ , . . . , u ^ from U n (1) f ( k ) ( u . , u k ) = . n f( ^(f^ ^ (u-j , . . . , u ^ v _ i , * • • , u k - l ^ n n +1 2n f ( k _ 1 ) ( u u )) . ( n - l ) ^ 1 n k We w i l l c a l l U f - s o l v a b l e i f there i s a k > 0 such that f o r 8. a l l sets u-j_i • • ' k o f elements, from U , f ( u . . , )Vi ) _ Q . n " ' . . n • Hencefor th , we w i l l w r i t e the r i g h t hand s ide of ( l ) as f ( i ) ( f ( k - i ) ( , , f ( k - i ) ( ' ) ) . . . ) f ( k - D ( ), . We note tha t f^x-^Xg) = x-^Xg i s an element i n the f ree n o n - a s s o c i a t i v e a lgebra on the two generators x-^  and x^ f o r any f i e l d F . . Hence, f - s o l v a b l e and f - n i l p o t e n c e w i t h respect to t h i s element are j u s t o r d i n a r y s o l v a b i l i t y and n i l -potence. We b e g i n our study o f f - s o l v a b i l i t y w i t h n k k Lemma 1 .1 .1 L e t k > 0 and tuj_3j__i be n elements from U . Then f o r any i , 1 <_ i < k , f ( k ) " ( v - ^ - k ) = f U ) ( f ( k _ 1 ) ( ) , . . .^ ( k" i )( )) • n P roof : The p r o o f i s by i n d u c t i o n on k . I f k = 1 or k = 2 , there i s n o t h i n g to prove . By d e f i n i t i o n , we have ( i ) f W ^ u k ) a f W ( f ( u ) $ ) , . . . , f ( k - 1 ) ( )) . . n Hence, i f i = 1 , ( l ) i s the d e s i r e d r e s u l t . Assuming 1 < i , then 0 < i - 1 < k -1 , and by the i n d u c t i o n h y p o t h e s i s , f ( k - l ) ( J = f ( i - l ) ( f ( ( k - l ) - ( i - l ) ) ( ) ^ > ^ f ( ( k - l ) . : ( i - l ) ) ( } ) = f ( i - D ( f ( k - i ) ( ) , . . . , f ( k - i ) ( )) ; By ( l ) , we have 9. ( 2 ) f ( k ) ( ) = f ( l ) ( f ( i - D ( f ( k - i ) ( ) , . . . , f ( k - i ) ( ) ) , . . . , f ( i - l ) ( f ( k - i ) ( ) j < < ^ f ( k - i ) ( ) ) ) m Since l + ( i - l ) = i < k , we may app ly the i n d u c t i o n hypothes is to the r i g h t hand s i d e of ( 2 ) to o b t a i n f ( k ) ( } = f ( l + ( i - l ) ) ( f ( k - i ) ( ) ^ . . j f ( k - i ) ( )) = f ( i ) f ( k - i ) ( ) a . . . j f ( k - i ) ( . )) i which was to be p roved . Q .E .D . \ As an immediate consequence, we have Lemma 1 . 1 . 2 I f A i s an i d e a l of U and- bo th U /A and A are f - s o l v a b l e , then U i s f - s o l v a b l e . P roo f : S ince U /A I s f - s o l v a b l e , there i s a ku, > 0 such that n k l • f o r a l l sets C u i ^ i = i o f elements from U , f ( k l ) ( u n , . . . , n , ) e A . But A i s f-solvable, hence there i s a k„ > 0 such that f o r k / a l l sets ( U j J i J ^ of elements from A , f ( k 2 ) ( ) - o ' . Thus f o r a l l sets °^ elements from U , (k . ,+k j (k ) (k n ) (k n > .. f 1 . 2 ( ) = f 2 (f 1 ( ),...,f 1 ( )) = 0 which implies U i s f-solvable. Q .E .D . 10. Lemma 1.1 .3 Suppose U i s f - s o l v a b l e and V i s another f i n i t e d imens iona l a lgeb ra over F . I f a i s a homomorphism of U onto V , then V i s f - s o l v a b l e . P roof : S ince U i s f - s o l v a b l e , there i s a k > 0 such that f o r a l l sets o f n k elements {u .}^_ n from U , f ^ ( u , , . . . , u , ) = 0. k k n • Suppose ^ v i ^ i _ i l- s a n a r b i t r a r y set of n elements from V . S ince a i s onto V , there are elements w i i n U such k tha t v i = a (w 1 ) f o r I = 1 , — , n . Then, f ( k ) ( v v . . . , v . ) = f ^ k ) ( a ( w ) , . . . , a ( w k ) ' ) = a f < k ) ( w , . . . , w k ) = 0 n • n n \ which i m p l i e s N V i s f - s o l v a b l e . Q .E .D . As a r e s u l t of Lemma 1 . 1 . 5 a we have Lemma 1.1 .4 The sum of two f - s o l v a b l e i d e a l s o f U i s an f-s o l v a b l e i d e a l . P roof : Suppose A and ' B are f - s o l v a b l e i d e a l s of U . By an isomorphism theorem, we have (A+B)/B = A/(ARB) . A p p l y i n g Lemma 1 . 1 . 3 , we see tha t A/(ADB) i s f - s o l v a b l e , thus (A+B)/B i s f - s o l v a b l e . By Lemma 1 . 1 . 2 , (A+B) i s f - s o l v a b l e . Q .E .D. We now prove Theorem 1.1 .5 For f i n i t e d imens iona l a lgebras over" F , f - s o l v a b l e i s a r a d i c a l p r o p e r t y . P roof : (A) i s a consequence of Lemma 1 . 1 . 3 . Now w r i t e f (U) 11. for the f-solvable i d e a l of U of maximum dimension. I t follows from Lemma- 1.1.4 that f(U) contains a l l f-solvable ideals of and (B) follows: Now suppose f(U/f(U)) i s not zero. Then there i s a nonzero i d e a l I/f(U) of U/f(U) such that I/f(U) i s f-solvable. By Lemma 1.1.2, I i s an f-solvable i d e a l of U , hence I a f(U) , and I/f(U) = 0 , a contradiction. We conclude that f(U/f(U).) = 0 , which proves (C). Q.E.D. 1.2 Properties of f - s o l v a b i l i t y I t i s known that i f U i s niipotent, then U Is solv-able. An easy induction argument on k shows that f ( k ) ( U ) a. f k ( U ) , where f k ( U ) denotes the subspace of U spanned by f N u i * • • • * u k ( n _ i ) + i ) * a n d s i m i l a r l y f o r f^ k^(U) . Consequently, i f U i s f-nilp o t e n t , then f (U) =0 f o r some k , so f v ;(U) = 0 and U i s f-solvable.. We have Property 1.2.1 f-nilpotence implies f - s o l v a b i l i t y . . The converse of Property 1.2.1 Is known to be f a l s e . Indeed, l e t f ( x ^ , x 2 ) *= x-^Xg i n the case of L i e algebras. Property 1.2.2 f - s o l v a b i l i t y Is a hereditary, r a d i c a l i n the sense that ideals of f-solvable algebras are f-solvable. Proof: The proof i s immediate, and notes that subalgebras of f-solvable algebras are f-solvable. Q.E.D. 12. Property 1 . 2 . 3 If" K i s an extension of. F , then U i s f-solvable (f-nilpotent) i f and only i f U K = K ® F U i s f-solvable ( f - n i l p o t e n t ) . Proof: Now f(x^,...,x n) i s an element i n the free non-associative algebra on the n generators x-^,... over P . Since K i s an extension of P , c e r t a i n l y f i s an element i n the free non-associative algebra on the generators x^,...,x over K . Therefore i t makes sense to apply f • to elements of U K . Thus i f U K i s f-solvable ( f - n i l p o t e n t ) , c l e a r l y U i s f-solvable (f.-nilpotent). Conversely, i f {uj_3j_=i i s ' a basis f o r Uj, , then [1 ® u i ^ i _ i i s a °asis f o r Ug . Therefore, since f i s multi-l i n e a r , we see that TJj, f-solvable (f-nilpotent) implies U K i s f-solvable ( f - n i l p o t e n t ) . Q.E.D. Property 1 . 2 . 4 I f U i s a direc t sum of ideals TJi i = l , . . . , p , then U i s f-solvable (f-nilpotent) i f and only i f U i i s f-solvable (f-nilpotent) for a l l i . Proof: If U i s f-solvable ( f - n i l p o t e n t ) , c l e a r l y each i s f-solvable ( f - n i l p o t e n t ) . The converse follows because f i s mul t i l i n e a r . Q.E.D. Throughout the rest of this thesis, we will~assume .the degree n of f ( x 1 , — , x ) i s greater than one. Property 1 . 2 . 5 I f U i s solvable, then U i s f-solvable. 13. Proof: Suppose U = 0 . ..Then clearly. U. is f-solvable. We now define = U , and for k > 1 , u(k) = u ( k ~ l ) u ( k - l ) ^ Then, since U is solvable, the derived series terminates after k steps, that i s , where U^ k _ 1^ ± 0 . Observe that (U^"" 1)) 2 = 0 :. Thus U^ k - 1^ is f solvable. Now u ^ ^ ^ U ^ " 1 ^ and since ( u(k - 2 ) ^ 2 = t U ( k - l ) ^ w e h a y e ( u ( k - 2 ) / u ( k - l ) ) 2 = 0.; consequently U^ k" 2Vu^ k _ 1( is f-solvable, and by Lemma 1 . 2 . 2 , U^ k" 2V i s ( k - 3 ) • (k-2) f-solvable. Repeating this process with IP ' and U v ' , f k - 3 ) ' ' we see U v ' i s f-solvable. Since this process must term-inate after a f i n i t e number of steps, we conclude that . TJ is f-solvable.* ' \ Q.E.D. Suppose L i s a semi-simple Lie algebra, and l e t f(x-^x :) = x 2 . x 2 + X 2 X I * Then L is an f-radical algebra. This example shows that the converse of Property 1 . 2 . 5 i s false. What Property 1 . 2 . 5 shows i s that i f 'S(U) i s the solvable radical of U , then S(U) c f(U) . This example above shows that this, inclusion w i l l in general- be proper. However, we can show Property 1 . 2 . 6 If f.(x^fxn) Is a monomial, then S(U) = f(U) Proof : Let f(x 1,x 2) = x.^ . If n = 2 , we may, without loss .of•generality, suppose f = g , and the result i s immediate. 1 4 . Suppose n - ~ . Then without loss of generality, we may l e t f(yi^.x^ 'X-y) = x^Xg'X-^ . For elements u-^UgjU-^u^ of U ,. we observe that ( u ^ u ^ u ^ , ^ ) = f (u^u^u^,^) , and indeed any 2 elements u 1,...,u k from U can be grouped into three elements y1>y2>Y~ f r o m U s u c h t h a t ( ui> • • • >u k^ ~ f(y^,y 2,y^) . An easy induction argument shows thi s i s true for any n . Hence, suppose U i s f-solvable. Then there i s a k^ such that f o r a l l sets of k^ elements from U , f ( ) = 0 . The above argument shows that there i s a k^ \ 2 2 k„ such that any set ^ u i ^ j _ _ i °^ 2 elements from U can be n k l k l grouped into a set [y±^±~± o f n elements from U such ( k j O O g (:un,... ,.u k ) = f (y-,,... ,y k ) . 2 2 x n 1 O O ' ^ (k ) -Since f ( ) = 0 , this shows g ( ) = 0 , and we have (2k ) U d = 0 . - ... Q.E.D. 1.3 f - n i l algebras Suppose u^,...,u n_^ are n-1 a r b i t r a r y elements' from U . We w i l l write' S(u^,... * u n„T_) 'for the map from U to U defined by xS(u 1,.. .,u n_ 1) = f ( x , ^ , . . . ,u n_ x) for a l l x i n U . 15. We observe that U i s f-nilpptent i f and only i f S(u 1,..-,u n_ 1)-S(v 1,...,v n_ 1)'...-S(w 1 J...,w n_ 1) = 0 f o r a l l u i * v i * ' * * 3 w i l n u • l n p a r t i c u l a r , i f U i s f-nilpotent, then S(uj,. . . ,un^_j) i s a nilpotent map on U . Therefore, we w i l l c a l l an element u of U f-nilpotent i f and only i f S(u,...,u) i s a nilpotent map. U i s said to be f - n i l i f and only i f each element of U i s f-nilpotent. Lemma 1 .3-1 I f the dimension of U i s m , then u i s f - n i l -potent i f and only i f S ( u , — , u ) m = 0 . Proof: I f S(u,...,u) m = 0 , then u i s f-nilpotent. Con-vers'ely, suppose S(u, ...,u) = 0 . Then the c h a r a c t e r i s t i c polynomial cp(\,u) of S(u,. .. ,u) must be Xm . I f not, there i s an Irreducible factor ir(\) ^ X i n the f a c t o r i z a t i o n over F of cp(X,u) , which implies there i s a v e U such that vS(u,.. . , u ) k 4 = 0 f o r a l l k , which i s impossible. Since S(u,...,u) i s a root of cp(X,u) = X m , we have S(u,... , u ) m = 0. Q.E.D. It i s clear that I f U i s f-nilpotent, then U i s f - n i l . We are interested i n the converse, that i s , when does f - n i l imply f-nilpotent. To study t h i s problem, we need to know that under suitable conditions on F , f - n i l is"preserved under f i e l d extensions. As i n Property 1 . 2 . 3 , we see that i f K extends F , i t makes sense to tal k about U^ . being f - n i l . Lemma 1 .3 -2 I f the dimension of U i s m and F has at le a s t 16. nm+1 elements, then U i s f - n i l i f and only i f U^ . i s f - n i l for a l l extensions K of F . Proof: I f Ug. i s f - n i l , i t i s clear that U i s i t s e l f f - n i l . Conversely, suppose u-^,...,um i s a basis of U over F . We consider the map S(a 1u 1+... 4 ^ , 0 ^ + . . . -ny^,... , 0 ; ^ + . . . - K y i J where a. e F and u = ^ ^ + . . . + 0 JI . Since S i s l i n e a r i n 1 1 1 m m each of i t s arguments, we have \ 0 = S(u,.. . , u ) m =• = ( S a ...a S(u ,...,u ) ) m = T ± 1 ,. . . , i =1 1 m 1 m - • Let T' equal the sum of the terms of T where a, appears to x l • • 1 the i , power. Set T' = (a^l)T. . Then mm i-, T = E a, T. i 1 = 0 x 1 1 However, T = S(u^,... , u ^ ) m = 0 and T = S(a^.u„+... +a u ,... ,a_u~+. .. +a u ) m = 0 . o ^ 2 2 m nr ' 2 2 mm/ nm-1 i Therefore T = S a. T. = 0 . "Since F has at l e a s t nm+1 i 1 = l 1 xl elements, we choose nm-1 d i f f e r e n t non-zero values f o r . 17 . This y i e l d s a system of nm-1 homogeneous equat ions i n T. , . ^1 i ^ = l J . . . J , n m - l , whose m a t r i x of c o e f f i c i e n t s i s a Vandermonde m a t r i x . Thus we conclude T^ = 0 f o r i ^ = 0 , . . . , n m . Now l e t Tf . be the sum o f the terms of T. where .V -2 xl i 2 a „ appears to the i _ power and set T' . = a 0 T. . . Hence 2 • 2 . 1 1 1 2 2 1 1 1 2 nm-i-, i p T. = E a T. . = 0 . I f 1-, 4= 0 > w e choose nm-i 1 +1 x l i 2 = 0 d 1 1 1 2 ± X d i f f e r e n t va lues f o r a 2 and conclude as i n the p rev ious case that T. . = x0 . I f i , = 0 , we have T = 0 , or i 1 i 2 1 - o S(a~u_+.. . +a u , . . . ,a ,_u„-t- . . .+a u ) m = 0 . Repeat ing the above 1 2 2 m nr * 2 2 m m ' ^ to process on t h i s map, we conclude T Q ^ = 0 . Con t inu ing t h i s method, we have T. = 0 f o r a l l 1 m m - t u p l e s ( i n , . . . , i ) such that i , + . . . +i = nm .' * ^ 1 m • l . m Now { l g a i ^ } ^ ^ i s a b a s i s of . . Thus, i f v e U^. , m m we may w r i t e v = E § . ( l ® u . ) = E ? . u . , where §. e K . i = l 1 x i = l 1 1 1 Then S ( v , . . . , v ) m = - - r ^ V V ...,5-^+.-.+? m%) m • i , i E §, • • • § m T. - , . . * i ^m i , . . . i 1-^ +. . . +im=nm But T. . = 0 . Consequent ly , S ( v , . . . , v ) m = 0 f o r a l l - v 1 1 " * m 18. i n Ug , and Ug i s f - n i l . Q.E.D. As a consequence of the method of proof of this lemma, we have Lemma 1 . 3 . 3 I f V i s - a subalgebra of U , where U. and P s a t i s f y the hypothesis of Lemma 1 . 3 * 2 , and i f S(v,...,v) i s a nilpotent transformation of U f o r a l l v i n V , then f o r a l l extensions K of P , S(v',...,v') i s a nilpotent trans-formation of ^ Ug for a l l v'' i n Vg . \ We now set B „ = {x e U : xS(u,...,u) m = 0} where the dimension of U i s m . When no ambiguity over the f arise s , we w i l l simply write B u . We see that u i s f-nilpotent i f and only i f B u = U . Lemma 1 .3 -4 Suppose every maximal subalgebra of U i s an id e a l of U and that B u i s a subalgebra of U containing u . Then B u = U . Proof: We note that the only i d e a l of U that contains B^ i s U i t s e l f . For S(u,...,u) = S i s a transformation of U , hence by F i t t i n g ' s lemma, U = U Q S + U l g where m^ U n o = {x € U : xS = 0 } = B„ , U. c, i = 0,1 , are invariant Ut> U l o under S , and S r e s t r i c t e d to U^g i s an isomorphism. Now, suppose B c I<j U . Since u e B , IS < I . But UL „S < I as u e I . Consequently, XT'= U Q S & U l g <. I , which implies 19 . U = I . Thus i s a maximal subalgebra, and = U . Q.E.D. A class of f i n i t e dimensional algebras i s c a l l e d an E-class over F i f and only i f the following three conditions are s a t i s f i e d : (A) If' U , then the ground f i e l d of U i s either F or an extension K of F . " (B) I f H e ] ( 3 then ell f o r a l l extensions K of F. (C) I f \ V i s a subalgebra of an algebra i n , then VeU. We w i l l say that f i s an Engel function f o r the. E-class i f and only i f f o r a l l algebras U e "IL , (D) U i s f-nilpotent i f and only i f U i s f - n i l . (E) B^ i s a subalgebra of U containing u . If f i s an Engel function, the B • are c a l l e d Engel subalgebras. We now give a c o r o l l a r y to Lemma 1.3.4. Corollary 1 . 3.5 I f f i s an Engel function for the E-class 7/,, U € \L , and every maximal subalgebra of U i s an i d e a l , then U i s f-nilpotent. Proof: By Lemma 1.3-4, B^ = U for a l l u e U . Consequently U i s f - n i l . But f i s an Engel function, so U i s f-nilpotent. Q.E.D. 20. 1.4 Car tan Subalgebras Suppose L i s a n i l p o t e n t L i e a lgeb ra of l i n e a r t r a n s -format ions of an m-dimensional v e c t o r space ¥ over F . Then we can w r i t e ¥ = ¥ Q © ¥^ where ¥ Q and W^.. are r e s p e c t i v e l y the F i t t i n g n u l l and one component of ¥ r e l a t i v e to L . ¥e have ¥ Q = {w e ¥ : w£ m = 0 f o r a l l I e L} and ¥ = n ¥ ( L * ) i i where L * i s the subalgebra- generated by L and the i d e n t i t y i n the enve lop ing a s s o c i a t i v e a lgeb ra C(L) of L [18] . I f K i s an ex tens ion of F , then L g i s a n i l p o t e n t L i e a l g e b r a of l i n e a r t rans format ions on ¥ ^ . S ince ( ¥ Q ) ( L * ) m = 0 and elements o f L ^ are K l i n e a r combinations of elements of L * , we have that ( W Q ^ K = ( ^ K ^ O * S i m i l a r l y , = ( ^ K ^ l ' Furthermore, we have Lemma 1.4.1 (Jacobson) ¥^ can be c h a r a c t e r i z e d as any compli-mentary subspace of ¥Q which i s i n v a r i a n t under L . P roof : Suppose N i s such a compliment. Then ¥ = ¥ Q © - N . However, L ac t s on N as a n i l p o t e n t L i e - a l g e b r a of l i n e a r t r ans fo rma t ions . Therefore N has a F i t t i n g decomposi t ion N = N Q Q r e l a t i v e to L r e s t r i c t e d - to N . But N Q must be ze ro . For i f n e N Q then n ^ = 0 where m' i s the dimension of N and i s the r e s t r i c t i o n , to N o f £ e L . 2 1 . This implies that n e ¥ Q n N Q = 0 since N Q c N. = ¥ . Thus N = N-|_ £ ¥ 1 and N = ¥ 1 . Q.E.D. Let ^ he an E-class of algebras over F , and f = f(x^,...,x n) an Engel function for ^ . Let U and •V be a subalgebra of U . By L - r j ( V ) w e w i l l mean the L i e algebra of l i n e a r transformations on the vector space U generated by S ( v 1 , . . . . v n _ 1 ) , v i e V , i = 1 ,...,n-l . ¥e w i l l be interested i n subalgebras V of U f o r which Ly(V) \ i s nilpotent.. Indeed, i f I»rj(V) i s nilpotent, we can decompose U i n t o U Q & U ^ where U Q and are the P i t t i n g n u l l and one components of U r e l a t i v e to Ly(V) . ¥e would l i k e to be able to say that under these circumstances, V c U Q . ¥hat w i l l happen i n our theory i s that i f V i s f -nilpotent, then Ly(V) i s nilpotent and V c U Q . Prom the remarks preceding Lemma 1.4 . 1 , we see we w i l l be able to study"-this problem by extending P to i t s algebraic closure. Motivated by these observations, we make .the following d e f i n i t i o n . ¥e say a subalgebra H of U i s a Cartan sub-, algebra i f and only i f : (A) H i s f-nilpotent. (B) Lg(H) i s nilpotent. (C) H coincides with the P i t t i n g n u l l component of U • r e l a t i v e to Lg(H) . 22. In addition, i f U contains a unity, then (D) H contains the unity of U . The class of f i n i t e dimensional Lie algebras i s an E-class for any F , and f(x 1,x 2) == x^x 2 i s an Engel function for a l l members of this class. As i s well-known, (A), (B), and (C) give the definition of Cartan subalgebras of Lie algebras. Let 1L be an E-class over F and f an Engel function for 26 . The subalgebra B^ w i l l be called minimal Engel in U i f the dimension of B„ i s minimal. Clearly i f B i s minimal-\ u u Engel in U and B y c B u , then B y = B^ . In the Lie theory, Barnes [8] has shown that i f L i s a Lie algebra of dimension m and P has at least m . elements, then H i s a Cartan subalgebra of L i f and only i f H is min-imal Engel in L , where the Engel function for L i s the one given above. We would l i k e to develop a similar theory for our Engel class ^U. . To study this problem, we need some more information on extending P . If V i s a subalgerba of U e 1L and i f K i s an extension of P , then ( L U ( V ) ) K ; = ( V K ^ " Hence, i f K Ly(V) is nilpotent, then L^ (Vg) i s nilpotent. Since we K have ( U 0 ) K = ( U K ) Q a n d v ffnilpotent implies V^ i s f - n i l - . potent, i t follows that V Is a Cartan subalgebra of U i f and only i f Vg i s a Cartan subalgebra of Ug. . 23. 1 . 5 Construction of Cartan Subalgebras As i n the case with L i e algebras, a fundamental problem i s constructing Cartan subalgebras. We begin.our in v e s t i g a t i o n as follows. Let be a basis f o r ah algebra U In the E-class 2C. over P .. For u e U , write u = a^u^+.. . - K* mu m . Let cp(\,u) be the c h a r a c t e r i s t i c polynomial of S(u,...,u) , where S(u,...,u) i s the map defined above for the Engel function f . An easy, but tedious, computational argument shows Lemma m ( i ) 1 . 5 . 1 u iS(u,...,u) = £ i«k ' (0^,.. . j C t m ) u k=l where ) i s a homogeneous polynomial of t o t a l degree n-1 is. J. m In a-^,.. . , a m \ We now consider the matrix of S(u,.._. ,u) acting on_ U . Since the matrix i s determined by the action of S(u,.*.,u) on the basis elements u^, i = l,...,m , we have, l e t t i n g i ~ "'I ^ l>>-°'am} 11) (1) , ( D 10. (1) m UD (2) . J 2 ) 1 (2) (m) . 0J = matrix of S(.u,...,u) 24. Lemma 1.5.2 cp(x,u) = X m' + 0 1(u)X m~ 1+.. .+6 s(u)X r a~ S where either B^(u) = B^(a^,.. . ,a m ) i s a homogeneous polynomial of t o t a l degree ( n - l ) i or the zero polynomial. Proof; cp(x,u) i s the c h a r a c t e r i s t i c polynomial of the matrix of S(u,...,u) . Hence P j ( u ) a r e J u s t products of the uoj^ . Therefore, i s either zero or homogeneous of t o t a l degree ( n - l ) j i n the c^,. . . ,a . Q.E.D. For an algebra U l e t s' be the maximal integer such that Pe>/(.u) 4= 0 £°r some, u e TJ . An element v € U i s ca l l e d f-regular i f and. only i f B v(v) 4= 0 • Lemma 1.5.3 Suppose U e Vi . An element v e TJ i s f-regular i f and only i f " B y i s minimal Engel i n TJ . Proof; cp(X,u) = X m + B ^ u j x ™ - 1 +...+ B s, (u)X m~ s' . = X m _ s (X a +...46 ,(u)) . s I f u i s f-regular B /(u) 4= 0 . Thus the m u l t i p l i c i t y of the s root 0 i n cp(X,u) i s m-s' ,. so the dimension of i s m-s7, where s' _< m-1 since u € B u . I f B^ i s not minimal Engel, then there i s a v e U such that the dimension of B v Is less than m-s7 , say dim B y = m-s" where s" > s' . Then cp(X,v) = X m + PiCvJx" 1" 1 +...+ B * ( v ) X m _ s where B // (v) 4= 0 contradicting our choice of s' . .Hence B^ i s minimal Engel i n U . Conversely, suppose. B u i s minimal Engel i n ' TJ . 25 . Then there i s an s' <_ m-1 such that dim B u = m-s" and cp(\,u) = X m + 3 1 ( u ) X m ~ 1 +...+ p s ^ ( u ) \ m " s / / where pg//(u) 4> 0 . I f s" > s' , then we have a contradiction of our choice of s' , whereas i f s" < s' , then c l e a r l y B u i s not minimal Engel. Consequently, s" = s' , and u i s f-regular. Q.E.D. We w i l l now show that i f our ground f i e l d i s s u f f i c i e n t l y large, then U contains f-regular elements. Recall that 7/ i s an E-class over F where f i s an Engel function f o r 2^  • Lemma 1 . 5 . 4 ^Suppose U el£ and dim U = m . I f P has at le a s t (n-l)(m-l) elements, then U contains f-regular elements. Proof: We w i l l prove f i r s t that i f g(x-^,. .. ,x^) i s a non-zero homogeneous polynomial i n x^,... ,x^ . of t o t a l degree (m-l)s' over a f i e l d K of at l e a s t (m-l)s' elements, then there are elements i n K such that -• - . gC?!*. . - ,^) * 0 • ...[8] The proof i s by induction on k . If k = 1 , we may take g(x^) = x^ and c l e a r l y there i s a non-root of g . Suppose we have v e r i f i e d the r e s u l t for a l l k' < k , and suppose next that g(x^,... ,x^._^,xk) i s not i d e n t i c a l l y zero. Thus there are elements ? i J * * * ^ k - l "*"n ^ * such that g( ..., ? k_ 1,x k) has a" non-zero c o e f f i c i e n t . This i s simply a polynomial i n x^ . , and i f the maximal power of x^ . appearing Is les s than s'(m-l) , then there i s a non-root "5^ 2 6 . i n K . Hence g(§-j_j...,?k) =|= 0 . If the maximal power of x^ i s s'(m-l) , then we have g(x 1 J...,x k) = Bx£ ( m _ 1 ) + terms where "terms" involves lower powers of x^ . , each term of "terms" has at l e a s t one x^, i ^ k > i n i t , and 0 f B e K . Then g(0,...,0,l) = B 4= 0 • In either case, we see g has a non-root, which was to be proved. Now B /(u) p /(a ,.. . ,a ) Is a homogeneous poly-nomial of degree s'(m-l) , and i s not i d e n t i c a l l y zero by assumption. Since s'(m-l) _< (n-l)(m-l) , the above r e s u l t shows there are elements i n P such that ^1* ' m B / (?.. > • • • 4= 0 • Thus the element u = §-,u '+—+§ u . i s s j_ m _L _L m m an f-regular element of U , where u i > ' , * * J u m i s a basis f o r U . Q.E.D. Lemma 1.5.5 Suppose P has at l e a s t m(n-l) elements and V i s a subalgebra of U . I f i s minimal with respect to dimension i n the set {B y : v € V} and V C B u , then B u c B y f o r a l l v € V . Proof: We w i l l consider TJ , B^ , and U-B^ as vector spaces over P . For a fi x e d element c e V , we write u^ = u + uc , p. e P . Since V c c U and u^ € B^ , we may consider S(u^,.. . ,u^) as a l i n e a r transformation on TJ , B^ , and U-B^ . Let §(X,u ) be the c h a r a c t e r i s t i c polynomial of the l i n e a r transformation induced by S(u^,... ,u^) on B^ , ijr(X,u^) be the ch a r a c t e r i s t i c polynomial of the l i n e a r transformation induced 27 by S(u a..•,u ) on U-B u . Since u e B , B i s invariant under S(u ,...,u ) / - • Let ^ 1 ^ = 1 b e a oasis f o r B u and extend i t to a basis C u ^ } ^ _ l f o r u • Relative to this basis, the matrix of S(u ,. .. ,u ) i s block triangular. r* H* A . 0 B C where A may be regarded as the matrix of the l i n e a r trans-formation S(u , ...,u ) induces on B . Hence the charact-e r i s t i c polynomial of the block' A i s 8(X,u ) . We have 9(X,u^) = X m + a*(u)X m . .+a*/(u) where o t * ) i s a poly-nomial i n u- of degree at most ( n - l ) i . Now (uf + B / _ i s a basis for U-B , so C may be regarded as the matrix of the transformation induced on U-B^ by S(u ,. . . ,u ) • Hence the c h a r a c t e r i s t i c polynomial f o r C is_ , /, \ •j.m-m7 a * r \>m-m/-l, , a* /,.\ * ( X ' * V + ^ ( u J X +**-+Pm-m/^^ where i s a polynomial of degree at most ( n - l ) i i n (i . Since the c h a r a c t e r i s t i c polynomial of A* i s cp(X,u ) i t follows that cp(X,u ) = 6 ( X , u H U J U ) • We claim that a * (u ) i s i d e n t i c a l l y zero f o r a l l i . By construction, 0 i s not a ch a r a c t e r i s t i c root of the l i n e a r transformation induced on U-B u by. S(u,...,u) . Therefore, P L ™ ' ( ° ) '4= 0 • Since p* ^ /(u) has degree at most fn-l)(m-m' 'm-m x ' m-m s ' \ /\ P m_ m/(u) has at most (n-l)(m-m') roots i n F . Consequently 28 . there are p = m'(n-l) d i s t i n c t elements i n F such p that B* /(§,) f 0 . Hence B . c B since 0 i s not a characterstic root of S(u+§ .c,...,u+? .c) on U-B . By the o <J u minimality of B u ', we have B u = , c * Therefore, <] 0(X,u§j) = X m j or a i ^ j ) = 0 f o r 1 ~ lj«-', m' and j = l , . . . , p . But. cc*(u.) has at most i ( n - l ) d i s t i n c t roots i n P . Therefore, i f i < m' , then ( n - l ) i < p , and i t follows that a^(n) i s i d e n t i c a l l y zero. Since u + u.c i s i n B and B ^ , a*/(ti) must be i d e n t i c a l l y zero, u u+tic ' m v ' J \ Now a^(n) i d e n t i c a l l y zero f o r a l l i implies that B u - Bu+uc f o r a 1 1 c € V ' ^  € F * H e n c e f o r "b e V , l e t c=u-b . ' Then B u c . Q.E.D. Barnes [8] shows the following. Let f (x - ^ X g ) = x^x 2 be the Engel function f o r the class of Lie algebras, and suppose L i s a L i e algebra over • P . I f the dimension of_ L i s m , then Lemma 1 . 5 - 4 says that i f P has at l e a s t m-1 elements, then L has regular elements. Lemma 1 . 5 . 5 i n this case says that I f P has at l e a s t m elements and i f L-^  i s a subalgebra of L , B a Is minimal i n {B^ : b e L-^ 3 and L-j_ £ B& , then -B a £[ B^ for a l l b e L^ . He' then gives an example to show that f o r these lemmas, the r e s t r i c t i o n s on F cannot be removed. Now suppose B i s minimal Engel i n L . Then B a a i s c e r t a i n l y minimal i n {B^ : b Ba3 , and hence B & c f o r a l l b i n B . Thus S(b) i s nilpotent on B for a l l b i n B Q and i t follows since f . i& an Engel function that B a, a 2 9 . i s n i l p o t e n t . In t h i s case , i t Is c l e a r tha t L ^ ( B a ) i s n i l p o t e n t and B c o i n c i d e s w i t h the P i t t i n g n u l l component of L r e l a t i v e to L T ( B ) . Consequent ly , i f P has a t l e a s t m elements and j-i a B^ i s min imal Engel i n L , then B o i s a Car tan subalgebra . o a In chapter two and th ree , we w i l l use Lemma 1 . 5 . 4 and Lemma 1 . 5 -5 to cons t ruc t Car tan subalgebras f o r a l t e r n a t i v e and Jordan a lgebras u s i n g e s s e n t i a l l y the same method desc r ibed above f o r the L i e case . 1.6 The Inner Automorphism Group of an A l g e b r a To show that Car tan subalgebras of U are conjugate under a c e r t a i n c l a s s of automorphisms on U seems to be a d i f f i c u l t problem. Cheva l l ey [11] has so lved the problem i n the case of L i e a lgebras over a l g e b r a i c a l l y c l o s e d f i e l d s of cha rac t -e r i s t i c ze ro . In t h i s s e c t i o n , we cons t ruc t a method that p a r a l l e l s C h e v a l l e y ' s that w i l l a l l o w us to so lve the problem i n Chapters Two and Three when the ground f i e l d i s a l g e b r a i c a l l y c lo sed and of c h a r a c t e r i s t i c z e ro . Where proofs are not g i v e n , the reader i s r e f e r r e d to Cheva l l ey [ l l ] . We beg in by d e f i n i n g the Z a r i s k i topology on an•• T r i -dimens ional v e c t o r space V over P where the c h a r a c t e r i s t i c o f F i s .zero. Suppose v ± > ' - - > v m i s a b a s i s f o r V ." Then for^ each v e V , there i s a unique set . . . . , § m of elements i n P such that v = § v ^ . ..+? v •. Now suppose f ( X - , , . . . , X ) i s an 1 1 mm . 1 m . element i n the p o l y n o m i a l r i n g F [ x n X ] . Then 30. f ( X - , , . . . , X ) and the b a s i s v, , . . . , v determine a map from V ^ ± ' 3 m' 1' m ^ to F def ined f ( v ) = f ( § ; , _ , . . . , § m ) • We c a l l f a p o l y n o m i a l f u n c t i o n on V . I f f and g are two po lynomia l f u n c t i o n on V and a e P , we set (f+g)(v) = f ( v ) + g(v) ( f g ) ( v ) = f ( v ) g ( v ) ( a f j ( v ) = a ( f ( v j ) . Thus the set ^ F[V] o f po lynomia l func t ions on V i s an F -a l g e b r a . Moreover , s ince the c h a r a c t e r i s t i c of. F i s z e ro , the map i s an isomorphism of F [ X ^ , . . . , X ] onto F[V] . This map sends X^ i n t o the po lynomia l f u n c t i o n 77\ where T T 1 ( § 1 V 1 + . . . + § 1 V 1 + . . . + § m V m ) = 5 ± • Since 77\ i s a homomorphism from V to F , i t f o l l o w s that V = Hom-p(V,F) and the constant func t ions generate F[V] . I f W i s a non-empty subset o f V , the po lynomia l func t ions on W , F[W] , are def ined to be the r e s t r i c t i o n to W of elements i n F[V] . Suppose E. i s a subset of V . We say E i s an a l g e b r a i c set i f and on ly i f there i s a subset B of F[V] " such that 31. E = {x e V : b(x) = 0 f o r a l l b e B} . The union of two algebraic sets i s algebraic, and the i n t e r -section of a family of algebraic sets i s algebraic [11; I I I , page 169 ] . Furthermore, V and 0 may be considered algebraic. We w i l l c a l l a subset of V closed i f and only i f i t i s algebraic. The above properties on algebraic sets show that these closed sets determine a topology on V . This topology i s c a l l e d the Z a r i s k i topology. I f E i s a subset of V , the closure of E , E , i s defined E = {x e V : p(x)=0 f o r a l l p e F[V] such that p(y)=0 f o r a l l y € E} . Now i f v^,...,v^ i s another basis f o r V , then f o r v e V , there are sets of elements 5 ^ , . . . , § m , . . . , § ^ i n F such that But v^ = r l j _ i v i + ' • • + r l j _ m v m where r\±1,.. . , n ^ e F . Consequently, i f f(X-^,...,X ) i s an element i n F[X^,...,X m] , then there i s another element f ' (X-j_,. . . ,Xjn) i n F[X 1,...,X j n] such that f ( § 1 , . . . , ? m ) = f'(§£,....,?;). . - * If F[V] and F [ V ] / are the polynomial functions on V r e l a t i v e to v^,. . . ,v and Vp... 3v^ respectively, i t follows f o r f an element i n F[V] , there i s an element f ' i n F[V]' such that f(v) = f'(v) f o r a l l v i n V . In this respect, polynomial 3 2 . functions are independent of the basis chosen f o r V , hence the Z a r i s k i topology f o r V i s independent of the basis. I f ¥ i s a subspace of V , I t follows from the remarks above that P[W] i s the r e s t r i c t i o n to ¥ of P[V] . I t i s clear that ¥ i s algebraic, thus closed, and the Z a r i s k i topology on ¥ i s the topology on ¥ induced by the Z a r i s k i topology on V • Suppose E i s a non-empty subset of V . Then E i s ir r e d u c i b l e i f and only i f P[E] i s an i n t e g r a l domain. Under the topology induced on E by the Z a r i s k i topology on V , we observe that E i s i r r e d u c i b l e i f and only i f a l l non-empty r e l a t i v e l y open subsets of E are dense i n E . [11; I I I , page 175 ] . Since P i s of c h a r a c t e r i s t i c zero, we claim that a l l open subsets of V are Z a r i s k i dense i n V . - To see t h i s , we begin with Lemma 1 .6 .1 Suppose f and g are elements i n F[X^,..., X^] with g non-zero. Let C = {(a^,...,a m): e F and g(a 1,...,a m) ={= 0} . I f f ( a 1 , . . . ,a m) = 0 f o r a l l ( a p . . . . j o J c C , then f i s i d e n t i c a l l y equal to zero. Proof: For m = 1 , the re s u l t i s clear, since g '• has i n f i n i t e l y many non-roots. For m > 1 , we may write f and g as poly-nomials i n \ m with c o e f f i c i e n t s from F[X-^,...,X ^] . ¥e may then proceed by induction on m to complete the proof. Q/.E.D. 33. C o r o l l a r y 1 . 6 .2 I f E i s a non-empty Z a r i s k i open subset of V , then E i s Z a r i s k i dense i n V . P roo f : The compliment of E , cE , i s c l o s e d . Consequent ly , there i s a subset B of P[V] such that b(a) = 0 f o r a e cE and a l l b e B . S ince E i s non-empty, B i s non-empty. L e t b be a non-zero element i n B . Then E-^  = {v e V: b (v) 4= 0} i s an open set of V conta ined i n E . Suppose p e P[V] and p(v) = 0 f o r a l l v e E^ . . B y Lemma 1 .6 . 1 , i t • f o l l o w s that p (v ) = 0 f o r a l l v e V . Hence, E = V , and E i s Z a r i s k i dense i n V . Q .E .D . C o r o l l a r y 1 . 6 . 3 I f W i s a non-zero subspace of V , then ¥ i s i r r e d u c i b l e . P roo f : S ince the Z a r i s k i topology on ¥ c o i n c i d e s w i t h the topology induced by the Z a r i s k i topology on V , the r e s u l t f o l l o w s immedia te ly from C o r o l l a r y 1 . 6 . 2 and the remarks p reced ing Lemma 1 . 6 . 1 . Q .E .D. ¥e w i l l c a l l a subset E o f V epais i f and on ly i f E i s i r r e d u c i b l e and con ta ins a non-empty r e l a t i v e l y open sub-set o f i t s Z a r i s k i c l o s u r e . ¥ e have C o r o l l a r y 1 . 6 . 4 I f ¥ i s a non-empty subspace of "V then ¥ i s epa i s . P roo f : ¥ i s c l o s e d and i r r e d u c i b l e s i nce ¥ i s a subspace. S ince ¥ i s non-empty, P [ ¥ ] f 0 J Thus i f p i s a non-zero 3 4 . element i n F[W] , the set [v e W : p (v) =j  0} i s open and the . r e s u l t f o l l o w s . Q .E.D. We r e t u r n f o r a moment to our a lgeb ra U over F . R e c a l l tha t an element u TJ i s f - r e g u l a r i f and on ly i f 6 / (u) f 0 [Lemma 1 . 5 - 4 ] . But B / e P[U] , and we have s s C o r o l l a r y 1.6.5 The . se t of f - r e g u l a r elements of- U form a dense open subset of U . U l • W e x w i i l denote the set of f - r e g u l a r elements o f U by Since subalgebras of U are subspaces, we have C o r o l l a r y 1.6.6 I f H i s a subalgebra of U . then H i s epais i n U . ' To see how the Z a r i s k i t opo logy w i l l help us to solve, the conjugacy problem, we must def ine what we mean by the group of i n n e r automorphisms of TJ . . We beg in as f o l l o w s . L e t C = Hom F (V,V) . Then P[C] i s generated by the constant func t ions and C = Hom F (C,F) . A group G of auto-morphisms o f V i s - c a l l e d an a l g e b r a i c group i f and on ly i f there i s a subset B of P[C] such that G = [a : a i s an automorphism of V and b(a) = 0 f o r a l l b e B} . 3 5 . We wish to define what i s meant by the L i e algebra of an algebraic group G [ 1 1 ; I I , page 1 2 5 - 1 3 6 ] . We observe that F[C] can be made into a two-sided C module by making the following d e f i n i t i o n s : f o r e,x e C and p e F[C] , (p-e)(x) = p(ex) right t r a n s l a t i o n (e»p)(x) = p(xe) l e f t t r a n s l a t i o n . If F^ i s the set of constant functions, then F^ fl C* = { 0 } . Thus, i n F[C] , the sum F^ + C* i s di r e c t . Consequently, the l i n e a r map , e e C , which i s l e f t t r a n s l a t i o n by e on C* can be extended to a l i n e a r map D g on F^ + C by setting Dg(k) = 0 f o r a l l k € F^ . Chevalley [ 1 1 ; II,:page 2 1 - 2 6 ] has shown this map can be uniquely extended to a derivation D g of F[C] . I f e^ and e 2 are elements of C , since D and D + D both map F- onto 0 and agree e l + E 2 8 1 2 1 on C* , i t follows that D = D + D . , or the map e -* D e-^ +eg e^ e 2 e i s l i n e a r . S i m i l a r l y , since Dr . -i and [D ,D ] are Le^,e 2J e-^  e 2 derivations of F[C] which coincide on C* and map F^ onto. 0 , we have = ^ ^ e ^ ' Now l e t G be an algebraic group and l e t Q be the subset of F[C] such that i f q i s an element of Q ", then q r e s t r i c t e d to G i s zero. Clearly Q i s an i d e a l of F[C] , and Q i s c a l l e d the i d e a l of polynomial functions associated  with G . I f Q i s a prime i d e a l , G i s c a l l e d i r r e d u c i b l e . I t i s easy to see that G i s i r r e d u c i b l e i f and only i f F[G] 36. i s an i n t e g r a l domain. I f G i s ah a l g e b r a i c group and Q i s i t s a s s o c i a t e d i d e a l of po lynomia l f u n c t i o n s , we see that the set (e e C : D g (Q) c Q} i s a L i e subalgebra of the L i e a lgeb ra of C . This L i e a l g e b r a , denoted G^ , i s c a l l e d the L i e a lgeb ra of G . Now l e t L(U) denote the L i e a lgeb ra over F generated by R^, L ^ f o r u e U . L(U) i s c a l l e d the L i e m u l t i p l i c a t i o n  a lgeb ra of TP . A d e r i v a t i o n D of U i s c a l l e d an i n n e r  d e r i v a t i o n i f and o n l y i f D e L(U) . I t ' I s known- that the set <^'(U) of i n n e r d e r i v a t i o n s of U i s an i d e a l i n the d e r i v a t i o n a lgeb ra c^(U) of TJ [ 2 5 ] . Now l e t g(x, , . . . , x ) be a non-zero l i n e a r homogeneous element i n the f ree n o n - a s s o c i a t i v e a lgeb ra over F i n p generators x-^,. . . , x . • Suppose, f o r a l l elements u ^ , . . . J u p _ ] _ i n U , the map x D ( u 1 , . . . , u p _ 1 ) = g ( x , u 1 , . . . , u p _ 1 ) x e U i s a d e r i v a t i o n of TJ . Then D ( u ^ , . . . , u _ e #^(TJ) . . We have We set 37 . = {G : G i s an algebraic group of automorphisms on the vector space U such that G^ _> • "Chevalley [11; I I , page 179] shows that i f A i s the group of automorphisms of the algebra U , then A^ =^(U) . Of course, A eM . Then l(U) = fl{G : G } i s an i r r e d u c i b l e algebraic group such that I ( U ) L => ( u) [ H i 1 1 > P a S e 165-172] and A => l(U) . Consequently elements i n I(U) are automorphisms of the algebra U , and we c a l l I(U) the .group of inner auto- morphisms of U . We how return to our a r b i t r a r y vector space V . Recall that F[V] and FtX^...,*. ] are isomorphic. I f a = ?n v.. +.. . +§ v i s an element i n V where v.,,...,v i s a ' 1 1 'mm l J . m basis for V , and i f f i s an element i n P[V] , we define a l i n e a r function d f on V by a l a f ( n 1 v 1 + . . . + T l m v m ) = (|f ) t i 1 +...+(|f. ) aT! m 1 m We c a l l d f the d i f f e r e n t i a l of f at a . This map has the following properties: (A) d a(f+g) = d a f + d ag (B) d a ( a f ) = a ) d a f ) a e F (C) d a ( f g ) = f ( a ) d a g + g(a)d af . Furthermore, f o r a e V , the map y - d a f ( y ) i s l i n e a r . 3 8 . Let E be an i r r e d u c i b l e subset of V and x e E . The tangent space to x at E i s defined to be T ( E J X ) = {y e V : d f(y) = 0 f o r a l l f e F[V] such that f(a) = 0 for a l l a 6 E ] Since y d & f ( y ) i s l i n e a r , T ( E;x) i s a subsapce of "V . Furthermore, we. have Lemma 1 .6 .7 I f ¥ i s a subspace of V , then W c T(¥;x) f o r a l l xe¥ . \ Proof: Let v-,,...,v / be a basis for ¥ , and extend i t to 1 J ' m • a basis v i ^ ' * ' - ' v m ^ o r ^ * Relative to this basis, i f f e F[V] and f(v) = 0 f o r a l l v e ¥ , i t follows that ||- S 0 f o r i = l,...,m' . I f x e ¥ , y = T 1 1 v 1 + - • • + T l m v m 1 KL I f y e ¥ , y = ru v +. ..+ri /V / , "and i f f ( v ) = 0 f o r a l l u. j in jTi v e ¥ , d f(y) = 0 . Thus ¥ c T(¥;x) for a l l x e ¥ . . Q . E . D . . A . *•"" I f G i s an i r r e d u c i b l e group of automorphisms of V and E i s a subset of V , then = (y e V : y = xa f o r some • x e E , a e G} i s c a l l e d the or b i t of E with respect to G . I f E i s ir r e d u c i b l e , then Q i s i r r e d u c i b l e [11; I I I , page 192-193]. In order to apply the theory given by Chevalley i n what 39. follows, we must assume F i s a l g e b r a i c a l l y closed. Suppose H i s a subalgebra of our algebra U . Since H i s epais, by Lemma 1 . 6 .7 and Chevalley [11; I I I , page 192 -193 ] , the o r b i t n of H under l ( U ) i s i r r e d u c i b l e and T(fi;x) contains XdiV (U) + H for a l l x e H where x $ (U) = (xD: X € /#'(U.)} . From Chevalley [11; I I I , Proposition 13 , page 180 and the Corollary, page' 1 9 2 ] , we have Theorem 1 . 6 . 8 I f H contains a non-empty r e l a t i v e l y open sub-set 0 such that T(0;a) = U f o r a l l a e 0 , then fi contains a non-empty open subset of U . We now give Theorem 1 . 6 . 9 Suppose minimal Engel subalgebras of U are Cartan subalgebras. I f and Hg are two Cartan subalgebras of U such that and Hg s a t i s f y the hypothesis of Theorem 1 . 6 . 8 , then and Hg are conjugate under I(u) i n the sense that there i s an element a e I.(U) such that = HgO" . Proof: Let Q 1 and fig be the orbits of H^ and Hg under l ( U ) . By Theorem 1 . 6 . 8 , 0-^  and fig contain non-empty, open subsets of U . I t follows that U-j_ l~l (o^flfig) =f= 0 . Let b be a non-zero element i n this i n t e r s e c t i o n . As b € fi^ , i = 1,2 , b e f o r some a± i n I(U) . But "b i s f-regular, so B^ i s a Cartan subalgebra. Therefore B^ <= H^cr^ , hence B^ = H ^ C N . Thus H-^ = HgC where a = crgC^ 1 e I(U) . Q.E.D. 40. Consequently, when F i s a l g e b r a i c a l l y closed and of c h a r a c t e r i s t i c zero, to show Cartain subalgebras are conjugate, we must show that minimal Engel subalgebras are Cartan subalgebras, and that Cartan subalgebras s a t i s f y the hypothesis of Theorem 1 . 6 . 8 . With respect to the conjugacy problem, we f e e l there must be a self-contained approach. We f e e l this could be develpped following the methods given by Jacobson i n [18; Chapter IX]. However, to date we have been unsuccessful. CHAPTER TWO ALTERNATIVE ALGEBRAS 2.1 Introduction An algebra A over a f i e l d P i s c a l l e d an a l t e r -native algebra i f and only i f (x,x,y) = (y,x,x) = 0 f o r a l l elements x and y of A . From this d e f i n i t i o n , i t i s clear that homomorp.hic images of al t e r n a t i v e algebras are al t e r n a t i v e and that a d i r e c t sum of alte r n a t i v e algebras i s alt e r n a t i v e . Since the associator i s m u l t i l i n e a r , we have that A^ . i s a l t e r -native f o r a l l extensions K of F . Throughout Chapter Two, we w i l l assume our al t e r n a t i v e algebras are f i n i t e dimensional. In.this section we w i l l give the necessary theory to develop a Cartan theory for al t e r n a t i v e algebras. In what follows, where proofs are not provided, the reader i s referred to Schafer [ 25 ] . We begin by l i n e a r i z i n g the defining i d e n t i t i e s of . A , and obtain (1) (x,y,z) = -(x,z,y) = (z,x,y) (2) (x,y,x) = 0 ' -For right and l e f t m u l t i p l i c a t i o n R and L , x e A , of A , X X these become 42. (~) R E = R = L - L L = L R - R L„ x y xy xy y x y x x y = L L - L = R L - L R = R - R R x y yx y x x y yx y x and CO R x L x = L x R x We next give Theorem 2.1.1 (Artin) The subalgebra generated by any two elements of A i s associative. \ . As a consequence of this r e s u l t , we have Corollary 2.1.2 ( i ) Alternative algebras are power-associative, i n the sense that the subalgebra generated by a single element i s associative. ( i i ) For a l l x e A , R . = ( R J 1 and x L . = '{L)1 f o r i = 0,1,. . . x Now suppose e i s an idempotent of "A . . By (4) and the above c o r o l l a r y , R_ and L are commuting idempotent operators. I t follows that A i s a vector space d i r e c t sum A = A00 + A10 + A01 + A l l where A ± J = {x e A : ex. . = lx±. x±.e = Jx. .} i , j = 0,1 . Hence i f x e A , we write the decomposition of x : x = exe + (ex - exe) + (xe' ~ exe) + (x - ex - xe + exe). 43-where exe e A - ^ , ex - exe € k^Qi xe - exe e A Q 1 , and x - ex - xe - exe e A Q Q . A set e^,....e g of idempotents i s c a l l e d pairwise  orthogonal i n case e.e, = e.e. = 0 f o r i f j, i , j = l , . . . , s . I f A has a unity and e, ,...,e ds a set of pairwise orthogonal idempotents whose sum i s the unity of A t we get a refined decomposition of A as a vector space dir e c t sum s • (2.1.3) A = £ Ok.. i , j = l 1 J \ A . . * = fx. . e A ; e. x. . = 6, . x. . x. .e, = 6 ., x. .§ k = 1,...,s \ . where 6. . i s the Kronecker delta. This decomposition i s c a l l e d the Pierce decomposition of A r e l a t i v e to e^,...,e g , and we have Theorem 2.1.4 Let (2.1.3) he the Pierce decomposition of A r e l a t i v e to the pairwise orthogonal idempotents e,,...,e . Then: ( i ) A i j A i k - A±3 ±>3>* = 1 " " ' s ( i i ) A..A..CA.. I,J - l.....,s ( i i i ) A . j A ^ = 0 j ± k- ( i , j ) f (k,L) I j j j k , ! . = -1, •. ., s (iv) ( x ^ . ) 2 = o A . / I =j= J . 44. We now need some information on nilpotent a l t e r n a t i v e algebras. We begin with Theorem 2 . 1 . 5 The following are equivalent: ( i ) A i s nilpotent ( i i ) A i s solvable ( i i i ) A i s n i l An element z*A i s c a l l e d properly nilpotent i f and only i f za i s nilpotent f o r a l l a e A . I f 2 i s properly nilpotent, by Theorem 2 . 1 . 1 , we see az i s nilpotent f o r a l l a e A . I t i s known that the r a d i c a l of A can be characterized as the set of properly nilpotent elements of A . Suppose A i s semi-simple i n the c l a s s i c a l sense. Then A has a unity, and we have Theorem 2 . 1.6 A non-zero a l t e r n a t i v e algebra- i s semi-simple i f and only i f i t i s a direct sum of simple i d e a l s . Thus the study of semi-simple a l t e r n a t i v e algebras i s reduced ~to studying the simple ones. We say an idempotent e e A i s p r i m i t i v e - i f and only i f e cannot be written e = e' + e" where e' and e" are non-zero orthogonal idempotents. I f A has a unity, then this unity element can be expressed as a sum of pairwise orthogonal pr i m i t i v e idempotents. Let t be the maximal integer such that 45. the unity of A i s expressible as a sum of t pairwise orthogonal p r i m i t i v e idempotents. Then the degree of A i s defined to be t . Now suppose F i s a l g e b r a i c a l l y closed and (2.1 .3) i s the Pierce decomposition of A r e l a t i v e to a set e-^3...9e^ of pairwise orthogonal p r i m i t i v e idempotents. Since e^ i s the unity of A ^ and i n f a c t i s the only idempotent i n A ^ , we have i f x e A ^ and x i s not nilpotent .then the subalgebra of A ^ generated by x, F[x] , i s commutative, associative, and contains x e^ . By Wedderburn's theorem, F[x] = F^ + N where N i s a n i l algebra and F^ i s semi-simple and, i n this case, simple. Then F^ i s a matrix algebra over a d i v i s i o n algebra. Since e^ i s the only idempotent i n F^ and F i s a l g e b r a i c a l l y closed, i t follows F-^  = Fe^ . Consequently, i f x e A ^ , we may write x = ae^ + n where a e F and n i s nilpotent. Now l e t n € A ^ and n be nilpotent. We claim i f a e A.. , then na i s nilpotent. I f na i s not nilpotent, by the above remarks i t follows that na has an inverse (na) ~ , Let p be the integer such that rP = 0 =)= n^ ""1* . Then 0 ^ n p _ 1 = n p" J"[(na)(na)" 1] = n p a ( n a ) _ 1 = 0 , a contradiction. S i m i l a r l y an i s nilpotent, and thus the set of nilpotent . elements of A ^ form an i d e a l . I t follows that A ^ = Fe i + N where N i s nilpotent. Jacobson c a l l s an a r b i t r a r y algebra U over a f i e l d K almost n i l i f and only i f TJ has a unity 1 and TJ = F l + N where N i s a n i l i d e a l . 46, Hence, we have shown Lemma 2 . 1 . 7 I f F i s a l g e b r a i c a l l y c l o s e d and ( 2 . 1 . 3 ) i s the P i e r c e decomposi t ion of A r e l a t i v e to p a i r w i s e o r thogona l p r i m i t i v e idempotents e 1 , . . . , e g , then the are almost n i l . Now suppose A i s s imple . I f the degree of A i s 1 , then A i s a d i v i s i o n a l g e b r a . I f the ground f i e l d i s c l o s e d then Lemma 2 . 1 . 7 says that A = F l . I f the degree of A i s g rea t e r than two, then A i s a s s o c i a t i v e . I f the degree of A i s two, x then A i s e i t h e r a s s o c i a t i v e or a Cayiey a lgebra [25l • To def ine the i n n e r automorphism group of an a l t e r n a t i v e a l g e b r a , we r e q u i r e some i n f o r m a t i o n on i n n e r d e r i v a t i o n s of a l t e r n a t i v e a lgeb ra s . We beg in by d e f i n i n g the nucleus N(A) of an a l t e r n a t i v e a lgeb ra as the set -[g e A : ( g , x , y ) =-0 f o r a l l x , y e A} . • I f the c h a r a c t e r i s t i c of F i s d i f f e r e n t from 3 , then R - L • ' • . g g i s a d e r i v a t i o n of A i f and on ly i f . g e N(A) [25, page 7 6 ] . I f we set . -(5) D(b , c ) = [ L b , L c ] + [ L b , R c ] + [ V R c ] then D(b , c ) i s a d e r i v a t i o n of A [25, page 7 7 ] - " Schafer [25, page 78] shows that i f the c h a r a c t e r i s t i c of F i s d i f f e r e n t than 2 and 3 and i f A has a u n i t y , then any i n n e r d e r i v a t i o n D of A can be w r i t t e n 47. i = l g e N(A) , b i,c j L e A . 2.2 The Universal M u l t i p l i c a t i o n Envelope of an Alternative  Algebra. Suppose A i s an alt e r n a t i v e algebra over P and k' i s i t s anti-isomorphic image. I f a e A , we w i l l write a 7 f o r the anti-isomorphic image of a . We set B-^  = B = A © A y and i n d u c t i v e l y B n = B n _ j ® B 2 ' L e t T ( B ) b e t h e associative algebra defined by T(B) = B± ® B 2 © B^ where the vector space operations i n T(B) are as usual and m u l t i p l i c a t i o n i n T(B) i s denoted by ® . Let S be the i d e a l of T(B) generated by elements of the form (1) a^ ® a 2 - a 2 ® a^ - a 1 a g + a 1 ® a g (2) a ^ g -• a 1 ® a 2 - ( a ^ ) ' + a^ ® (3) ( a ^ ) ' - a 2 ® a 2 - a 1 ® a 2 + a g ® a 1 where a^ e A . The associative algebra U(A) = T(B)/S Is c a l l e d the universal m u l t i p l i c a t i o n envelope of A . If i ' i s the canonical homomorphism from T(B) into U(A) , then the 4 8 . r e s t r i c t i o n o f i ' to B = B^ def ines a l i n e a r map from B i n t o U(A) . We c a l l t h i s map i , and i f a e A , we w i l l w r i t e a i = a, , a ' i = a ' where a , a ' are the cosets of a and a ' i n U(A) . From ( l ) , (2) and ( 3 ) , we have the f o l l o w i n g .:• r e l a t i o n s In U(A) : ' ( 4 ) a,^  ® a 2 - a 2 ® a,^  ' = a^a^ - a^ ® a g = Xa^a^X' - a^ ® a_2 = ^ ® Hp - &2 ® a i • Lemma 2.2.1 'Let p be a l i n e a r map from B i n t o an a s s o c i a t i v e a lgeb ra V such that ' , [a-j_p,a2p] = ( a 1 a 2 ) p - (a-Lp)(a 2p) = ( a 2 a 1 ) 'p - (a-^pHa^p) = [ a ^ - a ^ p ] . Then there i s a unique homomorphism p* from ~U(A) i n t o V such that ap* = ap and a/p* = a'p , Proof : Suppose {b^ : i e 1} i s a b a s i s f o r B . . Then the d i s t i n c t elements b . ® b . ®. . . ® b . forwa b a s i s f o r B where b . ® , . . ® b . = b . ® . . . ® b . i f and on ly i f i , = j . k = 1,..-. , n . 1 n j d n Consequent ly , the set of a l l these elements f o r a b a s i s f o r T (B) . We now def ine the map p" . from T(B) i n t o V where (b. ® . . . ® b . )p" = (b. p ) . . . ( b . p) . C l e a r l y t h i s i s a homo-xl n . 1 • . x n • morphism from T(B) i n t o V such that ap" = ap i f a € B . 49. By h y p o t h e s i s , the generators of S are mapped i n t o the k e r n e l of p" , consequent ly p" induces a homomorphism p* o f U(A) i n t o V . I f a e A , then a i p * - ap* = ap" = ap ; s i m i l a r l y a ' i p * = a 7 p , and p = i p * as d e s i r e d . S ince T(B) i s generated by B , U(A) i s generated by B i , and i t f o l l o w s that p* i s unique . Q . E . D . Lemma 2 . 2 . 2 I f K i s an i d e a l of A and D i s the i d e a l i n U(A) generated by (K <© K 7 ) i ? then there e x i s t s an isomorphism of U(A / |0 onto U(A) /D such that a + (K Q K 7 ) i s mapped onto a + D and a 7 + (k Q K 7 ) i s mapped onto a 7 + D , a e A . ' P roo f : We def ine a map a*: B-*>U(A)/D where, I f b € B , then ba* = b i + D . S ince (K $ K 7 ) i C D , a* maps K Q K 7 onto 0 . Thus a* induced a map a : B / K 0 K 7 - U(A) /D such that (b + (K QK'))a = b i + D . I f 6 i s a l i n e a r map from B / K ;®K' . i n t o an assoc-i a t i v e a lgeb ra V where B / K © K 7 and 9 s a t i s f y the c o n d i t i o n s of Lemma 2 . 2 . 1 , i t f o l l o w s by examining homomorphism and u s i n g Lemma 2 . 2 . 1 tha t there i s a unique homomorphism 9 7 from U(A) /D i n t o V . -I f i ^ i s the n a t u r a l map from B / K ^ K 7 i n t o U ( A / K ) i t now f o l l o w s that the diagram 5 0 . U(A/K) . „ ^ ^ ^ B/K + K' - > U(A)/D i s commutative, and i£ i s the desired isomorphism. Q.E.D. We w i l l use the universal multiplication•envelope of A to prove the following r e s u l t . Suppose A^ i s a subalgebra of A , and V i s the subalgebra of HomF(A,A) . generated by the maps R , N L , a e A-, , acting on A . Then i f J i s a a a -L solvable i d e a l of A-, , the maps R , L , a e J , acting on A _L a a generated a nilpotent i d e a l i n C . We claim that i t i s s u f f i c -i e nt to show that (J Q J')i-|_ generates a nilpotent i d e a l i n U(A^) , where i ^ i s the natural map from A^ © A^ i n t o U(A) . Indeed, by (3) i n § 2 . 1 , the l i n e a r map p from A^ © A-j\ into C where, i f a e An , ap = R and a' p = L , and C s a t i s f y the conditions of Lemma 2 . 2 . 1 . Thus there i s . a homomorphism from U(A n ) into C . Since C i s generated by. R and L , _L a a i t follows that this homomorphism i s onto, and maps the i d e a l generated by (J Q J ' ) i onto the i d e a l of C generated by R , L_. , a e J . Consequently i f (J ©J')i, generates a a a j. nilpotent i d e a l i n U(A-,), R , L , a e J generated a nilpotent i d e a l i n C . We begin by showing 51. Lemma 2 . 2 . 5 Suppose I i s an ide a l of A and -I* i s the sub-algebra of U ( A ) generated by (I © l ' ) i . Then D = I* ® U ( A ) i s an i d e a l of U ( A ) . Proof: I t i s clear that D i s a right i d e a l . Hence, suppose a,b e A . Prom ( 4 ) , we obtain the following i d e n t i t i e s : (5) a ® b' = b' ®. a - ba + b ® a ( 6 ) &' ® b' = (ba)' + (ba) - b ® a, (7) a ® b = (ab) + (ba) - b ® a (8) a' ® b = b ® a' + b ® a - (ba) . Since ( A ($> A' ) i generates U ( A ) and. ( l © I ' ) i generates I* , i t follows from (5) - (8) that D i s a two-sided i d e a l of U ( A ) . Q..E.D. Corollary 2 . 2 . 4 I f I* i s nilpotent, D i s _ n i l p o t e n t . Proof: Since U ( A ) ® 1 * C I* + I* ® T J ( A ) , an easy -induction argument shows that D n £ ( l * ) n + ( I * ) N ® U ( A ) . • The re s u l t i s now immediate. Q.E.D. Lemma 2 . 2 . 5 I f A . i s solvable and the dimension of A i s 1~, then ( U ( A ) ) 3 = 0 . ' o Proof: I f dim A = 1 , then A = Pe where e = 0 .. Prom (5) -(8 ) , we obtain ee = e'e' = 0 and e'e = ee' . Since ( A ( £ ) A ' ) i generates • U ( A ) , e and e' generate U ( A ) i n this case. Consequently, ( U ( A ) ) - 5 = 0 . Q.E.D. 52. Lemma 2.2.6 I f A i s solvable, then U(A) i s nilpotent. Proof: The proof i s by induction on the dimension of A . By Lemma 2.2.5, we assume dim A = n > 1 . Since A i s solvable, 2 there i s an n-1 dimensional subspace I such that A c I | A , and i n fac t , I i s an i d e a l of A . Consequently, as dim A/I = 1 and ( A / l ) 2 = 0 , U(A/I) i s nilpotent. Let 1^ b f the i d e a l i n U(A) generated by (I ££)l')i . By Lemma 2.2.2, U(A/l) i s isomorphic to U(A)/I^ . I f I i s the subalgebra of U(A) generated by (I © I ' ) i , then 1^ = I* + I* ® U(A) . Since dim I\< h , U(l) i s nilpotent. I t follows from Lemma 2.2.1 that I* i s nilpotent since I* i s a homomorphic image of U(I) . Consequently, I 1 i s nilpotent, and since U(A)/I^ Is too, U(A) i s nilpotent. Q.E.D. Theorem 2.2.7 Suppose I i s a solvable i d e a l of A . Then ( I © l ' ) i generates a nilpotent i d e a l i n .U(A) . Proof: Suppose I* i s the subalgebra of U(A) generated by (I © l ' ) i . Since I i s solvable, U(l) i s nilpotent, thus I* i s nilpotent, and the re s u l t follows by Corollary 2.2.4. Q.E.D. Corollary 2.2.8 I f A^ i s a subalgebra of A , I : i s ' a solvable i d e a l of A^ , and C i s the subalgebra of Hom.p(A,A) generated by Rg^I^ > a £.A]_ , then ^,1^ , b e I , generate a nilpotent i d e a l i n C . Proof: 2.2.3-The proof i s immediate by the remarks preceding Lemma Q.E.D. 53. 2 . 3 Existence of an Engel Function f o r Alternative Algebras. We w i l l now apply the theory developed In Chapter One to a l t e r n a t i v e algebras. In this section we w i l l show that, i f the ground f i e l d F of an alte r n a t i v e algebra A has enough elements, then there i s an Engel function for the algebra. " We w i l l assume throughout the rest of the chapter that A- i s f i n i t e dimensional and contains a unity element 1 , and that the charact-e r i s t i c of F i s d i f f e r e n t than 2 . We define \ \ (1) a ( x 1 , x 2 , x 5 ) = K x y X p X g + x ^ - x ^ - x 2 x 5 - x 1 - x ^ x ^ } and observe that a(x n,x 0,x_ J i s a l i n e a r homogeneous element i n . 3 the free non-associative algebra on the generators, x^, x 2, x^ over F . We'begin our study with a test f o r the a-nilpotence of an alt e r n a t i v e algebra. . ' Lemma 2 . 3 . 1 I f F. i s a l g e b r a i c a l l y closed, then A i s a-nilpotent i f and only i f A i s a direc t sum of almost n i l i d e a l s . Proof: Assume f i r s t that A i s a-nilpotent and 2 . 1 . 3 i s the Pierce decomposition of A where the idempotents e^ are primi t i v e . By Lemma 2 . 1 . 8 , the are almost n i l , and from Theorem 2 . 1 . 4 , A i i A j j = A j j A i i ~ 0 ' 1 ^ J* * W e c l a i m A j _ j = 0 w h e n 1 =j= J • Indeed i f x^ . e A i . , i 4= j t then 54. - i x Consequently, a k ( x i j , e ^ e y - • • t ) = ( % ) k x i j • Since A i s a-nilpotent, for some k' , a k (x±ye±ey">e±ej) ~ (ia) k x j _ j = 0 > hence x.^  . = 0 . Thus A^j = 0 , and i t follows A = S © A ^ . Conversely, suppose A = 2 <Q B^ where the B^ are almost n i l . By the linearity of a(x^,x2,x^) , i t i s sufficient to consider a(x 1,x 2,x^) on one B^ . Since B^ = Ff^ + where f^ Is\a primitive idempotent, for b^,c^ € B^ , we compute a(f i , b i , c i ) = a(b i , f i , c i ) = a(b j L,c i,f i) = 0 . Again by the lineari t y of a(x 1,x 2,x^) , we see i t i s sufficient to consider a(x^,Xg,x^) acting on . Since i s nilpotent by Theorem 2.1.5, i t follows that Is a-nilpotent, hence A i s a-nilpotent. Q.E.D. For the alternative algebra, A ,. the maps' S intro-duced in §1.3 become xS(b,c) = a(x,b,c) for a l l b,c e A . We can now show that a(x^,x2,x^) satisfies the f i r s t condition to be an Engel function. [See 1.3. We also note that alter-native algebras form an E-class over F]. Lemma 2.3-2 Suppose dim A = n and F has at least 2n+l elements. Then A is a-nilpotent i f and only i f A i s a-nil. 5 5 . Proof: Since F has at least 2n+l elements, by Lemma 1 .3.2, i t i s sufficient to prove the result when P is algebraically closed. Clearly i f A i s a-nilpotent,. then S(b,b) n = 0 for a l l b € B and thus A i s a-nil. Conversely, suppose A i s a-nil and ( 2 .1 .3 ) i s the Pierce decomposition of A where the idempotents e^ are primitive. Then the A ^ are almost n i l , and we claim that A^ . = 0 , i f j . Indeed, i f x^. e A^ . , i 4= 3 > w e compute x i JS(e i,e i) = -(i)x±. Then ^ j S C 6 ! * 6 ! ^ = ( " ^ ^ i j »• Since S ( e i , e i ) n = 0 , i t follows that x^ . = 0 . Thus A i s a direct sum of almost n i l algebras, and by Lemma 2 . 3 . 1 , A i s a-nilpotent. Q.E.D. Recall that i f R i s a subalgebra of A , by L ^ ( R ) 'we mean the Lie algebra of linear transformations on A generated by S(b,c) , b,c,e R . Furthermore, B^ = {x € A _ : xS(b,b) n = 0) where dim A = n . We have Theorem 2.3*3 Suppose R i s an a-nilpotent subalgebra of A containing 1 . Then LA(R.). is nilpotent, and i f A = A Q C^ A-^  is the F i t t i n g decomposition of A relative to L A(R) > then (i) A Q i s a subalgebra of A containing R ( i i ) A Q ^ C ^ A 1 A Q C A 1 Moreover, If F i s algebraically closed, 5 6 . ( i i i ) A Q = n{Bb : b e R} Proof: By the remarks i n §1 . 4 , we see that we may assume F i s al g e b r a i c a l l y closed. Since 1 e R , there i s a set of pairwise s orthogonal p r i m i t i v e idempotents e n,...,e such that R = E fiP R. i s i = l 1 s where R± = Fe± + N± . Then N = S c© N i i s the r a d i c a l of R . Let C be the subalgebra of Hom1i1(A,A) generated by R^, L^, b € R . By Corollary 2.2.8, Rc, L c , c e N generate a nilpotent i d e a l i n C . Therefore, i f z e N , b e R i t follows that S(z,b) - z(~Rz\ + L z R b - L 2 D - \ z ) i s i n t n i s nilpotent i d e a l . S i m i l a r l y S(b,z) i s i n this i d e a l . Thus S(z,b) and . * - * S(b,z) are i n the r a d i c a l N of C . I f (2.1 .3) i s the Pierce decomposition of A r e l a t i v e to e^,...,e g , we see that R^ c A ^ , i =l,.. . , s . Further-more, by Theorem 2.1.4, we see that A^j i s invariant r e l a t i v e to R^, L b,,b € R . Therefore, to show L A(R) i s nilpotent, we claim i t Is s u f f i c i e n t to show that f o r a l l i and j , the r e s t r i c t i o n S(b,c)"^ of S(b,c) , b,c € R to A^ . generates a nilpotent L i e algebra of l i n e a r transformations on A. . . For (2.I . 3 ) i s a vector space dire c t sum of A , hence there i s a basis f o r A r e l a t i v e to which the matrix f o r S(b,c) i s block diagonal with each block representing S(b,c) i j* f o r some i and j . When the L i e product of S(b,c) and S(d,e) , b,c,d,e e R , i s considered, we see the product i s determined by the L i e product on the i n d i v i d u a l blocks. Therefore, .• 5 7 . L A ( R ) - E©L„ (R) . . i d - • where by (R) we mean the L i e a lgebra of l i n e a r t r a n s -format ions on A. . generated by S ( b , c ) 1 ^ , b , c e R . Con-J_tJ sequen t ly , i f L ^ (R) i s n i l p o t e n t f o r each i and j , i t f o l l o w s that L ^ ( R ) . i s n i l p o t e n t . I f b , c e R •, we w r i t e b = E ( 8 k e k + Z ] c ) 6 k , Y k e F \ • c = s < Y k e k + V V w k 6 N U s i n g the l i n e a r i t y of S ( b , c ) , we compute S ( b , c ) = S ( E ( B k e k + z k ) , S ( y k e k + w k)) = k ^ W ( e k e x ) + T . where, by the p rev ious remarks, T e N * .. Thus ( 2 ) S ( b , c ) l j = S k Y 1 S ( e k , e l ) i j + T i J ' . Suppose x . . e A . . . Then we compute i J i j x i j S ( e k , e ^ ) 1 J ='px i- where . p i s a s c a l a r . Prom ( 2 ) , we have. S ( b , c ) i j ' = T i j ' where \x. . € P and I i s the i d e n t i t y t r ans fo rmat ion of A . S ince T 1 J e N * , i t i s c l e a r that S ( b , c ) 1 ^ generate a n i l p o t e n t L i e a lgeb ra of " l i n e a r t ransformat ions on A . ., hence L f l ( R ) i s n i l p o t e n t . 1J ii .58. From Theorem 2.1.4, we readily compute that S(e k,e 1,) i i = 0 for a l l i,k, and 1. By (2) we have S(b,c) i j- = T 1 1 e N* . Since N* i s the radical of C , i t follows that each element of L A(R) acts nilpotently on . Therefore R c S A 1 ± c A Q . If b e R , then S(b,b) e L A(R) • By definition of A Q , i f x e .AQ then xS(b,b) n = 0 where dim A = n . Since e-,....,e are elements of R , we have I s R £ 2 A i ± £ A 0 n {Bb : be R} c n B 1=1 1 Suppose x e n B . By (2 . 1 . 3 ) , we write i=l e i s S i=l X — £ X • • H~ S X . . A 1 1 i+j i j We compute xS(e k,e k) = 2 ^ ( e ^ e * ) + ^ x i j S ( e k ' e k > • Now x j _ i S ( e k ^ \ ) = 0 f o r a l l i r x i j s ( V e k > = \ " K j o - i x 8 X i k For ±. 4= J , we compute i , j 4= k i = k 4= j 1 4= J = k 5.9. Thus xS(e,,e k) = (-£) Z x . . .. K . • i?k=j=j 1 J But x e B implies xS(e, , e v ) n = (~|) n S x. . = 0 . Since .. e k . K- i=k=f=j . 1 J ( 2 . 1 . 3 ) i s di r e c t , i t follows that x^ . . = 0 , k tj= j and x i k = 0, i 4 k . Since x e B for a l l k , i t follows that x = I] x. ., ek 1 1 s which implies fl B c £ A. . . Consequently, i = l e i 1 1 s I = n{a : b e R } = n B I I D i = l e j R C A 0 = S A i . „ , , ^ • Since S A . . i s invariant under L. (R) and since A = A A © £ A . . , we have, by Lemma 1 . 4 . 1 , that A N = S A . . . 0 ± + J i j 1 ± + J- i j That A Q i s a subalgebra of A and A Q A ^ c A ^ and A - ^ A Q C A ^ follow immediately from the properties of the Pierce decomposition. Q.E.D. Recall that we wish to show that a(x^,x^,x^) i s an Engel function f o r A . We have yet to show that f o r b e A , B^ i s a subalgebra of A containing b . Since a(b,b,b) = 0 implies bS(b,b) = 0 , i t is'immediate that b e B^ . Using Theorem 2 . 3 . 3 , we can now shpw Lemma 2 . 3 . 4 For any b e A , B^ i s a subalgebra of A . Proof: Prom (P -2) and (2) of § 2 . 1 we have the following . i d e n t i t y for a l l x,y,z e A : 60. (3) [xy,z] = x[y,z] + [x,z]y + 3(x,y,z) . Interchanging the x and y , and subtracting gives (4) 6(x,y,z) = [xy,z] - x[y,z] - [x,z]y - [yx,z] +' y[x,z] + [y,z]x .= [[x,y],z] + [[y,z],x] + [[z,x],y] . Now l e t x,b e A . By Theorem 2.1.1, x and b generate, an associative subalgebra of A , and hence, from (4) we compute (5) xS(b i,b J) = Kb^xb1 + b^b 0' - b V x - xb^'b1) = -K-b^b0' - b^'xb1 + xb^'b1 + bVx) = - I t t x ^ 1 ] ^ ] . Prom (4) we compute [ [xVhb 1 ] + .[[b J,b i],x] + [[b^x],^] =6(x , b V ) and hence [[x,b^"],bJ] = [[x,b^'],b^] . Consequently (6) S(b i,b J) = S(bJ",b1) ; Now suppose i >_ 2 . Then xS(b i,b J)-= ^ [ [ x ^ 1 ] , ^ ] = *[ [b i _ 1 ,b ,x] ,b J ] = *Ub l T - 1 [b ,x] ,b J ] +-[[b i"1,x]b,bJ]};by (3) - ^ ^ [ [ ^ x l ^ b J ] + [[b 1" 1^],^^]^} by (3 ) 61. = -\ b i - 1 [ [ x , b ] , b J ] - fcUx.b1-1],!)^ = x{S(b,b J)L ^ + S ^ 1 - 1 ^ ' ^ } . We now have (7) S(b i,hJ) = S(b,b J)L ^ + S ^ 1 - 1 ^ ) ! ^ i > 2 . . We observe that when the exponent i Is lowered i n (7 ) that j t ' t remains unchanged. Also, since R t = (R^) and L t = (L^) we may apply (6) and (7 ) to obtain \ ; (8) S(b I,b J) = S(b,b)cp(R b,L b) • • • i where cp(u,v) i s a polynomial i n u and v . Next we compute xR bS(b i,b J) = (xb)S(b i,b J) = - i t t x b . b 1 ] ^ ] = -|[[x,b i]b,bJ] = - i [ [ x , b i ] , b J ] b = xS(b i,b J')R b . Consequently, (9) R bS(b i,b J) = S C b 1 , ^ ) ! ^ and s i m i l a r l y (10) J^SCb 1,^) = S ( b i , b J ) L b From (9) and (10) we see that S(b"*",b^ ') commutes with any polynomial i n Rb and L D . From (8) i t follows 62. (11) S(b i,b J') k = S(b.,b)cp'(Rb,Lb) where cp'(u,v) i s a polynomial i n u and v . We now define' B*_ = {x € A : xS(b i,b J') n = 0 , i , j = 0,...,n} where dim A = n . From (11) we have immed-i a t e l y ' that B£ = . Now l e t F[b] be the subalgebra of A generated by b and 1 . Since F[b] i s commutative and associative, i t i s a-nilpotent. Therefore, by Theorem 2 . 3 . 3 , L A ( F [ b ] ) i s n i l -potent and the F i t t i n g n u l l component A Q of A r e l a t i v e to L.(F[b]) i s a subalgebra containing F[b] . But elements of L A ( F [ b ] ) are sums of products of S(b ,bJ) . Hence i t follows that A Q = B £ *= B^ and • B^ i s a subalgebra of A . Q.E.D. Summarizing our re s u l t s , we have Theorem 2 . 3 . 5 Let OC be the class of alte r n a t i v e algebras such that ' ( i ) i f A e <K, then A has a unity ( i i ) i f A eOl , dim A - n and F i s the ground f i e l d , then F has at l e a s t 2n+l elements. Then a.(,x^,,x^) i s an Engel function for Gt . As a corollary, we give Corollary 2 . 3 . 6 I f A i s r e s t r i c t e d as i n Theorem 2 . 3 * 5 and every maximal subalgebra of A i s an i d e a l , then A i s 63. a-nilpotent. Proof: The proof i s immediate from Corollary 1.3.5. Q.E.D. 2.4 Cartan Subalgebras of Alternative Algebras We w i l l now assume A i s an n-dimensional a l t e r n a t i v e algebra with unity 1 where the ground f i e l d F has at l e a s t N elements, where N i s the maximum of 2n+l and 2 n~^ . We note, therefore, that the results of 2.3 are v a l i d . In this section we w i l l show that the class of alt e r n a t i v e algebras described above have Cartan subalgebras [§1.4]. As an immediate c o r o l l a r y of Theorem 2.3.3, we give Lemma 2.4.1. Suppose F i s a l g e b r a i c a l l y closed. Then H i s a Cartan subalgebra of A i f and only i f there i s a set e-,,...,e of pairwise orthogonal primitive Idempotents whose sum A. , S S -i s 1 and H = E A. . . 1=1 1 1 From Lemma 1.5.4, we see that A contains a-regular elements, hence by Lemma 1.5-3, A contains minimal Engel sub-algebras. Furthermore," by Lemma 1.5-5, i f A' i s a subalgebra of A and i s minimal i n {B c : c € A'} where A' c B^ , then Bfe £ B Q f o r a l l c e A' . We can now prove 6 4 . Theorem 2 . 4 . 2 H Is a Cartan subalgebra of A i f and only i f H Is minimal Engel i n A . Proof: Suppose H = B^ i s minimal Engel i n A . Clearly 1 € . Now B^ i s minimal In {Bc : c e H} and since H c j B^ c B h f o r a l l h e H . Thus S(h,h) i s nilpotent on H f o r a l l h e H . I t follows from Lemma 2 . 3 - 2 that H i s a-nilpotent. By Theorem 2 . 3 - 3 , i s n i l P o t e n t and i f A Q i s the P i t t i n g h u l l component of A r e l a t i v e to » then B^ = H c A Q . However, A Q £ n{B h : h e H] c ^ since S(h,h) e L A(H)^ f o r h e H. . Thus AQ = and B^ i s a Cartan subalgebra. i Conversely, suppose H i s a Cartan subalgebra of A and B^ i s minimal with respect to dimension i n {B^:.h e H} . We claim H = B^ . Since H i s a-nilpotent, we have H c B^ , consequently ^ c B h f o r a l l h e H . We now extend P to i t s algebraic closure K and make the following observations: ( i i ) H K Is a Cartan subalgebra of A ^ . [ § 1 . 4 ] . Consequently from Theorem 2 . 3 - 3 , Hg = n{B h /: h' € Hg} . Since \ £ B h f o r a 1 1 h e H , S(h,h) i s nilpotent on B^ f o r a l l h e H . By Lemma 1 . 3 - 3 , we have that . S(h',h') acts n i l p o t e n t l y on ( B b ) K for a l l h' € Hg . Therefore ( B ^ g c Hg , and by (i ) H K = • But dim Hg = dim(B b) K implies dim H = .dim B^. Since H £ Bfe , i t follows H = B^ as required. Q.E.D. 65. We have as a c o r o l l a r y Corollary 2.4.3 A contains Cartan subalgebras. Proof: The proof i s immediate from the theorem and the remarks preceding Lemma 2.4.1. Q.E.D. 2.5 Properties of Cartan Subalgebras The problem we wish to study i s the one concerning the conjugacy of Cartan subalgebras. Barnes [8] gives a successful proof f o r solvable L i e algebras. We f e e l that a similar r e s u l t could be obtained f o r a-solvable alternative algebras, but we have not been successful. In any event, such a proof appears to "4 require the following two lemmas. Lemma 2.5-1 Suppose H i s a Cartan subalgebra of A and I Is an i d e a l of A . Then (H+l)/I i s a- Cartan subalgebra of A/I . Proof: For extensions of K of P we have (A/l)^. = ^ K/l^ and ( ( H + I ) / I ) K = ( H K + I K ) / I K • Consequently (H+l)/I i s . a ^  Cartan subalgebra of A/I i f and only i f ((H+l)/l)^. i s a Cartan subalgebra of (A/l)^. . I t follows we may assume P i s al g e b r a i c a l l y closed. Since (H+l)/l = H/HT1I , we have that' (H+I)/I i s a-nilpotent. Thus by Theorem 2.3-3, L. / T ((H+L)/Ij = t. i s . -66. nilpotent and i f kQ i s the P i t t i n g n u l l component of A/I r e l a t i v e to t 3 then (H+I)/I £ . By Lemma 2.4.1 there i s a set of pri m i t i v e idempotents t e.,,...,e, such that H = E A., where 1 = e n+. ..+e, . Arrange the e i so that e^,...3e^i $ I and e t ' - • • • • > e t e I t follows (e-j^+l)-!-.. . +(e t/+l) = 1., where 1 i s the unity of t A/I . Since K A , i f e^ e I , E A ± J. £ j=l I Let e^ = e^+I } i = l - , - . . . , f . Using the argument given i n Lambeck [ 2 0 ] , we see that we can l i f t the e^ to idempotents i n A, and i t follows that the e^ are pairwise orthogonal p r i m i t i v e idempotents i n A/I . Let A/I = E A. . be the Bierce "decomposition of A/I r e l a t i v e to the e^ , and write BA, = {x e A/I : x S ( ^ i , e i ) m = 0 } . By Theorem 2 . 3 - 3 , A fc/ we have (H+l)/I c A n c n B A . As i n the proof of Theorem 2 . 3 - 3 ' ~ u 1 e i i f we choose x e fl B A and write x = "E x. . + E x.„ we can e. i i . . . 1 + j i5 show x^j =. 0 for i 4= j , and i t follows that A Q = (H+L)/l . Therefore (H+I)/I i s a Cartan subalgebra of A/I . Q.E.D. Lemma 2 . 5 - 2 Suppose I. i s an i d e a l of A ,. <I i s a subalgebra of A containing 1 and such that I £ J £ A , and J/I i s a Cartan subalgebra of A/I . I f H i s a Cartan subalgebra of J , then H i s a Cartan subalgebra of A . Proof: Since H i s a Cartan subalgebra of J . H i s a-nilpotent 67. and contains 1 . Let A Q and A Q be the P i t t i n g n u l l compon-ents of A and A / I respectively r e l a t i v e to L A(H) and L A y j ((H+l)/l) . Since (H+l)/I i s a Cartan subalgebra of J/I' and since J/I i s i t s e l f a-nilpotent, i t follows that (H+I)/I = J/I , hence H+I = J and iQ = J/I . I f x+I e A Q then x e J . But H i s a Cartan subalgebra of J and i t follows that x e H . , Consequently, A Q = H and H i s a Cartan subalgebra of A . - Q.E.D. With respect to the conjugacy problem, we must be content with "s Theorem 2.5-3 Suppose P i s a l g e b r a i c a l l y closed and of ch a r a c t e r i s t i c zero. I f and E^ are two Cartan sub-algebras of A , then there i s an inner automorphism s e 1(A) such that ' Bp = H^s. Proof: In §1.6, we defined the group 1(A) of inner auto-morphisms of A . Prom (5) i n §2.1, we see that D(b,c) = [ L b , L c ] + [L b,R c] +[R b,R c] i s i n the Ideal D'(A) of inner derivations of the derivation algebra D(A) of A . Hence, to apply Theorem 1.6.9, we must show that H^ and E^ each -contain r e l a t i v e l y open subsets 0-^  and/' 0^ such that T(fi^ : a) = A for a l l a e 0^ , 1 = 1,2 , where 0^ - i s the orb i t of.' E± under 1(A) . t By Lemma 2.4.1, we "write H, = S A., where . i = l ± x e-j_,..-,et are pr i m i t i v e idempotents i n A^^ = Pe^ + . Let 68. 0, = {a e R, : a = T, a. e.+z. a. e F , z. € N. , and r (a .-a.) 4=0 3 i X' i< i 1 i i i i=j=j I f f(x, ,...,x, ) = TT (x.-x.) , i t follows f :(:xn ,.. .. ,x, ) -... 1 • ±h J- • • f e F[H^1 and 0^ i s an open subset of . Let ( 2 . 1 . 3 ) 'be the Pierce decomposition of A r e l -ative to e 1,...,e t . For b ^ e A ^ , k 4 = I , and a = E(a^e^ + z^) e 0^ , we compute, using Theorem 2 . 1 . 4 . • aD< V V - (ak-a^)'Dk^ + \ \ i " \ i z t ' Since D( ek> b}^) € D ' ( A ) > a D ( ^ ^ i ^ ) € T ( f i l > a ) D y t h e r e m a r l c s preceding Lemma 1 . 6 . 8 . Define Sk^ = <ak " + L z k * % where I i s the i d e n t i t y transformation on A . Then the above computations show k ^ S ^ € T(n^;a) . Now S^^ maps Ak£ i n f c o i ; t s e l f s i n c e AfciAa e ^ 8 1 1 ( 1 VtAw^ \ l ' From § 2 . 2 , L and R are nilpotent, hence L - R . i s zk z£ zk. z l nilpotent, and since a^. - a^ 4 = 0 , i t follows that S ^ i s i n v e r t i b l e . Therefore A k ^ £ T ( n i ; a ) for k ±'l . By Lemma 1 . 6 . 7 , we have Hp £ T(0p;a) , hence T ( f i 1;a) = A . Thus Hp and Op s a t i s f y the hypothesis of Lemma 1 . 6 . 8 . S i m i l a r l y there i s a r e l a t i v e l y open subset 0^ of H^ s a t i s f y i n g the hypothesis of Lemma 1 . 6 . 8 . The proof i s now immediate by Theorem 1 .6 .9. Q.E.D. CHAPTER THREE JORDAN ALGEBRAS 3 .1 Introduction An algebra J over a f i e l d F i s c a l l e d a (commutative) Jordan algebra i f and only i f [x,y] = (x-,y,x) = 0 for a l l elements x and y of J . Prom the d e f i n i t i o n , i t i s clear that homomorphic images of Jordan algebras are Jordan algebras and a dir e c t sum of Jordan algebras i s a Jordan algebra. Since the commutator and associator are m u l t i l i n e a r , i t follows that j i s a Jordan algebra for a l l extensions K of F . Through-out Chapter Three, we w i l l assume our Jordan algebras are f i n i t e dimensional. In this section, we w i l l give the necessary theory of Jordan algebras to develop a Cartan theory f o r these algebras. In what follows, where proofs are not provided, the reader i s referred to Albert [ 2 ] , o We begin by l i n e a r i z i n g (x ,y,x) = 0 and obtain (1) J(w,x,y,z) = (wx,y,z) + (xz,y,w) + (zw,y,x) = 0 Then computing 0 = J(x,y,w,z) ^ J(x,y,z,w) , we have (2) D(w,x,y,z) = (w,xy,z) - x(w,y,z) - (w,x,z)y = 0 . As a consequence of ( 2 ) , we have that the map 70 . (3) xD(a,b) = (a,x,b) x e J f o r a l l a,b 6 J i s a derivation of J . In terms of right m u l t i p l i c a t i o n R , (3) implies that [^a**^ ^ 3 a derivation °^ J > a n ^ further-more, by ( 2 ) , we have [R y, [R W,R Z]] = R ( w ^ z ) • Hence, the L i e m u l t i p l i c a t i o n algebra L(J) of J" i s R(J) + [R(<l) ,R(J) ] where R(J) = {R : a e J} . I f D i s an a r b i t r a r y derivation of J , then f o r x e J , [RX,D] = R ^ . Suppose D i s an inner derivation of J . Then D = R + E[R ,R ] . I f J X A X i has a unity, i t follows that x = 0 , since - ID = 0 . Therefore, fo r Jordan algebras with unity the inner derivations are expressible i n the form E D(b^,c i) by (3) • We next give Theorem 3 -1 .1 Jordan algebras are power associative i n the sense that single elements generate an associative subalgebra. In the course of the proof, we note (4) R ±R . = R .R . i , j = 0 , 1 , . . . a a a a Also, we have Theorem 3 . 1 . 2 The following are equivalent i n J : ( i ) J i s n i l ( i i ) J i s solvable ( i i i ) J i s nilpotent ' 7 1 . Now suppose e i s an idempotent o f J .. . We compute 0 = J ( a , e , e , e ) = 2(ae-e)e - 3ae-e + ae . Consequent ly , a(2R - 3R + R ) = 0 and the t r ans fo rmat ion R s a t i s f i e d the x e e e e po lynomia l 2X"5 - 3 X 2 + X = X ( 2 X - l ) ( X - l ) . Thus the c h a r a c t e r -i s t i c roo ts of R g are among 0 , ^ , and 1 . I t f o l l o w s that J can be w r i t t e n as a v e c t o r space d i r e c t sum N J = J n ( e ) + J i ( e ) + .J - (e) where J ( e ) = [x e J : xe = Xx] ^ = and 1 . I f x e J , then we w r i t e x = (2e-ex - ex) + (4ex -~ 4e-ex) + (x + 2e-ex - 3ex) . I f \ J has a u n i t y and e, , . . . , e are a set of p a i r -\ J . s wise o r thogona l idempotents whose sum i s the u n i t y of J , we get a r e f i n e d decomposi t ion of J as a v e c t o r space d i r e c t sum: (3.1.3) . J = I J , , i < j = l 1 J J . . = ^ ( e ^ ^ J . . = J i ( e . )nJ 4 ( e . ) i ± j . This decomposi t ion i s c a l l e d the P i e r c e decomposi t ion of J • r e l a t i v e to e, , . . . , e . We have 1 ' 3 Theorem 3 .1 .4 I f (3 .1-3) i s the P i e r c e decomposi t ion of J, r e l a t i v e to e ^ , . . . , e g , then (i) j f l £ j u J u J M r 0 1 + j (ii) J y J y C J y Jfj £ J ± 1 + J J J i + d ~ ( i i i ) J ^ E J i i J i i V = ° JiJJwV° I f i , j , k , < t are d i s t i n c t . i 7 2 . Suppose J is semi-simple. Then J contains a unity. Furthermore, we have Theorem 3 . 1 . 5 A non-zero Jordan algebra is semi-simple i f and only i f J is a direct sum of simple ideals. Thus the study of semi-simple Jordan algebras is reduced to the study of simple algebras. If J is simple and contains a unity, then in.the case when the characteristic of F i s 0 , the corresponding J. . , i 4= J > have a common dimension, necessarily greater than zero. If F is algebraically closed and ( 3 . 1 . 3 ) is the Pierce decomposition of J relative to a set of pairwise orthogonal primitive idempotents e ,...,e , we may argue as in the alter-1 s native case that i f x e J ^ , then -x = ae^+n where a e F and n i s nilpotent. By McKrimmon [ 21 ] , we have -Lemma 3 . 1 . 6 If F Is algebraically closed and ( 3 . 1 . 3 ) i s the Pierce decomposition of' J relative to a set of primitive idem-potents, then the J ^ are almost n i l . 3•2 The Universal Multiplication Envelope of a Jordan Algebra In [ 16 ] , Jacobson proved the analogue for Jordan algebras of Corollary 2 . 2 . 8 . We wil l now sketch the proof. Suppose J is a f i n i t e dimensional Jordan algebra 7 3 . over P . We define inductively J-L = J and J n = J n_i® Ji*' n > 1 • Let T(J) be the associative algebra defined by T(J) = J x © J 2 © . . . and l e t S be the i d e a l of T(J) generated by elements of the form (1) x ® x 2 - x 2 ® X p p (2) x y + 2 x ® y ® x - y ® x - 2x ® xy where x,y e J . The associative algebra U(J) •= T(J)/S i s ca l l e d the universal m u l t i p l i c a t i o n envelope of J . I f l' i s the canonical homomorphism from T(J) into U(J) , then the r e s t r i c t i o n of i ' to J = defines.a linear.map from J into U(j) . We c a l l this map i . and i f a e J , write a i = a where a i s the coset of a i n U(J) . Prom ( l ) and (2) we have the following identies i n U ( j ) : (3) x ® x = x ® x _ _ _ _ p p" (4) 2 x ® y ® x = y ® x + 2 x ® x y x y The analogues of Lemmas 2.2.1 and 2.2.2 now follow, and we state Lemma 3-2.1 I f p i s a l i n e a r map from J into an associative algebra V such that (5) ( x p ) ( x 2 P ) = (x 2p)(xp) 74. (6) 2(xp)(yp)(xp) = (yp)(x 2p) + 2(xp)(xyp) - (x 2y)p then there i s a unique homomorphism p* from U(j) into V such that xip* = xp for a l l x e J . Lemma 3 . 2 . 2 I f K i s an i d e a l of J and D i s the i d e a l i n U(j) generated by KI , then there exists an isomorphism of U(J/K) onto U(J)/D such that x+K i s mapped onto x+D . Furthermore, i t i s known that U(J) i s f i n i t e dimens-ion a l [ l 4 , page 519]-\ We now give Theorem 3 . 2 . 3 . I f I i s a solvable i d e a l of J , then I i generates a nilpotent i d e a l i n U(J) . Proof: The proof i s by induction on the dimension of I , and we assume I =j= 0 . By Penico [23 ] , we know that there i s an ide a l I' of J such that I c I' ^  I ., By the induction hypothesis, I ' i generates a nilpotent i d e a l 1J i n U(J) , and by Lemma 3 . 2 . 2 , U(J/I') = u ( j ) / l ' . Now suppose the image of I / l ' generates a nilpotent i d e a l i n U(J/I /) . Then by the isomorphism noted above, I i w i l l generate a nilpotent i d e a l i n U(J) . Therefore, i t i s s u f f i c i e n t to prove the theorem 2 ? f o r the case when I = 0 , since ( I / l ' ) = 0 . From the i d e n t i t i e s (3) and (4) we can prove the following results [ l 6 ] : 75. (A) U ( l ) i s n i l p o t e n t (B) I f e i ^ - - * ^ e m i s a . D a s i s f o r I a n d e2^> • • * > e m * • ' ' ' e n i s a b a s i s f o r J , and k i s a p o s i t i v e i n t e g e r , then any monomial i n U ( j ) of the form e. . . . e . J l 3>l i n which k+m of the j 1 s are i n the range Q = {1,2,...,m} i s a l i n e a r combinat ion of monomials of the form e. . . . e. . . . where i , , . . . , i v e Q . xl x k 1 ' K -From (A) i t f o l l o w s that i f I* i s the subalgebra of U(J ) generated by I i , then I* i s n i l p o t e n t . Suppose (I*)P t= 0 . From (B) we have that any product of elements a U(J) which i n c l u d e p+m elements of I i s z e ro . .. Consequent ly , i f I i s the i d e a l i n U(J ) generated by I * , ( i ) p + m = 0 . Q .E .D . C o r o l l a r y 5 . 2 . 4 I f i s a subalgebra of ..J , I i s a s o l v a b l e i d e a l of J-^, C i s the • subalgebra of Hom F ( J , J ) generated by R a , a e and C i s the i d e a l of G- generated by , b e i , then C i s n i l p o t e n t . P roo f : The map p : a -» R a i s a l i n e a r map from J-j- i n t o C . In d e r i v i n g ( l ) of 5 -1 , we note tha t p s a t i s f i e d the hypothes is of Lemma 5 . 2 . 1 . Consequently the diagram U ( J ; L ) /K v." A " - P* 1 1 K . 7 6 . i s commutative where i p i s the n a t u r a l map from Jp i n t o U(Jp) • I t f o l l o w s that p* i s onto C , and by Theorem 3 . 2 . 3 , as l i p generates a n i l p o t e n t i d e a l i n U(Jp) , C i s n i l p o t e n t . Q . E . D . 3.3 Car tan Subalgebras. of Jordan Algebras We w i l l now apply the theory developed i n Chapter One to Jordan a lgeb ra s . Since the development i s v i r t u a l l y i d e n t i c a l w i t h the a l t e r n a t i v e case , we w i l l f u r n i s h proofs o n l y when the method does not f o l l o w by an obvious m o d i f i c a t i o n of the a l t e r -n a t i v e p r o o f . In t h i s s e c t i o n , J w i l l denote an n -d imens iona l Jordan a l g e b r a w i t h u n i t y 1 where the ground f i e l d P has a t l e a s t N elements, where N i s the maximum of 2 n ~ and 2n.+l . We def ine (1) A ( x p , x 2 , x ) = x x x 2 - x ^ - X p - X p X ^ and observe that A ( x p , x 2 , x ^ ) Is a l i n e a r homogeneous element i n the f ree n o n - a s s o c i a t i v e a lgebras on the generators Xp , x 2 and x-j over P . We a l s o note that the maps S in t roduced i n § 1 . 3 become x S ( b , c ) '= A ( x , b , c ) f o r a l l b , c € J . We now g ive Lemma 3 . 3 . 1 I f F i s a l g e b r a i c a l l y c l o s e d , then J i s A--n i l p o t e n t i f and on ly i f J i s a d i r e c t sum of almost n i l 77 . algebras. Lemma 5 . 5 . 2 (Engel) J i s A-nilpotent i f and only i f J i s A - n i l . Theorem 5 . 5 . 5 Suppose R i s an A nilpotent subalgebra of J containing 1 . Then. Lj(R) i s nilpotent, and i f J = J Q ® i s the P i t t i n g decomposition of J r e l a t i v e to > (i ) J Q i s a subalgebra of J containing R ( i i ) J Q J ^ C J X Moreover, i f P i s a l g e b r a i c a l l y closed, ' ( i i i ) J Q = n{Bb : b e R} C o r o l l a r y - 3 . 3 . 4 I f P i s a l g e b r a i c a l l y closed, then H i s a Cartan subalgebra of J i f and only i f there i s a set e^,...,e t of pairwise orthogonal p r i m i t i v e idempotents whose- sum i s 1 and t ' H = s U., . 1=1 1 1 . We can now prove Lemma 3 . 5 . 5 Por a l l a e J , B i s a subalgebra of J . ' a Proof: The proof i s patterned a f t e r the proof of Lemma 2 . 3 - 4 . By (4) In § 3 - 1 - we have that R i and R . commute a a J f o r a l l i and j , hence we have (2) S ( a \ a J ) = SCa^a 1) i , j = 0,1,... I—? i Computing D(x,a ,a,a d) for i >_ 2 gives (3) S(a i,a J') = S(a,a J)R ± _ ± + S ( a 1 " 1 , a J )R a . • a -From (P-l) i n the Introduction, we compute ( a k x , a \ a j ) - (a k,xa i,a j) + ( a k , x , a i + J ) - a k(x,a i,a J) - (a k,x,a i)a J* = 0 . From (4) i n §3-1, we have ( a ^ x a ^ a 0 ) = (a^x^a 1"^) = (a k,x,a 1)a J' = 0 . Hence (xa k,a i,a J') - \(x,a i, a? ) a k = 0 and we have (4) R ^ ( a ^ a * 1 ) = S(a 1,a J)R fc i , J , k >_ 0 . ' a a By an easy induction proof on i , i >_ 2 , we can prove (5) S(a 1,a k) = S(a,a k) cp± where i-2 ^ i = . E R i - j R a " J + 2 R a _ 1 1 > 2> Ra = 1 > and i f j =0 a i = 2 , cp. = 2R a . From (-2) we have that S'(a,ak) = S(a,a)cp k . Therefore, ( 6 ) S(a 1,a k) S(a,a)cpicpk . By ( 4 ) , cp. and cp, commute with S(a^,a^) , hence ( 7 ) S ( a i , a k ) m = S(a,a) m(cp icp k) m . I f B* = {x e J : x S ( a i , a J ) n = 0 , i , j . = 0,...,n] then from ( 7 ) we have B a = B* . 7 9 . The rest of the proof i s the same as the conclusion of the proof of Lemma' 2.3.4. Q.E.D. Summarizing the above r e s u l t s , we have Theorem 5 - 5 . 6 Let be the class, of Jordan algebras such that ( i ) i f J e ^ ., then J contains a unity ( i i ) i f J e °3 , d im J = n , and P i s ' the ground f i e l d , thai P has at l e a s t 2n+l elements. Then* A ( x ^ , X g , x ^ ) i s an Engel function f o r t£ . i Applying Lemmas 1 . 5 . 5 , 1 .5-4, 1 - 5 - 5 and following the proof of Theorem 2.4.2, we have ' • Theorem 5-5-7. H Is a Cartan subalgebra of J i f and only i f _ H i s minimal Engel i n J . Corollary 5 -5 -8 J contains Cartan subalgebras. We remark that we have the counterparts of Lemmas 2 .5-1 and 2.5-2 for Jordan algebras. As i n the alt e r n a t i v e case, our attempt to follow Barnes [8] In obtaining conjugacy results was not successful. Again, we must be content with Theorem 5 - 5 . 9 Suppose P i s a l g e b r a i c a l l y closed and of ch a r a c t e r i s t i c zero. I f H-i and H p are two Cartan subalgebras 8 0 . of J , then there i s an inner automorphism s e l ( j ) of J such that ^ s = H 2 . Proof: Using D(b,c) = [R^R ] as our inner derivations, the proof proceeds i d e n t i c a l l y with the alte r n a t i v e case. • Q.E.D. Remark: I t should be noted that i f A i s an al t e r n a t i v e algebra, then we can make A into a Jordan algebra A + by defining a new product a»b of A by the equation a«b =• i(-ab + ba) . Using A (Xp,x 2,x^) i n A + , we might expect to develop a Cartan theory for a l t e r n a t i v e algebras. We compute (x^x^) -X 1 ° ( X 2 ° X 3 ^ = 2 a ( x i ^ x 2 , x 3 ^ + 2 A( xi> x2 > y Lj>} ' Therefore, there i s no reason to expect that the Cartan theory f o r A developed i n + Chapter Two arises from i t e r a t i n g the associator i n A Remark: Using the defining i d e n t i t i e s , we f i n d that the Engel functions are used f o r alte r n a t i v e and Jordan algebras are i d e n t i c a l . In the case of L i e algebras, a(x^,x 2,x^) = -x^xyx^ . By v i r t u e of Property 1 . 2 . 6 , i t follows that a(x^ , X g,x^) i s an Engel function for L i e algebras and that the Cartan theory developed from this function coincides with the c l a s s i c a l theory. Thus we have the one function that gives the Cartan theory .for. L i e , Jordan and al t e r n a t i v e algebras. 8 l . 3.4 A - s o l v a b l e Jordan Algebras Suppose j ' i s a f i n i t e d imens iona l Jordan a lgeb ra over a f i e l d P where P i s a l g e b r a i c a l l y c l o s e d and of char -a c t e r i s t i c z e r o . I f J ' i s semi-s imple i n the c l a s s i c a l sense, then J ' has a u n i t y l ' and i s e x p r e s s i b l e as a d i r e c t sum of s imple Jordan a lgebras (see § 3 . 1 ) . L e t J be one of these s imple summands. L e t 1 be the u n i t y of J , and l e t S J . . be the P i e r c e decomposi t ion of J r e l a t i v e to a set e ^ , . . . , e ^ of p a i r w i s e o r thogona l p r i m i t i v e idempotents whose sum i s 1 . A l b e r t [2] has shown that i f i 4= j , then . ={= 0 and i n f a c t a l l the J . . , i 4= 3 have a common dimension 6 . Fur thermore, -<- j each J . . , i | j j has an or thonormal b a s i s , tha t i s , a b a s i s x ^ , . . . , x g such that ' x^x^ = 0 i s k 4= ^ > and x^. = e^ + e. [ 2 ] . Of course , i f t = 1 , then J = PI .. There fore , we w i l l l e t t > 1 . Consider J . . , i 4 j , and l e t x . . be an element i n p ' an orthonormal b a s i s f o r J . . . Consequent ly , x . . = e. + e . . L e t y = e. - e . . ¥ e compute ( x l t J , y , y ) = ( x ^ e . - e ^ e . - e . ) = - x . . (x. . , x ± . , y ) = ( x i . , x . J , e . - e . ) = y . Suppose p i s an a r b i t r a r y p o s i t i v e i n t e g e r . From the above computations we see we may cons t ruc t a 3^ - tup l e ( z 1 , . . . , z ) .. 1 3P"-82. where the z. are either x. . or y such that ' ( Z l , • - • , z ) i I J i. -^p = x.. . Si m i l a r l y , we may construct a 3 P - t y p l e (w,,...,w ) 1,3 yp. such that A ^ (w-,,. . . ,w ) = y . It follows that J i s not ~ 3 P A solvable, hence J' i s not A-solvable. We have shown. Lemma 3.4 .1 Let F be ah a l g e b r a i c a l l y closed f i e l d of ch a r a c t e r i s t i c zero and J a Jordan algebra over F . Then J i s semi-simple with respect to A - s o l v a b i l i t y i f and only i f J i s a di r e c t sum of simple Jordan algebras, each ofie of which has degree greater than one. Proof: I f J i s semi-simple with respect to A - s o l v a b i l i t y , by Property/-1.2 .5 , J i s semi-simple i n the c l a s s i c a l sense. Consequently * J i s a di r e c t sum of simple algebras, and the above ca l c u l a t i o n shows that at l e a s t one of these must have degree greater than one (since simple Jordan algebras-of degree 1 over F are ass o c i a t i v e ) . The converse i s immediate. . Q.E.D. As a co r o l l a r y , we have Corollary 3 -4 .2 Let F and J be as i n Lemma 3 - 4 . 1 . I f - J i s A--solvable, and semi-simple i n the c l a s s i c a l sense, then J i s a direc t sum of copies of F . In what follows J w i l l denote an n-dimensional Jordan algebra with unity 1 over a f i e l d F where F i s of character-i s t i c zero. From (~), § 3 - 1 , we know that D(b,c) = [R^,^] i s 83-a derivation of J f o r a l l b,c e J . I f D(b,c) p = 0 for 0 0 1 k some p , then exp D(b.c) = E T - . D(b,c) i s an automorphism k=0 K 1 . of J . [18].. Albert [2] has shown that i f K i s an extension of F, since char F = 0 , Jp is- semi-simple i n the c l a s s i c a l sense i f and only i f J^- i s semi-simple i n the c l a s s i c a l sense. ' Since k P(b,c) i s m u l t i l i n e a r for a l l k , we see i t i s s u f f i c i e n t to investigate the nilpotence of D(b,c) when P i s a l g e b r a i c a l l y ' closed. We now prove Theorem 5.4 .3 I f j ' i s an A-solvable subalgebra of J , then the derivations D(b,c) , b,c e j ' , of J are nilpotent. Proof: By the remarks preceding the theorem, we see i t i s s u f f i c i e n t to prove the theorem when F i s a l g e b r a i c a l l y closed. ' Let us denote the c l a s s i c a l r a d i c a l of J' by S(J').' Case 1: S(j') = 0 . By Lemma 3-4 . 2 , there i s a set of pairwise orthogonal p r i m i t i v e idempotents e-,,. .., e e j ' such that j ' = Fe-^  Q ... ^ Fe g . Prom the i d e n t i t y J(w,x,y,z) = 0 [ ( 1 ) , § ( 3 . l ) ] - we compute J ( e ± , e ±,x, e..) = (e±,x,e;.) + 2(e ±e j.,x, ) = 0 . I f i =}= j , i t follows that D(e. ,e.) = 0 , - Qf course, D(e i,e i) = 0 . Thus f o r a l l b,c e j ' , D(b,c) = 0 . Case 2: S ( J 7 ) / 0 . By the Wedderburn p r i n c i p a l theorem f o r Jordan algebras [ 23 ] , we have J ; = J 7 ® S(j') where 84. Jp s J / / S ( J / ) • Consequently by C o r o l l a r y 3 . 4 . 2 , Jp = Fe^ ffi . . .ffi,Fe where the e. are p a i r w i s e o r thogona l p r i m i t i v e idem-3 J_ . . . . potents ' whose sum i s the u n i t y of J 7 [20; page 7 2 ] . Thus J ' = Pep $ . . . © F e s © S ( j ' ) . . L e t C be the subalgebra of Horn ( J , J ) generated by R a , a e J ' . Then the maps R^ , b e S ( j ' ) generate a n i l -potent i d e a l N i n C , by C o r o l l a r y 3 - 2 . 4 . Consequent ly , i f a e j ' and b e S ( j ' ) , D(a ,b) = e N * . Prom Case 1 , we see tha t D ( e . , e . ) = 0 f o r a l l i and j ... There-fore i f a ,b \e J and we w r i t e a = E aj_e^ + a]_ . a i , P i e F . b = S + bp a p , b p e S ( J ' ) i t f o l l o w s that D(a ,b) = S D(e f c , e^) + T = T £ N * . Consequent ly , f o r some p , D ( a , b ) p = 0 , and the d e r i v a t i o n s are n i l p o t e n t as d e s i r e d . Q .E .D. As a consequence o f t h i s theorem, we have that i f J i s A - s o l v a b l e , the f o r a l l b , c £ J , D(b , c ) i s n i l p o t e n t . Suppose we l e t I* ( J ) be the group of automorphisms of •' J generated by exp D(b , c ) , b , c e J .. ¥ e have not been ab le to answer the f o l l o w i n g : ( i ) What i s the r e l a t i o n s h i p between I ( J ) and I* ( J ) ? ( i i ) Are Car tan subalgebras o f J conjugate under I* ( J ) ? I f the answer to ( i i ) i s yes , i t may be p o s s i b l e to use the t ech-niques g iven i n Barnes [8] to improve our conjugacy r e s u l t s . -CHAPTER POUR . COMMUTATIVE POWER ASSOCIATIVE ALGEBRAS 4 . 1 Introduction A commutative algebra X over a f i e l d F i s c a l l e d power associative i f and only i f the subalgebra generated by each element of X i s associative. Examples of such algebras are the Jordan algebras studies i n the previous chapter. X i s c a l l e d s t r i c t l y power associative i f and o n l y . i f Xg i s power associative f o r a l l extensions K of F.. In this chapter, we w i l l l e t X denote a f i n i t e dimensional power associative a'lgebra over a f i e l d F where F has at l e a s t four elements and c h a r a c t e r i s t i c d i f f e r e n t than 2 . We note that (x ,x,x) = 0 i s an i d e n t i t y i n X . Since F has.at l e a s t four elements, we'may l i n e a r i z e this i d e n t i t y and obtain P P "5 ( l ) 4wX ' X = 2(wx.x)x + (wx -x) + wx^ and ( 2 ) 4[wx>yz + wy-xz + wz-xy] = (wx-y- + xy-w + yw.x)z + (xy.z + yz.x + zx-y)w + (yz«w + zw-y +'wy-z)x + (zW ' X + wx«z + xz-w)y . We begin our discussion by showing that solvable " commutative power associative algebras are nilpotent [ 6 ] . We 8 6 . define X 1 = X , and inductively X n = X n_ 1 ® X± . Let T(X) be the associative algebra defined T(X) = Xj © X g © ... , where the vector space operations are as usual, and m u l t i p l i c a t i o n i s denoted by ® . Let S be the i d e a l of T(X) generated by elements of the form P P 5^ (3) 4x ® x - 2x ® x ® x - x ® x - x^ (4) 4[x ® yz + y ® xz + z ® xy] - [x ® y + xy + y ® x] ® z - [xy-z + yz-x + zx«y] - [yz + y ® z + z ® y] ® x - { z ® x + x ® z + xz] ® y x,y,z. e X The'associative algebra U(X) = T(X)/S i s c a l l e d the universal  m u l t i p l i c a t i o n envelope of X . I f i ' i s the canonical homo-morphism from T(X) i n t o U(X) then the r e s t r i c t i o n of. i ' to X = X-^ . defines a l i n e a r map from X into U(X) . We c a l l this map i , and i f x e X , we write x i = x where x i s the coset of x i n U(X) . Prom (3) and (4), we have the following i d e n t i t i e s i n .U(X) : (5) 4x ® x 2 = 2x ® x ® x - x 2 ® x •- x? (6) 4[x ® y z + y ® x z + z ® xy] = [x ® y + xy + y ® x] ® z + [xy-z + yz-x + zx.y] + [yz + y ® z .+ z ® y] ® x + [ z ® x + x z + x ® z ] ® y . The analogue of Lemma 2.2.1 now follows, and we state Lemma 4.1.1 Let p be a l i n e a r map from X Into an associative algebra V such that 8 7 . ' ( i ) 4(xp)(xp) = 2(xp) 5 + (x 2p)(xp) + (x^p) ( i i ) 4[(xp)(yz)p + (yp)(xzp) + (zp)(xyp)] = [(xp)(yp) + (xy)p + (yp)(xp)](zp) + [xy-z + yz-x + zx-y]p + [(yz)p + (yp)(zp) + (zp)(yp)](xp) + [(zp)(xp) + (zx)p + (xp)(zp)](yp) . Then there i s a unique homomorphism p* from U(X) into V such that xp* = xp for a l l x e X . We also observe that U(X) i s generated by X i . We now prove Lemma 4.1.2 Let P be an a r b i t r a r y associative algebra over F and M a subalgebra of P such that c M + PM + MP . Then f o r each Integer k >_ 1 , P^ k c M k + PMk + MkP + PMkP . Proof: [b] The proof i s by induction on k , the r e s u l t being obvious f o r k = 1 . In general, we have Fp = Fp P^ P^ <= p^ p^p^ : 3 k Applying the induction hypothesis to P v , we obtain P 5 c M ^ ^ + M V M ^ P + PM^P^ + PMkPdMkP where a,b,c and d are integers greater than 2 . Since P n £ M + PM + MP for n >_ 3 , the above r e l a t i o n implies p 3 k + c M k + 1+PM k + 1+M k + 1P+PM k + 1P, which completes the induction proof. Q.E.D. 88. Lemma 4.1.5 If X i s solvable and the dimension of X i s 1 , then U(X) i s nilpotent. Proof: Since X = Fx where x = 0 , from (5) we obtain x ® x <8> x = 0 . Since x generates U(X) i n this case, we have (U(X)) 5 = 0 . Q.E.D. Lemma 4.1.4 I f X i s solvable, then U(X) Is nilpotent. Proof: The proof i s by induction on the dimension n of X . By Lemma 4.1.3, we assume n > 1 and that U(x') i s nilpotent f o r a l l solvable algebras X' of dimension l e s s than n . 2 c Since X ^ X , there i s an n-1 dimensional subspace p B such that X c B c X , consequently B<» X . Since B i s solvable, U(B) Is nilpotent. Let B* be the subalgebra of U(X) generated by B i , and D = B* +'B* ® U(X) + 'U(X) ® B* . We claim (7) U(X) ® U(X) ® U(X) c D S ince U(X) i s generated by X i and X = B + Fw for some w £ B , to prove ( 7 ) , i t i s s u f f i c i e n t to show x ® y- ® z e D-where x,y,z = B U {w} . If x or z i s i n B , then x ® y ® z £ D . There-fore we may assume x = z = w . From ( 6 ) and the fact that 2 X <= B , we have 2w ® y ® w = -2y ® w ® w - 2 w ® w ® y (mod p) 89. I f y e B , then w ® y ® w e D , and i f y = w , from (5) we have w ® w ® w e D . Consequently (7) i s v a l i d . Consider the diagram U(B) /\ B '• j >B* where j i s the map from B into U(B) and i : X - U(X) . Since i s a t i s f i e s the hypothesis of Lemma 4.1.1, i t follows there i s a homomorphism j * from U(B) into B* such that = b i • T h u s J* i s onto, and i t follows B* i s n i l -potent. From (7) and Lemma 4.1.2, i t follows that U(X) i s nilpotent. Q.E.D. Theorem 4.1.5 If x Is solvable, then X i s nilpotent. Proof: Let M(X) be the subalgebra of HomF(X,X) generated by R , a e X . The l i n e a r map p : X -• M(X) where p(a) = R , a a a e X , i s a l i n e a r map from X into M(X) which,' by ( l ) and (2), s a t i s f i e s the conditions of Lemma 4.1.1. Consequently, u(xk •" 'P i s X ^M(X) i s commutative, and p* Is onto. Therefore M(X) Is nilpotent, and by Schafer [ 2 5 ; p. 18], X Is nilpotent. Q.E.D. 9 0 . C o r o l l a r y 4 .1 .6 I f x i s a n i l p o t e n t element o f X , then R i s n i l p o t e n t . P roof : S ince the subalgebra F[x] of X generated by x i s n i l p o t e n t , U ( F [ x ] ) i s n i l po t en t , . Consequently. M(F[x ] ) Is n i l p o t e n t , and I t f o l l o w s tha t R i s n i l p o t e n t . Q .E .D . We have seen that n i l Jordan a lgebras are s o l v a b l e . However, f o r an a r b i t r a r y commutative power a s s o c i a t i v e a lgebra X , I t i s not knowtfif X n i l i m p l i e s X n i l p o t e n t . L e t x e be an idempotent of X . From ( l ) we see tha t 3 2 R s a t i s f i e s the equat ion 2R - 3R + R = 0 . As w i t h Jordan e e e e a l g e b r a s , we w r i t e X as a v e c t o r space d i r e c t sum X = X Q ( e ) + X | ( e ) + X 1 ( e ) where X ± ( e ) = [x e X : xe = Ix} I = 0 , § , 1 . A l b e r t [4] shows the f o l l o w i n g m u l t i p l i c a t i v e r e l a t i o n s : X±(e)X±(e) £ X ± ( e ) , i = 0 , 1 , X 1 ( e ) X 0 ( e ) = 0 , X i ( e ) X (e) c X : ( e ) + X n ( e ) , and X 1 ( e ) X _ ( e ) c X i ( e ) + X n ( e ) . We see these r e l a t i o n s are weaker than i n the Jordan case . We say the idempotent e i s s t a b l e I f and on ly i f X i ( e ) X n ( e ) c X i ( e ) and X 1 ( e ) X 1 ( e ) <= X x ( e ) . X Is c a l l e d s t a b l e i f and on ly i f every idempotent of X i s s t a b l e . I f X has a u n i t y 1 and e ^ , . . . , e g are p a i r w i s e o r thogona l idempotents whose sum i s 1 , we have a r e f i n e d decomp-o s i t i o n Theorem 4 .1 .7 [ A l b e r t ] X i s a v e c t o r space d i r e c t sum X = S © X where X = X (e ) , X = X j e ) D X ^ e ) i =|  j , •f / -? J-cl J - J - - L - L J-J S x 2 J 91. and ( i ) ( i i ) i + ( i i i ) = 0 q + i , j ( i v ) The decomposi t ion i n 4 .1.7 i s c a l l e d the P i e r c e decomposi t ion knowing whether n i l commutative power a s s o c i a t i v e a lgebras are s o l v a b l e and i n the f a c t that idempotents i n X need not be s t a b l e . F i n a l l y , i t i s known that i f X i s power a s s o c i a t i v e X need not be power a s s o c i a t i v e . To circumvent t h i s d i f f i c u l t y , we w i l l assume i n the r e s t of the chapter that. X i s s t r i c t l y , power a s s o c i a t i v e . • 1 4.g Car tan Theory of Commutative Power A s s o c i a t i v e A l g e b r a s . In t h i s s e c t i o n , X w i l l denote an n < <» d imens iona l commutative, s t r i c t l y power a s s o c i a t i v e a lgeb ra w i t h u n i t y 1 over a f i e l d F of at l e a s t 4 elements, char P f 2 . Thus the r e s u l t s of § 4 . 1 are v a l i d . - Our study begins w i t h o f X r e l a t i v e to e .. ., e s For our Car tan theory , we can expect d i f f i c u l t y i n not 9 2 . Lemma 4.2.1 I f P i s a l g e b r a i c a l l y c lo sed and 1 i s the on ly idempotent i n X , then X i s almost n i l . P roo f : U s i n g a recent r e s u l t of Oehmke [22], i f X i s s imp le , then X = F - l and we are done. Therefore we assume X i s not s imp le , hence n > 1 . L e t N be a maximal i d e a l i n X where N =}= x • C l e a r l y 1 N , and s ince 1 i s the on ly idempotent i n X , i t f o l l o w s that N i s n i l . Therefore N i s the n i l r a d i c a l o f X , and X/N i s s imp le . L e t t\ be the image of - 1 i n X/N . We c l a i m i i s the o n l y idempotent i n X/N . For suppose e = e^ + N i s an idempotent i n X/N . By Lambek [20, p . 7 2 ] , we may l i f t e to an idempotent e i n X such that the image of e i n X/N i s e . But e = 1 , consequent ly \\\ = e as d e s i r e d . • By Oehmke X/N = F*£ , and I t f o l l o w s X = F - l + N , or X i s almost n i l . Q.E.D. As i n the Jordan case , we def ine (1) A{xl3x2,Xj) = X p X g - x ^ - X p - X g X ^ We can now prove Lemma 4.2.2 I f X i s n i l and A - n i l p o t e n t , then X i s n i l -po t en t . P roo f : The p roo f i s by i n d u c t i o n on the dimension of X . We may assume X i s not a s s o c i a t i v e , hence n > 1 L e t N^ = {g e X : ( g , x , y ) = 0 f o r a l l x , y e X] be the nucleus of X . 95. We observe, s i nce X i s commutative, i f g e ^ , then ( x , g , y ) '= ( x , y , g ) = 0 f o r a l l x , y e X . S ince X i s A - n i l p o t e n t , N rfr {0} . As N-j^  i s a s s o c i a t i v e , i s s o l v a b l e . For u e ^ , we note tha t uXo X . I f uX = X , s ince X i s commutative, X n = ( u X ) n = u n X n . But u n = 0 , hence X n = 0 as d e s i r e d . I f uX = 0 f o r a l l u e ^ , then T^X = 0. i m p l i e s N^o X . S ince dim X / N < n , X / N 1 i s s o l v a b l e by the i n d u c t i o n hypo thes i s . .Thus X i s s o l v a b l e , and by Theorem 4 . 1 . 5 , X i s n i l p o t e n t . I f 0 =J= uX 4= X , we t r e a t X /uX s i m i l a r l y to conclude X i s n i l p o t e n t . Q .E .D . Lemma 4 . 2 . 5 I f F i s a l g e b r a i c a l l y c l o sed and X i s A - n i l p o t e n t , then X i s a d i r e c t sum o f almost n i l a l geb ra s . I f N i s the n i l r a d i c a l o f X , then N i s n i l p o t e n t . P roo f : The p roof i s i d e n t i c a l to the p roo f g iven i n Lemma 5 . 5 - 1 , -s ince the X ^ i n the P i e r c e decomposi t ion of X r e l a t i v e to p a i r w i s e or thogonal p r i m i t i v e idempotents are almost n i l . That N i s n i l p o t e n t i s an immediate consequence of Lemma 4 . 2 . 2 . Q.E .D. As i n the case of Jordan a l g e b r a s , we def ine maps S ( b , c ) by (2) x S ( b , c ) = A ( x , b , c ) f o r a l l b , c e X . 94. We now give Lemma 4.2.4 I f X i s n i l , then X i s A - n i l . Proof: For x e X , we wish to show that S(x,x) i s nilpotent. Since x i s nilpotent, the subalgebra of X generated by x , denoted by F[x] , i s nilpotent. I f M(F[x]) i s the subalgebra of HomF(X,X) generated by R^  , b e F[x] , we have seen from Corollary 4 . 1 . 6 that x U(F[x]) i ^ ^ P* F[x] > M(F[x]). commutes, where p : b -• R , b e F[x] . Since S(x,x) = R R - R o i s i n M(F[x]) , i t follows S(x,x) i s nilpotent. Thus X i s A - n i l . Q.E.D We would l i k e to show that the analogue of Lemma 3 . 3 - 2 , (Engel) holds i n X . I f we assume F has at l e a s t 2n+l elements, we know i t i s s u f f i c i e n t to study the problem when F i s a l g e b r a i c a l l y closed.• I f X i s A - n i l , we can write X = E X ^ where the X ^ are almost n i l . But to-say that X i s A-nilpotent, we need the converse of Lemma 4 . 2 . 3 . 95. Next we wi sh to d i scuss whether A - n i l p o t e n t subalgebras of X generate n i l p o t e n t L i e a lgebras of l i n e a r t rans format ions of- X . We w i l l suppose that F i s a l g e b r a i c a l l y c l o s e d , and tha t R i s an A - n i l p o t e n t suba lgebra•of X c o n t a i n i n g 1 . L e t C be the subalgebra o f HomT ; i(X,X) generated by R o , a e R. r a By Lemma 4 . 2 . 3 , t h e r e . i s a set e 1 , . . . , e t o f p a i r w i s e or thogonal p r i m i t i v e idempotents such that R = E Fe.^ + ^ , and N = E E± i s n i l p o t e n t . Consequent ly , by Theorem 4 . 1 . 5 , the subalgebra N' o f C generated by R^ , b e N , i s n i l p o t e n t . Thus there i s an i n t e g e r p such that ( N ' ) p = 0 . We have \ Lemma 4 . 2 . 5 N ' generates a n i l p o t e n t i d e a l of ; C . P roof : L e t X = E X . . be the P i e r c e decomposi t ion o f X ^ i<J 1 J r e l a t i v e to e, , . . . , e , . Hence Fe. + N , c X . . ' . I f x e X , we w r i t e x = E x . . + E x . . , and i f a e R , a = E a.e. + a. , a. e F , a . e N . . S ince R i s l i n e a r i n a , i t f o l l o w s that 1 . 1 1 a ' . C i s generated by R, , b e N , and R , i = l , . . . , t . e i L e t b e N and w r i t e b = E b . , b . e N . . We observe l ' i i " . that x ^ = x . ^ and x . ^ = x . ^ R ^ + ) , i ^ j . L e t b ^ , . . . , b ^ P ^ e N where b ^ = E b ± ^ , and l e t T l = R ( 1 ) " " R (p) e H ' " T h e n x i i T l = ° ' s i n c e x T i = Q •' Suppose 0 rj= x . . e X.. . , i } j . Then x . .R /- \ = 96. ( X l j b j 1 ^ ! ! • W r i t i n g T ' 2 = ^ ( 2 ) ' * * ^ (p) w e c l a i m ( x i j . b [ l ) ) J j . T 2 = ( X i j b j 1 ' ) ^ ^ = 0 • Indeed, we have (x. . b . X ) ) . . T 0 = (--.((x. . b ( i : ) ) . . b ( . 2 ) ) . . b ( . 2 ) ) . . . b ( . p ) ) . But 1 1J 1 ; J J 2 v 1 33 3 33 3 J 0 = (• • • ((x. .b? 1 ^)b^. 2 ^ )• • - b ^ ) c X. . + X. . + X . . . We observe i j 1 3 3 n 1J 33 tha t the component i n X . i s e x a c t l y (x. .b^ 1 ^) . . T p . S ince the P i e r c e decomposi t ion i s d i r e c t , (x. . b ' ^ ' K . T = 0 . S i m i l a r l y , (x. .bi1^) . .T = 0 as d e s i r e d . I f = R . .R ^ , we may prove in a similar manner that ( ( x ^ 1 ) ) ^ 2 ) ) ^ , = ••• - (^/^h/^)^, = 0 , and t h i a process con t inues . Hence, by i n d u c t i o n , f o r any k > 0 , we have x . .Tn = £ ( • • • (x. .b. ) . . ' • • -b ) . .T, where the summation i s over i j . 1 v i j m k ' l j k a l l p o s s i b l e combinations of m^ = i or j , and = R ^ ^ . . . R Suppose, next tha t T i s a product of R f l \ > I = 1,.. . ,p , and some R '.s. C l e a r l y , x. .T = 0 i f k rj=g> or k 4= j f o r some k . There fore , assume k = \l or k = j , and tha t T = T 'R^ . . . R T* % e k 1 z where T r = R ^ j . . . R ^ j and T* = R ^ + 1 j T * . From the " 97. computations above, we have x. .T = ( | ) Z s(...(x, -.b ( 1 /). . . . . b ^ ) . -T* i j ^ v v iO mp ' i j m^  ' i j where the summation i s over possible combinations of m = i or j . Repeating this process by spli t t i n g T*, and continuing, we conclude x j _ j T = 0 • What we have shown is that i f T e C and each term of T contains at least p elements R^  , b e N , then xT = S x^^T + Ex..T = 0 . Thus i f a e R and b e N , i t follows that K J 1 J x ( R ^ ) ^ a ( % R a ) p s 0 • A s t h @ i d e a l generated by I' i s N* = N' + CN7 + N'C + CN'C , It now follows that' (N*) P = 0 . Q.E.D. We now have. Theorem 4 . 2 . 6 If R i s an A-nilpotent subalgebra of X con-taining 1 , then L X(R) i s solvable. Proof: By the remarks in §1.4, we may assume that P i s algebraically closed. Hence, l e t R, N, N 7, N*, and C be as defined in Lemma 4 . 2 . 5 . Now i f b e N , a e R , we see that S(a,b) and S(b,a) e N* . Furthermore, S(e.,e .)S(a,b) and S(a,b)S(e.,e .) e N* , for the idempotents e^ and e. . Since [S(e i,e j.),S(e k,e^)] = 0 i t follows that i f . ^  , T g e L X(R) , then [Tp,^] e N* . As (N*)P = 0 , i t follows immediately that (L x(R))( 2 p' 1 = 0 . Q.E.D. . 98. Theorem 4.2.7. Suppose K i s the algebraic closure of F ,. and Xv i s stable. I f R i s an A-nilpotent subalgebra of X containing 1 , then L^R) i s nilpotent, and i f X = X Q © X ^ i s the F i t t i n g decomposition of X r e l a t i v e to L^(R) , then ( i ) X Q i s a subalgebra of X containing R ( i i ) X Q X ^ E \ • . Moreover, i f F = K , then ( i i i ) X Q = n {B^ : b e R} •i Proof: The proof i s i d e n t i c a l with that of Theorem 2.3,3, [ Q.E.D. The .following example shows that f o r A-nilpotence, r the analogue of Theorem 3 -3.3 f o r general commutative power associative algebras i s f a l s e . . . . Example 4 .2.8 Let X ' be the algebra with basis u,f,g,h over a f i e l d F whose c h a r a c t e r i s t i c i s prime to 30. Let X 7 be commutative, and the m u l t i p l i c a t i o n table determined by the 2 following: u = u , uf = f ', ug = -|g , f g = h , and a l l other_ products zero. Albert [4] shows that X ' i s power associative. Let X be the algebra obtained from X ' by adjoining a unity 1 . Then X i s commutative and power associative.- Let v = 1-u . Then u and v are pairwise orthogonal p r i m i t i v e idempotents, and X±1 = X 1 ( u ) = Fu + Ff , X±2 = X i ( u ) fl X|(v) = Fg, and X 2 2 = Fv + Fh . C l e a r l y R = X - ^ + X 2 2 i s an A-nilpotent 99. subalgebra of X c o n t a i n i n g 1 . Now L X ( R ) i s generated by S ( b , c ) , b , c e R . Thus S ( f , v ) and S ( u , v ) e L X ( R ) . We compute g S ( f , v ) S ( u , v ) = 0 and g ( S ( u , v ) S ( f , v ) = ^ h . Con-sequen t ly , g [ S ( f , v ) , S ( u , v ) I = - ^ h . By i n d u c t i o n , , we show ' . • g [ [ . . . [ S ( f , v ) , S ( u , v ) ] , . . . , S ( u , v ) ] , S ( u , v ) ] = ( - ^ ) P h . There fore , there does not e x i s t a p o s i t i v e i n t e g e r k such that [ L Y ( R ) ] k = 0 , . s o Ly(R) i s not n i l p o t e n t . \ 100. BIBLIOGRAPHY Albert, A. A., On Jordan algebras of l i n e a r transformations, Trans. Amer. Math. S o c , 59 (1946) , 524-555. • .. A Structure.theory f o r Jordan algebras, Ann. Math., 48 (1947) , 5 4 6 - 5 6 7 . Power associative rings, Trans. Amer. Math. S o c , 6 4 (1948) , 5 5 2 - 5 9 3 . A theory of commutative, power associative algebras, Trans. Amer. Math. S o c , 69 (1950) , 503-527. X "Structures ' of Algebras", '(Cplloq. . Publ., Vol. 2 4 ) , Amer. Math. S o c , Providence, 1939. Anderson, C. T., Foster, D., and Suttles, D., Enveloping algebras of commutative solvable algebras, to appear. Barnes, D. W., Nilpotency of L i e algebras, Math. Zeitschr., 79 (1962) , 237-238. On Cartan subalgebras of L i e algebras, Math. Zeitschr., 101 (1967) , 350-355. Bourbaki, N., "Groupes et algebres de L i e " , Hermann, Paris, i960.. Bruck, R,' H., and K l e i n f e l d , E., The structure of alte r n a t i v e d i v i s i o n rings, Proc. Amer. Math. S o c , 2 (1951) , 878-890. Chevalley, C , "Theorie des groupes de L i e I I " , (1951) , I I I (1955) , Act. S c i . , Paris. Divinsky, N., ''Rings and Radicals", Toronto: University of Toronto Press, 1965. 101. 1J>. Hochschild, G.P., "Representation Theory of L i e Algebras", University of Chicago, 1959. 14.. Jacobson, N., Structure of alte r n a t i v e and Jordan bimodules, Osaka Math. Jour., 6 (1954), 1-71. 15. A theorem on.the structure of Jordan algebras, • Proc. Nat. Acad. S c i . , 42 (1956), 140-147. 16. . • - .Cartan subalgebras of Jordan algebras, Nagoya Math. Jour., 27 (1966), 591-609-17. "Lectures i n Abstract Algebra", Vol. I I , Van Nostrand, Princeton, 1953-18. ' X'' "Lie Algebras", Wiley (Interscience), New York, 1962. 19. Kaplansky, I., On a problem of Kurosh and'Jacohson, B u l l . Amer. Math. S o c , 52 (1946), 496-500. 20. Lambek, J., "Lectures on R ings and Modules",-Blaisdell, Toronto, 1966. 21. McCrimmon, K., Jordan algebras of degree 1 , B u l l . Amer.. Math. S o c , 70 (1964), 702. 22. Oehmke, R. H., Commutative power associative algebras of degree one, Journal of Algebra, to appear. 2~.. Penico, A. J., The Wedderburn p r i n c i p a l theorem i f or Jordan algebras, Trans. Amer. Math. S o c , 70 (1951), 404-420. 24. Schafer, R. D., The Wedderburn p r i n c i p a l theorem f o r a l t e r -native algebras, B u l l . Amer. Math. S o c , 55 (1949), 604-614. -25. ; "An Introduction to Nonassociative Algebras", Academic Press, New York, 1966. 

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0080531/manifest

Comment

Related Items