PEIRCE DECOMPOSITION IE SIMPLE LXE-ALMISSIBU!

POWER-ASSOCIATIVE ALGEBRAS

DISSERTATION

Presented In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy In the Graduate School of The Ohio State University

B&r

William Eugene Coppage, B.A., M.S.

The Ohio State University

Approved by

Advisers Department of Mathematics This dissertation has been 64—1251 microfilmed exactly as received

COPPAGE, William Eugene, 1934- PEIRCE DECOMPOSITION IN SIMPLE LIE- ADMISSIBLE POWER-ASSOCIATIVE ALGEBRAS.

The Ohio State University, Ph.D., 1963 Mathematics

University Microfilms, Inc., Ann Arbor, Michigan ACKNOWLEDGMENTS

Research for this dissertation was

carried out under the guidance of Professor

Erwin ICLeinfeld, now of Syracuse University.

The author is grateful to Professor KLeinfeld for suggesting the topic considered here and for his many constructive comments made while carefully reading this dissertation.

The author also acknowledges the help given hy Professor D. Ransom Whitney who, when

Professor Klelnfeld left Ohio State University, helped coordinate the committee which read the dissertation.

Finally, thanks to Barbara Coppage who typed every draft of the manuscript and did a marvelous job of proofreading. TABLE OF CONTENTS

Introduction ...... 1

Chapter

1. Definitions and Identities ...... 5

2. A Structure Theorea ...... 22

3. Another Structure Theorea ...... 44

4. Examples ...... 57

Bibliography ...... 69

Autobiography ...... 71

iii INTRODUCTION

The class of Lie-admissible power-associative rlogs includes all associative rings, Lie rings, Jordan rings, rings of type ( Jf anti-flexible rings, and others. The structure theory of such rings is therefore likely to be most complicated.

Rings of type (X , & ) have been studied by Albert [$],

Kleinfeld [10, ll], Kokorls [15, 16], and Maneri [SO]. It has been shown that if e is an idempotent of a simple ( g , £) ring A then A has a Peirce decomposition,

A “ *11 + *10 + *01 + *009 where ■ £x € A | ex ■ ix, a ® Jxj . Simple ( K, S) rings with an idempotent e not the identity element have been shown to be associative.

Kleinfeld [l2] introduced the term assoclator dependent for a ring which is third-power-associative and satisfies an identity of the form

^(x,/,*) + <*2 (y,z,x) + «<3(*,x,y) + oC^(x,z,y)

+ «*5(s,y,x) + o^(y,x,z) - 0 where the are constants, not all equal, in some field of scalars and

(x,y,z) - xy*z - x*yz.

1 2

In [lk] It vas shown that every aaeociator dependent ring

•atlsflee (I) (x,y,s) + (y,i,x) + (z,x,y) - 0; (II) (y,x,x)

- or (III) ®/(y,x,x) - (c* + l)(x,y,x) + (x,x,y) - 0,

a fixed scalar.

Third-power-associative rings A which satisfy (III) with

oC / 1# - (l/2), - 2 were proved to have a Peirce decoaqposltlon

A “ ^ll + A10 + AD1 + AD0 with respect to an idempotent e of A. Simple rings with characteristic prime to 6 and having no idempotent e such that

*10 + *01 ^ ® were shown to be alternative. Kleinfeld [9 ] showed that a simple alternative ring is either associative or a

Caley-Dickson algebra over its center. Thus there are no new simple rings in this class.

Actually the authors note in the above that, except for the case Ql m - l, the hypothesis A ^ + A ^ / 0 can be replaced by the assumption that e is not an identity element. The case

- - 1 yields the so-called anti-flexible law.

Rosier [id] showed that anti-flexible power-associative rings are also Lie-admissible. He displayed examples of new simple algebras in this class. As indicated by the above these algebras have the property that

A “ *11 + *00 in every Peirce decomposition. 3

If o<. - 1 In (III) one obtains the flexible lav,

(x,y,x) « 0 .

In (21J Oehmke shoved that a simple flexible pover-assoclatlve finite-dimensional algebra over a field of characteristic zero has a unity element and therefore has a degree. He shovs that for algebras A of degree greater than tvo A+ is a simple Jordan algebra.

Kokorls [1 7 ] extended this result to algebras of degree tvo.

Kleinfeld and Kokoris [ 13 ] proved that the only algebras of degree one are one-dimensional fields.

Laufer and Tomber [19J proved that if A is a flexible power- associative algebra over an algebraically closed field of characteristic zero and if is a simple Lie algebra, then

A - a(~). The hypotheses of pover-associativlty and A^') being a Lie algebra imply also that A satisfies (I).

If o* * - (l/2 ) in (III) one gets the left alternative lav; o< » — 2 leads to the right alternative lav. Rings of these types are anti-isomorphic. Albert [6 , 7 ] has shown that any semi-simple finite-dimensional right alternative algebra of characteristic not tvo is alternative. Maneri [20] proved that a right alternative ring of characteristic prime to 6 which satisfies (I) and has an idempotent e, not a unity element, is alternative.

The identity (II) gives a class of rings anti-isomorphic to those satisfying (III) with 0. Thus we are left to consider k

(I). Ibis gives the class of Lie-admissible rings which, even if power-associativity is also assumed, is a very big class. In view of aame o f the known results on finite-dimensional algebras satisfying these assumptions, a reasonable additional hypothesis would seem to be to assume the existence of a Peirce decomposition.

It is the purpose of this dissertation to investigate simple power-associative finite-dimensional algebras A satisfying (I) and having an Idempotent e, not a unity element, which gives a

Peirce decomposition of A.

Chapter I of the dissertation is devoted to basic notions and deriving multiplicative relations satisfied by the modules A ^ in the Peirce decomposition of A.

In Chapter 2 it is proved that if A ^ ■ 0 in the Peirce decomposition of the finite-dimensional algebra A over an algebraically closed field of characteristic prime to 6 then A is isomorphic to a certain three-dimensional algebra first introduced by Weiner f 22J.

Another theorem is proved (without the assumption that the scalar field $ 18 algebraically closed) under the strong assumptions that - $ e , - 0, and A ^ / 0 in the Peirce decomposition. This is done in Chapter 3* The algebra in this case Is of dimension s + 2 where s is a positive integer.

The fourth chapter is devoted to examples of simple algebras. 1. DEFINITIONS AND IDENTITIES

Dy an algebra ve mean a finite dimensional vector apace equipped with a Multiplication which la bilinear over the field of acalara of the vector apace. Thia definition of an algebra ia the cuatoaaxy one except that ve delete the assumption that

Multiplication ia aeaoclative and ve alvaya aaauae finite dimensionality. Ey the characteristic of an algebra ve refer to the characteristic of the field of acalara of the algebra.

If x>y>z axe elements of an arbitrary algebra A « (A,+,») ve define the aaaoclator

(x,y,zj - (x*y)«z - x*(yz) and the commutator

(x,y) « x.y - y.x.

One easily verifies that these functions are linear in each variable and axe related by the identity

(1 ) (x»y,z) + (y.z,x) + (z«x,y) - (x,y,z) + (y,z,x) + (z,x,y).

A lie algebra is by definition an algebra (L,+,*) which satisfies the Identities

(x*x) * 0

(x«y)«z + (y*z)*x + (z*x)*y « 0 .

Every algebra A has an attached algebra A ^ which is obtained by introducing commutation as a multiplicative operation in the

5 vector space A. A is termed Lie-admissible in case A ^ is a

Lie algebra. A ^ always satisfies (x,x) - 0 so is a Lie algebra if and only if

(2 ) (xy - yx,z) + (yz - *y,x) + (zx - xz,y) - 0 is an identical relation in A.

An algebra A over a field is said to be power-associative provided each x in A generates an associative subalgebra A^.

Since the powers of an element in a power-associative algebra are unambiguously defined it makes sense to speak of nilpotence of an element. A non-nil power-associative algebra contains a non-nilpotent element x which is also a non-nilpotent element of the associative algebra A • Therefore by a well-known [8, p. 15 J theorem on associative algebras A^, hence A, contains an idempotent.

Albert [h, p. 560] has shown that if e is an idempotent element of a power-associative algebra A of characteristic different from two, three, or five then A may be written as the supplementary sum

(3) A - Ae(l) + Aft(l/2) + Ae(0) of submodules A^( X ) where

Ae( A ) - £x€A|ex + xe = 2AxJ.

Moreover the submodules A (0) and A (1) are orthogonal and such 0 £ that

e*xl - *j/e " *1

" V * " 0 for all elements x^ in Ae(l) and x^ in Ae(0). 7

Let A be a Lie-admissible power-associative algebra and euppoee that A contains an idempotent element e with the property that for every x in A

(*0 (e,e,x) - (e,x,e) - (x,e,e) « 0 .

The reason for imposing this last condition ia that it enables us to prove exactly as for associative algebras that A decomposes as a vector space direct sum

A “ All+AI0 +A0l + A00 where A^j « { x € A j ex - lx,xe « J x J for i,J - 0,1. This so-called Peirce decomposition can be an aid in determining the multiplicative structure of A. Of course if e is the identity element of A then A^q * A ^ - Aq q ■ 0 and A « A ^ so the Peirce decomposition tells us nothing.

Comparing the Peirce decomposition with the decomposition

(3) it is easy to see that A0 (l) « Au ,Aq(0) * Aqq and

Ae(l/2) « A^q + A ^ . However, we emphasise that the Peirce decomposition is assumed whereas (3 ) is provable from the assumption of power-assoclatlvity. For an example of a simple Lie-admissible power-associative algebra possessing an Idempotent for which (k) fails to hold see Chapter 4.

Albert [3, Lemma lj noted that when the field of scalars contains at least three distinct elements an algebra satisfying

(5 ) (*2,x) - 0 also satisfies the multilinear Identity 8

(6 ) (xy + yx,z) + (yz + zy,x) + (zx + xz,y) - 0 obtained by linearising (5)» Conversely putting s ■ y ■ x in (6 ) yields (3 ) so long as the characteristic of the algebra is not tvo or three.

An algebra of characteristic not tvo which is Lie-admissible and third-paver-associative satisfies the identity

(7) (x.y,z) + (y**x) + U*x,y) - o obtained by adding (2 ) and (6 ) and deleting a factor of tvo in the result. Conversely any algebra vhlch satisfies (7) is Lie-admissible and if the characteristic is not three is third-power-associative.

We assume throughout that the characteristic of A is not tvo or three. Then

H(x,y,z) - (x,y,z) + (y,z,x) + (z,x,y) - (x*y,z) + (y«z,x)

+ (z»x,y) - 0 for all x,y, z in A. Also the fourth-power -associativity identities 2 2 (x ,x,x) « 0 and (x,x,x ) ■ 0 may be linearized to yield

P(a,b,x,y) » 2L(a*b,x,y) « 0 and

Q(a,b,x,y) - £ (a,b,x-y) « 0 respectively. Hue £ appearing here in both cases indicates a sum to be taken over all twenty-four permutations of the elements a,b,x,y.

We write x ^ for the component of the element x in the module

A .. of the Peirce decomposition of A with respect to e. Our study 9 begins with the derivation of some basic multiplicative properties of the modules A^.

X0MA 1. If i / J then

<8 > ztirJ3 - °- (9) *11^ - W i ylJ + ^xH yi3^JJ £ *1J + AJJ' (10) yi / i i * € V (11 ) yjiXii “ ^y^ixi i ^ i + (yJixi i ^ J € AJi + V (12) xiiyJi " ^yjixii^JJ € AJJ*

(13) xijyij “ yijxij e Aii + Aj y

(14) xiJyji € ^ i + AJj'

(15) xiiyii ^ \i * Aay

(1 6 ) xiiyii + yiixii ^ Aii'

(17) xii ^ Aii>

(18 ) ^flJ,Aii + Aji + AJJ^ “ 0>

(19) ^AiJAJi,AJJ^ " °* (20) ^AJiAJ4,AiJ^ “ °* (21 ) ^afay^O ” °* (22) aii*xi / i J “ xijyij*aii " ^1/2 ^ ^ aiixij‘yiJ^ii + Ki’VVli*'

(23) aii’xJiyJi " XJiyJi*aii - d / 2 )((xJiau .yJi^ii + (24) ^aii^xiJyJi + yjixij^ii * ^aiixiJ*yJi^ii + ^yJi cud

(25) ((*ijyji + yjlXlj)aliAi ■ ^yJl'ailXipii + (Xij'yjial)Al' 10

PROOF. Since the function P Is Identically zero

0 - (l/2)P(x,y,e,e) - (aqr + yx,e,e) + (xe + ax,e,y)

+ (xe + ex,y,e) + (ey + ye,e*x) + (ey + ye,Xj«)

+ (e,x,y) + (e,y,x).

The assoclator (xy + yx,e,e) - 0 fcy (4). Hence the above becoaee

(26) (xe + ex,e,y) + (xe + ex,y,e) + (ey + ye,e,x)

+ (ey + ye,x,e) + (e,x,y) + (e,y,x) - 0 .

Similarly

0- (l/2 )Q(x,y,e,e) = (e,e,xy + yx) + (e,y,xe + ex)

+ (y,e,xe + ex) + (e,x,ey + ye) + (x,e,ey + ye)

+ (x,y,e) + (y,x,e)

and (e,e,xy + yx) « 0 , so

(27) (e,y,xe + ex) + (y,e,xe + ex) + (e,x,ey + ye)

+ (x,e,ey + ye) + (x,y,e) + (y,x,e) - 0 .

To prove (6 ) put x » x ^ , y = y ^ In (26) and (27) obtaining

(28) 2i(x±±*e>yj p + 2i(xu ,yj y e ) + 2^ yj y etX 11 ^

+ s*(rjj,xlt,‘ ) * ■ 0 and

(29) 2i(e*yj y xii) + 2i^yjJ,e,xU ^ + a^ * ' xU ,yjj) + 2j(x11,e,yJJ) + + (yjj>xu'e) ‘ 0 respectively. Subtracting (29) from (28) gives

(30) (21 - + 2(J - i )(yj^e *3Clt)

+ (1 - 2J)C®**11»yjj) + (2J -

+ 2(1 - j)(x+ (i - 2i)(e>y^*xii) ■ °* 11

(j - i)H(x^,yjj,e) m o yield* an Identity which we write as

(31) 0 - 1 )(xa>yjjte ) + (J -

+ (J - “ °* Also (1 - ,))H(yjj,xil,e) • 0 yields

(32) (1 - + U - M xii'e'yjj)

+ (1 - J ^ e»yjj,xu ^ “ °-# Now adding (30), (31)* (32) and using the fact that 1 + j * 1 we obtain

3(J - i)(yjj'e*xii^ + 3 U - J)^xii^e>yjj) " 0 which, since 1 - J ■ * 1 and the characteristic of A Is not three,

Implies

Therefore

0 - H(.,*u ,yjj) - (e,x11>yjj) + U tl,yj y .) +

- 2(1 - J)xliyjj + x ^ y e - e - x ^ . (21 -

+ (21 - 2J - I H x 113rjj )10 + (1 + 21 - 23 )(xIiyjj)01

+ (21 -

By equating each component to zero we obtain (8).

Next put x * x^, y « y^j In (26). Since 1 + J » 1 this gives

(33) 2i(xu ,e,yij) + 2i^xii,yij,e^ + ^yij,e,xu ^

♦+ ^y<7 iu J ' ,xi xii'e) l , e ^ ♦* (e<*'xu *x a ' 7 : ij> + ('•7 ij'xn > • Subtract frost (33) the Identity 12

H(x11,yiJ,e) + H(y±j,xu ,e) -

+ + <*'xu'3'u) + (yy ,xii'e) + + (e'yij'x ll) - 0 . There results

(3*0 (21 - l)((x11,e#y1J) + (xtl»yij*e )) ■

How (x1j»e »y1j) “ 0 aod since 21 - 1 - i 1, (3A) reduces to

(xu ’yu ' e) * 0 or

xliyij,e • This Is equivalent to (9).

Por every element a ve have

(35) («,.) - («u + + «Jt + .jj.e)

- (J - l)atJ + (1 -

By (9), (xiiy1 J « 0 and therefore

- t ^ xtiytj)« * Hence

0 - H(ytJJ.,xu ) . (y^e.x^) + ( « lt.yy)

* (xiiyij'e) ' (J - 1)(yijxu * xu ytj + 'V i j V' Since J - 1 • t 1 It follows, again using (9), that

yijXii * This Is (10).

There Is associated with the algebra A - (A,+, •) an anti-isomorphic (under the Identity napping) algebra - (A,4 ,#) in which Multiplication is defined by

x#y - y*x. 13 i A has all the properties of A including a Peirce decomposition.

Indeed ( A * ) ^ - and (A*)n - A ^ , but (A*)1Q - A ^ and

(A^)q ^ ■ A^q . This fact obviates the proof of (U) and (12) since i these are the analogues of (9) and (10) respectively In A .

Pros (33) ve see that (a,e) ■ 0 ispllea * a ^ - 0. Nov

0 ■ H(xu ' yij«e) + t o i j ' v > ■ (xi / u + yijxij'e) + (i + + (i + J)(yltj**ij)

- (xi / u + V u , ! ) 80 XiJyi;J + yiJXiJ iS Aii + Ajj* means in particular that

^ l / l A v - - (ylJxlJ >uv when u / v. Thus, again using (35);

0 " H(xij'y ij'e) ' (xi / i J ' e) + (J ' i)(yij'xij>

- (j - + (! -

+ (J “ («J - i^ xIJyij^ii - 2 (j - O U ^ y ^ ) ^ - 2(J - i)^xijyij)ji

+ (J - i )(y1jxitj)jj - (J - i X * 1jyij)jj« We find by equating components in the same module to zero that

W/ijC • W/lAv ■ 0 when u / v and

^ l / l A u * (ylJxlJ^uu for u - 0,1. This proves (13)*

Nov for arbitrary u,v *= 0,1,

0 - H(xuv'yvu'*) ' ( V ™ - * 1 + u^v»'xuv>

+ • ( V w * 1’ It follovs from (35) that ’WS r u ^ All + AJJ* Taking u - i, ▼ ■ J gives (lk) and taking u ■ v - i gives (15).

To derive (16 ) put x ■ y ■ y^ in (26) to obtain

(36) 2t^ xii,e,yii^ + (xii'yii,e) + ^yii,e,xii^ + (yii'xii*e »

+ + (#'yii#xii^ “ °* Subtracting the identity

21(H(xil,yu ,e) + HCy^x^e)) « 0

from (36) givea

(1 - 2i)((e,xli,yii) + ( e . y ^ x ^ ) ) - 0 which implies

*(xiiyu + iriixn ) ' + V u > ' Id view of (15) this aufficea to eatabllah (16). Putting yil " ( V 2 ) ^ ! in (16 ) gives (17). An observation which will be useful in the sequel ia that

(9 ) and (10) together imply

(37) (xii'yij) ^ and (U) and (12) together imply

(36) (xii,y3i) ^ Aji* To prove (id) we first write 11 0 in the form 15

• ((aiixu )jj'y u ) * Aiy and by (1 3 ), (8), and (15),

V u *1!!1 * ((X1 / 1 J )U + (xl / l J W i i *

" ^ XiJylJ^li,ati^ € * 1 1 + Thus putting a - ®tl In (39) gives a sub of three c(mutators, the first of which Is In A ^ + and the others In A^, equal to zero. We say equate the A ^ + component to zero to obtain

(xU y lJ>,1ll) - °* whence

< 4 j ’V - °- Putting a * a ^ In (39) we obtain three c(mutators which we now evaluate separately. We find that by (13),(37), and (36),

W / l j ' V * + er Ajji whereas by (lk), (37), and (38),

^ylJajl,Xlj^ " ^ yijaji^li + ^yljaJ l ^ J ,xiJ^ & A1J' and

(ajixij,yij) ■ ^ ajixij\i + (ajixij)jyyi p & Again equating caaponents to zero In (39) we find (xijyij*aji) “ 0 or

^ J '* J 1 ^ " °* To complete the proof of (18) we take a - a.. In (39). ftr (13), Jj (8 ), and (1 5 ), 16

■ ^ xijy i p j j ,aw ^ € Ai i + Aj y by (11), (13), and (37),

^yijaj;j'xi ^ " + ^ylJa,Jpii'xl ^

" « yU aM>ii'\j> ^ V and by (12) and (37),

(W 7U } “ ((* U a« )U ' 7U ) 6 V As before we m y conclude that ( r ^ y ^ a ^ ) » 0 or

(A« ' V ■ °- This completes the proof of (18).

The proofs of (19) through (21) depend upon the Identity

H f x ^ y ^ a ^ ) - 0, which we write as (ll0> + (yjivV * ■ °- By (14), (8 ), and (15),

* ((V j i )ii * (xu yji)jj^ajj)

" ^ XiJyJ i ^ J ,aJj^ € ^ 1 + AJJ; by (10 ) and (37)

*yJiaJJ,XlJ* " ^ aJJyj A i ,xl,J* A1J; and by (12) and (38),

^J^lJ^Ji* “ ((xljajj)ll'yji> ^ Aji* By equating the components in A ^ + A ^ , A ^ , and of the left- hand side of (40) to zero we obtain (19), (20), and (21) respectively. 17

To derive (22) we first write p («»a11*x 1j*3r1j) - H(aii»xij,3rij) - “ 0 which, since 2i - 1 ■ i - J, reduces to

(U) (i - J)(«u > * y . y y ) + (i -

* (aii*u + xy ‘u ' e'lru ) +

+ e)

+ * + a - °* Here we have used (13) to write

V u + yu xu “ ^ i / i j - By (18), (13), (8 ), and (16),

Xijyij*aii “ aii*XijyiJ “ (I/2 )^xijyij*an + aii’Xijyi p ^ Aii* Thus by (13) and (8 ),

( V i l l i ' * ) * ^x13y13,ejan^ * °- Moreover, since and £ Aii* we that

(xu * u ' e'yij) + < V ‘u' ,ri3',) - °- Since the sane relation holds with x^j and y^j Interchanged, (1*1)

becomes, again using - y^x^,

(*«) + ^aiixij,yij’e ^ + C1 • ^)aiixij'yjj + (au y1J,.,x1J) + ( a ^ y , * y , . ) + (i - J)attytJ-*1J

+ 2(J -D a u ^ i / y ■ 0 . Bat .inc. £ A y + ^ • * 6 *ii + V (“yXy.e.yy) + ^JTy ,. ) + (i - 3 ) . ^ ^

- (i - J)(*11>‘i3-yl3)11- Thus, and similarly, (1*2) becomes - w ■ ° > and (22) follows. (23) is the analogue of (22) In an ontl-lsoaozphlc algebra,

To obtain (2U) and (25) ve begin with the identity

P(au ,.,x1J,y3i) - H(a11(3c13,yjl) - ) - 0 which we record as

(^3) (i - + U - • W aii'yji'xij)

+ (au xij + xu * u ' e'yji> + a)

+

+ ^Xljyjl + yjiXlJ'e,all' + ^Xijyji + yJlxlJ,all,a^ “ °"

Slnca ailxij + x tJall 6 A1J + AJj>

^ailXlJ + xljatt'*'yjl^ " ° ’ sl"oe xu yji♦ yjixu e Au + V

(xljyJl + yJixU » * ' * U ) * °- A1*°' xlJaU £ AJ3'XlJail’yJl € AJ1 + Aii» *°

(xiJaU ' yJt'*) * and

aiiyji € AJJ,aiiyJi*xiJ € AiiJ so

(aiiyji,e,xij) + ^aiiy Ji'xiJ**) “ °* Thus (^3 ) reduces to (44) + ^ " J^*itxij'yji + + (yjiali’v e) + (l - + (xi/jl + yjlx14'ali'*> - (i - * yjixij^ * °* So. .lace .^xy-y^ € ( \ } * AjjJa^ S *1± ♦ A ^ + Ajj5 ve have

(45) (a!ixlJ,yJi,e) + (* “ ^ aiixiJ*yJi « (i - J^ ftiixij*y,ji^ii + ^ " J^ aiixij*yji^i* 3* (9) and (14),

(aiixiJ*yJ i ^ i “ ^ aiiXlJ^iJyJi^Ji + ^ aiixi J ^ J yJ i ^ l hy (9 ) and (10),

((aiixiJ)Jjyji)jl “ ^ alixl J ^ J ,y,Jl^; t* (10),

and lay (20 ),

(xM aii*y,l) - °- Thus

■ °> which, along with (45), enables us to rewrite (44) as

(46) (i - J)(a11x1J-yJ1)11 + (yji'ii^.Xy)

+ (yjiau ' xij'e) ♦ <1 - J>au yji-Xu

+ (x« yji + yjix« ' aii'a) ’ (1 * J)aii(xi / j i + yj i V w (ii), 20

^yJiaii,e,XiJ^ " 1(yjiaii>JixiJ + ^ yjiaii^JJXiJ

" 1 ^yJiali^ixiJ " 1^yJiai i ^ J XiJ “ ^ " i ^ y,3Iai i ^ J XiJJ and by (12),

(i - J)auyji*xi j " C1 - J^ yjiai i ^ xu* Thus

and (k6) bee ones

(1*7) (1 - J)(aiixij*yji)ii + (yjiaii'xij,e^

+ ^Xij yj i + yJlXlJ ,a l i ,# ^ - (i - J)aii(xijyji + yjixij) “ °*

Since € (Aji + AJj)Aij - An + a j j >

^ytJiaii,XiJ,e^ “ i^yjiaii‘XiJ^ii + ^ yjiaii*xij),)j

- J(yjiall*xlj)ii “ ^ yjiaii*xi ^ j j * (d - J ^ yjlaii*xi p i i and (1»7) reduces to

(48) (t - 3)(*11*1j ,Jrj1)li + (*• -

+

K + - *11 + * « »li » because of (8 ) and (16). Thus

(Xljyjl * yJlXiJ’*ll’'^ - (4 - J *aii^Xijyji + yJlXlJ ^

- 4*1! + J*jJ - lxli - U jJ - (1 - i)«u + (1 - J)*jj - ■ (i - J)(»u (xli)yi]1 +

Together vith (1*8) this laplies (2U). Then 21

0' + H(rji'xiy*u) *

* ^XilXiJ + XlJ*li’yJl^ * W l yJi + 5rJi*ii,xil) and Toy (20) and (21)

W A i ' V ■ (,ity ji'x tj) * °- Hence

+ yJl’ailXlJ " y Jiaii'XiJ + XlJ*yjiail' Equating the component* in and using (24) gives (25). Tills completes the proof of Lenna 1. 2. A STRUCTURE THEOREM

In this chapter we assume that A is a simple algebra and that

Aqq ■ 0. Ad example of such an algebra is the algebra B with basis

■^e,x,y } such that e2 ■ e, ex ■ x, ye » y, aqr ■ - yx ■ e, and all other products of basis elements are zero. Every element a of B has the form

a«o(e + ^x+ify where ot, p , Y are scalars. If a is a nonzero element of an ideal

I of B, then I also contains ea + ae - a - «< e, x(a - ea) « V e, and (a - ae)y ■ /®e. Since the scalars , and Y are not all zero, e 6 I. Hence also ex ■ x and ye « y are in I so that I ■ S.

Thus the only ideals of B are B and the zero ideal. Therefore B is simple.

If

for r ■ then

aiaJ + ' * i * I * i / V x + ^ ^ i * V y * 2 In particular a^ - <* ^a^, from which we see that B is power- associative and in fact

* r lai for every positive integer n. Moreover

22 23

+ « i * J - X j * ! * and

4 [2^ i * J " * i£ j ¥ k “ ^ k ^ i ^ J “

+ " K j ^ i ^ k " 2 ^ k ^ l ^ J " ^ l/3J^ly* A short computation shows that

+ ((aj»aic)*a1 ) + ^ V ai^,aj^ “ 0 or B Is Lie-admissible. La fact a closer look at in which

(e,x) ■ x, (e,y) ■ - y, and (x,y) ■ 2e, reveals that so long as the field of scalars has characteristic different from two, B ^ is a simple Lie algebra.

It is evident that B has a Peirce decomposition with respect to the idesrpotent e.

THEOREM 1. If A is a simple Lie-admissible power-associative algebra over an algebraically closed field <£ with characteristic prime to 6, and if A has an ideapotent e satisfying (If) and such that Aqq « 0 in the resulting Peirce decomposition, then either e is a unity element of A or A - B, where B is the three-dimensional algebra described above.

PROOF. The multiplication properties in Lemma 1 will play an important role in our proof. We shall make repeated use of the module multiplication table 2k

*11 *10 *01

0 *11 * u *10

0 *11 *11 1 _____ J *01 *01 *11 * which compactly exhibits the information in (9 ) through (lit) for the case that « 0 . Ve note also that (1 3 ) says

(50) (A^qjA^q) « (Aoi,Aoi) “ 0 and (IB) says

(si) < 4 , ^ * v - <4 , ^ + v - °.

From (19) «e have

(52) < W l O ' V * ° and (2 2 ) through (2 9 ) specialize to

(5 3 ) *11**10^1.0 ’ *10^10**11 “ ( V 2 )(*11*10^10 + ® n yio,acio^

(54) “li'Woi " W o i ' V “ (VaX^i-n-yQi +

(55) all^xlCpr01 + “ aU X10,y01 + ^bl*!!**^ and

< *> (xi

We assume throughout that e is not a unity element for A and begin by proving 25

XJ9MA 2. JLjj^ la as associative subalgebra of A and Moreover*

(57) ^ * ^ 11*^11) ■ ) " (^n*^n*A) “ ®* PSOCP. . (Xji,^..y) ♦

+ < * y ' 3Cu ' s,n ) - Os and if i / j then* by (*»9 )* tvo of the assoeiators are equal to aero*

whence all three are equal to aero. We also see from (1*9 ) that A ^

is a subalgebra of A. Thus it suffices to show that A11 Is associative.

Ve assert that the subspace

1 " ^‘*ll,All,All^ + ^^ll'^ll'^ll^^ll is an ideal of A. To prove this asaertlon we need the fact* readily

verified by expansion* that the function

T(a,x,y,b) « (ax*y*b) - (a,xy,b)

+ (a^>yb) - a(x>y*t>) - (a»x*y)t> is Identically aero in any non-associative algebra. Thus

0 - T<\n’ltU ’yll’V " (' W ‘ll'yI l ' V - bij>

+ ^am n ,xU ’yllbiJ^ " aim^I£ll’yU ' biJ ^ " (,«n'xn ' yl i ^ ‘lj- Ifm+n*i + J»2, then it follows that

A11^A11,A11,'^L1^ ~ I# If m + n « 2 and i + J - 1, then from (U9 ) and the observation in

the first paragraph above we get

If m + n ■ 1 and i + J ■ 2, we get

Am / ,All,All,All) * °* Thus 26

and

A^A11,^11,*11^ ~ ** Mow (A^A^A), (Ajj^ A jA ^ ) , and (A,Aj^,A^) are each contained

In I and consequently

^*ll,*ll,*ll^*ll*A ~ ^*U'*ll,*ll^,All,A^ + (A^A^A^A & I.

We have shown that IA fi I. Finally

A‘(*ll,All,*ll)All - ^A*^Ail,Ali,Al i ^ Ali^ + A(An ,A11 ,A1 1 )-A11 4 I + 4 I, and It follows that AI S I. Hence I Is an Ideal of A. If A - I

< A11 then e Is a unity element for A, which contradicts our assumption. Therefore I ■ 0 and In particular ( A ^ A ^ A ^ ) - 0 which proves t.*— »» 2 .

Next we show

USMMA 3« * 0 for i / j. 2 2 PROOF. We prove that A^q * 0. Then A ^ ■ 0 hy the anti-

Isomorphism principle. 2 2 First we prove that J - A ^ + ^o**10 l* M ideal of A#

I* (53),

*u4 - 4 ^ s 4* ty (49),

*10**10 - *10*11 “ 0; and hy (51) and (49), *Q1a210 “ *10*01 - *11*01 " 0> Since $ J lay definition, we have £ J and AA^^ S

* (*9>,

^*w*io^*n s *11*10**11 - *10*11 “ °* Hy (49), (57), and the paragraph above,

*u< 4 > V ■ <*n4 >>*io £ *10*10 s J- Again hy (49),

< * io V * io - *10^*10*10^ £ *fo £ J- totting ^ - V u in (55), « h a w ^ q V W o I + 7 ^ )

‘ ^u10T10*aC10^r01 + *y01-U10T10^t10- Sut * (51) “ d (**9),

yoi*“ioYio * uioYio‘yoi ' Hence

(“lo^o'^oV - 'V'io'W oi + e *10*11 - J hy the above, and ao

*i o ^.o **q i £ J - Finally,

0 ' H(u10Y10'X10’y01) * (uW T10**10>y01>

+

y01*U10V10 ’ 0 aa above. Also

(xiojroi,uiovio) € ( * l l ,A10^ “ 0 by (49) and (5 1 ), ao y01^1^lJOT10*x10^ ■ ^ ( T i o ’^ o ^ o i € *10*10**101 s J * Thus J Is an Ideal of A.

Since A is simple either J - O or J - A. If J - 4 o * 1 0 + 4 o * A then A£q . A^, *12*10 - and A ^ • 0. Since e 6 t^Q , w aagr write

e g j *10 y10

Hour for any ^ q ^ I O we have

(58) o - H(*10,*10.y10) - (4o*)rio) + (W io*Jtio)

+ ^lO^O’^O^ * *1 0 ^ 1 0 + 2xioy i o ’xio hy (^*9). Also

0 « (l/4)P(x^0 ,x^Q,y1o>5r1Q) “ (xio,y10,y10^

+ 2(x10ir10^x1o,yio^ + 2^xioyio,yio,xio^ + (yio,xio'3W

• W io*yio * W io + 2

“ ^*10^1 0 **10^10 + 2 ^X10yl0 *y10^X10

" 2x10Jrl0*y10X10 + ^ o ^ o ' ^ o “ ^ O ^ O *

Using (1*9), (58), and the fact, expressed In (51) that A^q

Is in the center of the associative sub algebra A ^ , we obtain

*10^10 * - 2< W io )2- It follows that

^*10^1 0 ^ ■ (iAX^io'2 - dA^orio - 0 eince *3, . - 0.

Sut then ve have 29 since every t e m in the multinomial expansion must contain, for soaw J, a factor * 0. Prom this contradiction me conclude 2 that J ■ 0. Thus A^q * 0 and the lemma is proved.

In view of Lemma 3 we may nov replace (1*9) with the table

*11 *M> *W

0 *11 * u *10

0 0 A o *11

0 % *01 *11

We observe in passing that if either A ^ or is zero then the other is an ideal of A. If both are zero then e is a unity element for A « A p .

UM 4A k. If is an ideal of Au then

L “ *11 + *11*10 + *01*11 + *0l‘*ll*10

+ *11*1 0 '^oi + .,B11A10^AU is an ideal of A.

PROOF. Let ^ 80 Ideal of A^. Then

*11*11 + *11*11 ~ B11 - L* By (59),

*10*11 + *11*10 " *11*10 £ L and 30

* o A i + *11*01 “ *01*11 - L* © w a

AJB^l ♦ Bj^A £ L.

By (59),

\l*lO**ll S *10*ll * 0i by (5 7 ) and tbe fact that is an ideal of A^,

* 1 1 ' * 1 1 * 1 0 “ *11*11**10 ~ *11*10 - Li and hy (5 0 ) and (59),

W h o ‘ auAio‘'*io - 4.0 m 0 - Noting that

*01**11*10 + ^UL*10**Ol " L hy the definition of L we conclude that

* • ^ * 1 0 6 1 and

W A £ L - *** bu € BU * * (55>'

(6°) y0 lbil,at10 ■ h l l ^ O ^ O l + y01X1 0 ^ “ h l l ^ O ^ O l * Since x ^ q l + y ^ i o ^ *11 and is an ideal of the above implies

*00*11**10 S *11 + *11*10**01 S L* Similarly, lay (56 ),

(61) *10^01*11 *

*10**01*11 & * U + *0l’BH*lO* Moreover by (57) end the feet that le ea Ideal of

A aA i#Aii “ Am/AllAli S A o A i ~ L* toy (59),

^ ‘ A o A i £ Al iAh. “ 0; and toy (5 9 ) end (50 ),

Axl’AmlAu * AaAi'Aoi - *8l " °* Thus AniB11 *A end A’A ^ B ^ are coatalned in L.

Nov

^Aa/AuAioAi £ L by definition of L, and since, hy (59 ) and (5 2 ),

*Asl*®uAu>'Ali) £ ^AwlAlo'Ali^ " °> It follows that

S L - We see hy (59) that

A m / ^ A lo s A ll and hence

WA q i ’A liA l o ) 5 A.o A li “ 0# Therefore

^A h /A u A l o ^ o “ ^ A & T A u A i o 'A lo) and hy using the function H,

(Am/AlAo'Alo) s (AuAlo’Alo'Am^ + AoAxl'AuAlo^

Kow *11*10 S *10 10

AlAo'Alo - * 1 0 " 0 32

toy ($9)* Moreover c A ^ and It follows from the second

paragraph of this proof that

(6 2 ) <***11*10^ L# Consequently ve hare

**toL,Bll*lo)*01 S *11*61 " 0 and since (6l) implies

*01#B11*10 ~ h i + *10**01*11 ve have

(63) W * O l ,Bll*10^ S *oAi + *01^*10**01*11** By (59) and use of the function H,

*d i ^*i o ‘*o i b i i ^ * " ^*i o a o i b h ,*d i ^ " (Aio‘*orBn ,*oi*

- (*Ql*ll’*ai,*10* + ^*01*10,*Ql\l^ (5 9 ),

( * o A l ‘* 0 1 ,A10* - ^*01,*10^ " 0 and

(*0 l*l0 ,*01*ll* - ^*11**01* 11*' and toy the third paragraph of this proof

( 0 0 (A'*oAl* £ L- Since A ^ B ^ £ L, (6 3 ) nov Implies

*01^*01**11*10* - L* Ve thus have A<*0i-®uV £ I and

(Ao i * * n V A £ L- Let € Bq. Then 6 and by (56 ),

(6 5 ) (*n*io*yoi + yoi*bu*iQ^*ii “ yoi^aii*bn xio^

+ *ii*io‘yoi*ii* Bsr (57) «nd the fact that B11 la an Ideal of A^,

“ yai*®iibii**io* € adi*biiaio - L * Also y^a^ 6 10 that

bU x10*y01all € *31/10**01 - L* Since

(yQl**11*10^*11 6 ^*01*11*10^*11 - h * it now follows from (6 5 ) that

^lAo^Ol^ll e L# Next we note that (55) implies

(66) ^ ( * 11*10**01 + y01#bll*lQ^ " (*11 **11*1 0 ^ 0 1

+ ^l/Ll'hlAo' end

*11 (y01 **11*10> " (y01*b3l/l0^®ll ^ (*01*®Ll*lo)*ll 5 Toy (59)* (52), and the definition of L* Since y ^ A ^ U ^ *Di

yQ l*ll**11*10 € Aoi'^uAlo ~ L* and hy (37 ) and the fact that B,-.JJL is an ideal of A-,, XX

**n'bn xioty». * ^*uhu'*ie^yoi & b u *i o ‘a d i £ l- It now follows from (6 6 ) that Aii(^ii*10*y01^ £ L or *11**11*10**01) £ L*

Now *11*10 £ *10 80 \ l*i o ’*o i - *10*01 s * u “ d conse

*10*^11*10* *01) “ *10*11 “ °* It then follows that

*^L1*10**01^*10 “ **11*10**01'*10^* Hy use of the function H we find

**11*10**01'*lo) - **01*10'*11*10) + **10**11*10'*01^’ Now (59),

S 4 ) * °» and hy (6 2 )

* *01*10'*11*10) ~ L * Thus

**11*1 0 **01^*10 £ L* Aiso c A^A^S. A^, and therefore

**11* 10**01**01 £ *11*01 " °* Now (60) implies that

*11*10**01 S *11 + *01*11**10* Therefore

*0 1 **11*1 0 * *0 1 ) £ *oi*®u + *01*11* *io)* Since AQ1B11 £ L, it follows that ^ ( ^ * 1 0 * ^ 1 ) - L if and only lf *01**01^11"*10) £ L* Now *01*11 £ *01 iB® lie8 *01*L1**10

S *01*10 - *11 iH!Plie8 **01*11* *10)*01 “ ° ’

*01 **01*11**1 0 ) “ " * *01*11* *10'*0 1 )' and by using the function H, 35

W A a A i ’fu)) - ^^i o ^d i *Aoi^ l i ^ + ^Am l 'AbA i 'Al o ^ But AgjBjj^ < A ^ 80 £ A ^ « 0 by Lemma 3* Alio, by

(610,

(AloA ji'AqA i * - L * Hence

Aw-AlAlO’^ ” L* This canpletea the argument that

4 » u W £ L and that

(Bn W A 5 L * Hy (59) and (52),

(ADl'3^LlA10,An ) - (Ao i A lO'Au .) “ °*

Thus ve see that A ^ A ^ R j j A ^ ) « (^Qi ’B h A l o ^Al l * ® ertfore suffices to shov that (Aqj/ B ^ A ^ J a^ A and A ' A ^ A ^ B jjA ^ ) are contained In L. Hy (59)* (57)* the fourth paragraph of this proof,

^*oi'En Aio^Aii'A * (*ca‘Bu Aio^’Au A

* ( ' W h i V * - L > and

A 'All^AOl'BllAljO^ ■ AAll‘(A0l'BllA10^ s Ada'^o) s 1.

This completes the proof that L Is an Ideal of A.

IEMMA 5* A ^ is a simple algebra.

PROOF. Leana h shows that If Is a non-sero Ideal of

A ^ then \l*10 " *10 and

*01*11 " *0 1 * Proa (5 7 ) It follows that for every positive Integer n

*11*10 " *10 and

*01*11 * *01 Thus A ^ can have no non-zero nilpotent Ideals. In particular then,

the radical (which is defined to he the maximal nilpotent ideal) of

A ^ is zero and consequently is a seal-simple (associative)

algebra.

By a well-known [8, p. 27 J theorem of Wedderburn, A [1 Is the

direct sun of simple algebras. If and are distinct (hence

orthogonal) simple components, we have, by (57 ) and the facts that

BU C11 * 0 “ 4 \ 1A 10 * Cll*10 ■ *10'

0 ■ Bu°u.-Aio “ Bn‘°ii*io ■ * ^ * 1 0 - ho' which as ve have seen cannot be the case. Thus the number of simple

components Is one, which means that A ^ is simple, proving Lemma 5 .

By Wedderburn's theorem there is a $ -Isomorphism between the

simple associative algebra A,-, and the algebra M^D) of n by n matrices over D, where D is a division algebra over £ . When <£ is

algebraically closed, D « <|S . We shall not Invoke the hypothesis

that $ Is algebraically closed until a later stage of the proof.

Doing this will enable us to prove a stronger theorem in Chapter 3* Let the matrix having a one In the 1-th row and J-th column and zeros elsewhere, correspond In the algebra Isomorphism to the element e ^ of A11» Then the Identity matrix corresponds to the element ^ e ^ and since e Is the unique Identity element of we must have n * - ^

We may identify D as a matrlc suhalgehra of Mb (d ) over <£> consisting of the matrices

diag(d,* • *,d) where d £ P.

Let d^,* • *,d^ he elements in A ^ corresponding to basis elements of the division algebra D over ^5 . By the above we may think of P as being a subalgebra of . Consequently we shall refer to the elements as being "In" D.

The arbitrary element of A11 may be written in the form

I £ 1 ^ p , m»l p,q*l where <£ . To multiply such elements we use the distributive laws and the rule

^dm * p q ^ dkeuv^ * ^ q u da^kepv* The conditions £ A^^ and ( A ^ A jj^ A ^ ) - 0 Imply that

A ^ may be thought of as a set of (left) linear transformations on the vector epace A^q over $ . Fron this viewpoint the elements e1A, I ■ 1,* * *,n, fore a supplementary aet of orthogonal projections.

Consequently A ^0 may be decomposed Into the vector space direct sue n r \ o ^ eii^' vbere R^Q(eij_) denotes the range space of the operator i*l eii. Therefore ve have a basis

4r V (V 1 ) / . • e • . V ^ 1 ^ • • • v ' ( 1 ) ' . • • • v ^n^? > ^ t > ,xn * * n i for over vith the special property

(67) .u xJk>- for ell k.

Note that if i / J, then for all m,k,

(6 8 ) emixj ^ * *nielixj ^ " 0# Here and in the following ve make free use of (57).

For all m,J,k,

. ,x(k) . e . ,*

(6 9 ) e V e„J- If 0 j< d £ D , then

xJk> ’ ‘ (4'1'4.)(d*«J)xJk>« and consequently,

(TO) d,»JxJk> * °- A ]1 nay also be viewed as a set of (right) linear transformations on A ^ and A ^ has a basis \ * Hy (52), la contained In the center of A^. The center of A^i conalsta of those elements of which correspond to scalar matrices with an element from the center of D down the diagonal.

Thus we may write

(72) y(k)x r^ where the •*« •

Now

and the last commutator is zero hy Lenma 3 . Using (59), the above reduces to m *uV>yJk)‘*r*) + ^ k)*'*)- ^ T) - 0 ; and substituting from (7 2 ) and (7 3 ) into this we obtain

r T x "*4 d e x(s) + d x(v) 0. fcl pfq. 1 *3*”r V ID PJNJ, * il&TiJkrs u U

Using (6 8 ) we have ko

(75) ^ F I 0 Jkuv 4 a * px* *(,) r + 4 Jkrs , _ *(T) u - 0.

Ve may now prove

LEMMA 6 . A ^ * D.

PROOF. The lean* asserts that n ■ 1. Asswe n > 1. Then ve may aelect a v such that v / u, and consequently

e™ (ajkr.xir)> - ew (dJkr.*uuIuV)) '

Thus aultlplieatlon of (75) on the left Toy #w yields

I o jkuv 4avrr * *(,)- 0 or

(T d )e x (s> - 0 * Jkuv a vrr m*l By (70) we conclude that

n ^ d . o £ l 'Jtaiv a and then, since {d^, • • • ,d^_ | is linearly Independent over <$>

This holds for all j,k,u,v,m, and r and all v / u. Thus (75) *ay now be written

^ + dJkrsXuV^ “ °*

Sjr (55),

( ) e (x(v)y(k) + y(k*x(v)) - e x(v).y(k) + y(k)e *x(r) v 7{ 8 i ww u yj yj u ; ww u yJ + yj ww xu * Since v u the right-hand side of (7 8 ) Is* toy (6 7 ), (71), and (72),

£ v j djkuv*’ (76), (73), «nd (7 2 ), ve have

and therefore the left-hand aide of (7 8 ) la

Thus (78 ) implies

(79) If v jf J this says

4Jtuve™ - °> which li^plle a

(8 0 ) d,. » 0 . Jkuv If v - J then (79) yields

dJ k u v ^ ell) * °* which again implies (8 0 ). Thus (80) holds In any case. Recall that there was no restriction an the subscripts J,k,u,v appearing in (60). Our only assumption is that there be a v different from u; that Is, that n > 1. In view of (80), (77) becomes

dJ « - 0, which, toy (70 ), implies 42

Since { d^, • * ’ } ie linearly independent over <£> , (81) In view of (60), (72 ) becomes

^ k)x

*

Thus AQ1A10 - 0 - But then hy (59),

**10 “ **11 + *10 + *01**10 “ *11*10 ~ *10 and

* 10* " *10**11 + *10 + *0 1 * " °* Thus A^q is an ideal of A. Ve noted earlier (following the proof of Lenma 3 ) that A^q / 0. Since clearly does not contain e,

f A. Thus the simplicity of A is contradicted. This contradiction arose from the assumption that n > 1. Therefore n ■ 1, and the proof of Lemma 6 is complete.

Ve now use our hypothesis that £ is algebraically closed to conclude that A ^ is isosK>rphlc to . Since e is the identity element of A | ^ ve have A ^ - e.

I£MMA 7 . Aj£ and Aq^ are each one-dimensional over <£ .

PROOF. Let x*v* he any non-rero element of A^Q. If x*** is another element of A ^ such that { x*v *,x*** } is linearly independent over $ , then for any y*k * in A ^ ve have **3 (8a) x(TV k >.x(,) ♦ y (k)x(,).*(v) . 0 , which is Just (74) with the (now) superfluous subscripts omitted.

Since both and y^kM 8^ are elements of $ e , (82) asserts a dependence relation over $ between and x^v\ which were taken to be linearly independent. Therefore x^vV ^ ■ y ^ x ^ * 0.

Interchanging v and s in (8 2 ) and applying the above argument again gives x<8 >y ■ y ^ x ^ « 0 .

How x was an arbitrary element of A^q and y^k ^ arbitrary in

and we have shown that if A^Q has dimension greater than one over £> then x(v)y(k) . y 0 0 x (v) . Q'

But this iaplies « AqjA^q * 0 which, as was shown in the proof of Lemma 6 , cannot be the case. Therefore A^0 is one- dimensional over £ . Also A ^ ** a£q is one-dimensional over .

We now have A ^ » $ e , A^Q ■ x, and AQ1 ■ c£y. A change of notation in (8 2 ) yields

(xy + yx)x ■ 0 .

Since xy + yx is in e, we conclude that

xy + yx ■ 0 .

Without loss of generality we may take

xy « - yx - e.

This completes the proof of Theorem 1. 3. ANOTHER STRUCTURE THEOREM

Suppose now that A is a simple algebra such that A ^ is one- dimen aional over <$. If Aqq - £> z then z2 € A ^ hy (17) ao 2 z ■ ol z, and without loss of generality, ve may take oC ■ 0 or 1 . 2 We shall assume that s - z and make free use of this as well as the fact, expressed in (6 ), that

zAn - A ^ z - 0.

Since (l/2)Q(e,z,z,x1Q) ■ 0 we have

(8 3 ) (e^zx^ + x^z) + (z^zx^ + x^z) + (x^e,*)

+ («*x10>*) + ( ^ s \ 0 ) - 0. Since zx10 £

(e,z,zx10) - (z,e,zx10) - 0 .

Also (x1Q,e,z) - 0 and since x ^ z 6 ^ + A ^ ,

( e ^ ^ z ) - 0.

Thus (8 3 ) reduces to

(e,z,x10z) + (z,e,x1Q z) + (z,z,x1()) « 0 .

H0» £ *!„ + A^, e l(Aw) + A^) £ A^, and

Z'Zx ^q £ zA^ « 0. Therefore

- A-*!*)* - + **10 * 0 or

(84) sx10 - 2Z-XJ0Z.

But

44 0 - (1/6- ( z ^ z x ^ + x ^ z ) - (s,s,a^o), and since zA^g 9 A^,

Z ' Z ^ q » z C z - ^ z ) - 0.

Thus we have

*‘*10* ‘ **10 ■ 0 or

“ 10 * z-xio*‘ Together with (84) this implies

(85) z^ q = 0.

Then also

(86) v e J^o hy (11) and (12). Consideration of the anti-isomorphic algebra gives

(87) - o and

(88) w 0i £ A m .- How f * e + z is an idempotent element of A and for every basis element x^0 of A^q we have

(f.f,^) - »10 - ^ - **10 • 0J

(f>*10f) * **10** - f -*10f - *10* - *10* ■ 0 and, since H(f,f,ac^Q)> 0,

(x^jf^f) - 0.

Similarly, for every element yQ1 of A ^, ve have U6

(t>t>y01) - (t,y01,e) - (yQ1,f,f) - o.

Since f acts as a unit element on 4 Aq q ve see that

(f,f,A) - (f,A,f) - (A,f,f) - 0.

That is, A has a Peirce decomposition,

A - Au (f) + A ^ f ) + A ^ f ) + ^(f), vlth respect to f .

If x € A, write x - x 11 + x10 + x01 + x00 where e A^®)*

Then f x - X11 4 x10 + ^ 4 X q o and xf - X;^ 4 Xi0 s 4 Xqi +Xqo*

Thus x € A ^ f ) if 0X14 only If x - 0. That Is, Aq q (*) * 0. Nov e and z are non-zero elements of A ^ ( f ) and ez - 0; so ve see that

A11(f) cannot he a division algebra. It follows from the proof of Theorem 1 that f » e 4 z must be an identity element for A.

This Implies In particular that x ^ z * x^Q and zyQ1 « yQ1 for every

*10,y01 1x1 *10* *01 reBP«ctiTe]y < Note also that A has a Peirce decomposition with respect to the idempotent z » f - e. Observe moreover that In this decomposition

A )1 (z) ■ Q z is one-dimensional. This means that our investigation may be completed by determining all algebras of this type.This is attempted next. We consider simple algebras A for which * ^jje

Is one-dimensional. For such algebras ve have

IZMtA 8. The modules A ^ multiply as indicated In the table which follows. * 7

$ e *10 *01 *00

$ e *10 0 0

0 $ e + Aq q *K> ^ • + *oo *10

*01 *01 + *00 $ e + *00 0

0 0 *00 *01 $ e + *00

PROOF. The only thing really nev here Is that

“ *01*00 " 804 *10*00 S *10* *00*01 £ *01* Use (12)> and the assumption that A ^ - ^ e to write

*00*10 " U e and

*10*00 * rfe + u10- Then, hy (18) and (13), 0 ■ ^(x^x^a^j) « (a^Q^aoo^ + (*00*10 + ^© “oo^o) - (2 ot e + i ^ , ^ ) - 2 erf (e,*^) - 2 *

It follows that o< » 0. Hence Aq q *10 “ 0 *10*00 “ *10* ^ # consideration of A it follows that A ^ A q q » 0, and A0q Aq1 £ A ^ .

OhBerve that if + A^ * 0 then is an ideal of

A * <£> e + AOQ, whence A * <£ e is a field with unity element e.

Next we consider an algebra A for which A^q - 0 and A ^ ^ 0

(algebras with ^ 0 and A ^ ■ 0 are anti-isomorphic to these).

An example of such an algebra is the algebra G which follows. * 6

Let Q be the algebra with basis * * ,y# sod with 2 2 multiplication defined by e « e, z ■ z, es ■ ze ■ y^z ■ ey^ ■ 0,

*y± - y4e - «4 y^j - - <*„(. + *) «h«. th. are scalars such that the symmetric matrix ( W ) is non-singular. XJ First we note that e + z is an identity element for G. Write e + z m l. Ve next show that G is staple*

Let I f G be an ideal of G and suppose

8 a - £ e + tfz + 21 is an element of I. Then eae ■ £e £ I and unless £ ■ 0, every yi " yie is 111 I# Bttt then every y^y^ ■ o^^'l is in I and since not all the oC can be zero, 1 6 I whence I - G. Thus £ ■ 0.

Similarly zaz ■ 3 z £ I and unless J « 0, every y^ * zy^ isin I.

This again implies that I ■ G, so we conclude that 3-0. Thus if

I contains a non-zero element a, a must have the form

8 a - E n iyi* i«i x x s But then for each j, ay^ * E Hi ■ °* S'** this says that

(fll'* * m l°>m * *'°) which, since det( ei ) yf 0 is possible only if each H t ■ 0. Thus a - 0 and consequently I ■ 0. Therefore G is simple. s Let a* - £'e + j'z + £ n Jy. with a similar notation for i-1 x 1 the elements a",a'" of G. Then ve compute + 1 & 1 n i n j Note that

(a')2 - < £ ')2« + ( J')2* + + S ' ^ + X. <£'+ •*■)•' -<£’ •S')-1

+ i S a nir,i‘‘«*l‘ This says that G is a quadratic algebra. Thus G is power-associative.

Next observe that

(a',.") - ♦ nie* - ry l e')yt and

( y ^ a " ’) «(£*"- V n }yv

Thus

. rts'ni - j"ni + nie" - nle'Kt"' - j " ’) ^ from which one finds that

((a',a"),a’" ) + ((a",a'"),a*) + ((a'",a* ),a") « 0.

Thus G is Lie-admissible

THEOREM 2. Let ^ be a field of characteristic not 2 or 3 and let A be a simple lie-admissible power-associative algebra over $ possessing an Ideapotent e such that (e,e,A) - (e,A,e) ■ (A,e,e) ■ 0.

If A ^ ■ <^ e, - 0, and A ^ / 0 in the resulting Peirce decomposition then A is an algebra of the type described above. 50

PROOP. Hote that / 0 under our hypotheses for otherwise, as noted following Leona 3# A^q ■ 0 Implies that e is a unity element for A. We shall show that Aq q is one-dimensional over $ .

Consider the subspace

I - $ e + A ^ + (4>oo.

By (8) and (22),

aoo(xoiyoi*oo " aoo‘xoiyoi - (I/2 ) ( ( aoox01 'y0i >00 + (^x/oi'^^OO*

“ xoiyoi*aoo * (’W o i *00*00' which implies

*00*(*01*00 “ (*oi*oo*0o & (*01*00* Using the multiplicative properties stated in Lemma 8, ($e ♦ V<&>00 * °*

< & w * e ♦ v £ * o i s *• $ e . A « <§e(

A* $e « ($e + A^ + A^Jfe I,

*01* “ *01 (^ 6 + *01 + *0 0 * * *01 + *01 " T> and

**01. “ ( ^ e + *01 + *00**01 “ *01 + *01 s lm Thus I is an ideal of A. Since e £ I, I f 0. Since A is simple,

I « A. Therefore

*00 " (*io W and ly (18),

(89) 0A*,, V - 0. 51

The nodule Multiplication table for A has the following form:

$ e *01 *00

f e $ e 0 0

(90) Aoi *01 * e + Aoo 0 0 *00 *01 *00

2 from which we see that If A^ £ A^, then A ^ + A ^ 1b an Ideal of

A. Since e ^ A ^ + i^, we can have only A ^ + A ^ ■ 0. But then

A ■ $ e is a field with unity element e, contradicting our 2 ± assumption that e la not a unity element. Therefore A ^ ^ A^.

Since a01t>01 - *01*101'

“D^Ol ' (V2)(V + b01)2 - (1/2)4 ' (V2)4> 2 and consequently every element of A ^ may he written In the form 2 g i 2, 0< ^y^ where y^ € A^. Since A ^ ^ A ^ we may assume that A ^ contains an element y ^ such that

yoi ■ o t e * *00 By associativity of cubes and (9 0 ),

(9 1 ) y0 1 - y01yoi - yoi ■ yoiyoi “ zooyoi* Therefore / 0. Moreover, by (90), fourth-power-associatlvity, and the above,

o< e + (*q q ) ■ y01y01 “ y01y01 m y01 ^01 “ e + *00’ 52

2 Hence (zqq) * oi Zqq* Since «< ^ 0, we are now able to write

(9 2 ) y ^ ■ ot(e + Zqq) where Zq q ■ oc'^Zq q and consequently

(93) ^ . <*-a(«&)2 - « ' 2( * *io)

■ * '1*oo * *00- Also it follows from (91) that

(9l° W o i ’ yoi' Our next step is to show that

A^x - $ ( e + Zoo) where is the idenrpotent element above.

Let aQ1 be an arbitrary element of A^. In view of our earlier 2 remarks it will suffice to show that a ^ £ ^(e+Zgg). Ve have

yoi ■ * (e + W - Let

(95) 4 l * P* + ‘’00 and

<96> W o i ■ yoi*oi ' * e + coo- Since

(l/6)Q(y0i,y0i,y0i,a0i) ■ 0, ve have .2 x . , _ 2 (97) (a0 l'y0 1 ,Jr0 1 ) + (y01 ,a01 'yQl> + (y0 1 'y0 1 'a01y01 + yox*oi> ■ °- (9 2 ), (9^), and (90), we have 53

(*01'y01'y01) ■ * <*01’y01'* + W

- * (*oiWe + W-Woi> ■ ■<<«•♦ <=00*00 - *■« - =0 0 )

- o( (“00*00 ■ c0 0 ^

(y01’ *0 1 'y01* ■ * V>' + W

- * « «• + co o )(* * Io o ) - y01*W.)

- <* <«• ♦ 000*00 * « • - «oo)

* "< (°00*00 ■ coo*; and

(y01’y01’*01y01 + y01a01 ^

* 2(y01,y01* < e * ° 0 0 } - a( <* (« + * • + « „ ) - y01- *y 0 1 )

- 2( U X e + ec z00c00 “ * «< e - If ct z ^ )

" 2 * (zoocoo " * soo)* Thus (9 7 ) becomes

2 o( (C00Z00 ■ c0 0 ^ + 2 ^Z00C00 " ^ z(Xp “ 0 which, by (8 9 ) and the fact that ot j 0 , Implies

(9 8 ) c00*00 “ ( V 2 )(Coo + ^ z0 0 ^*

Nov for any «ooytoo in tJie subalgehra Aq q and any ae^ in ve have, by (90),

^*ioi,8oo,too^ “ ^OO^Ol^OO^ “ °’ Therefore 0 - ^800**00**01^ whence

< " > (‘W V V 5 - °- By (99) and (9*0, 5*

coo*oo*yoi " coo* Wo i " C00y01i by (9 8 ) and (94),

c00*00*y01 “ ( V 2 )(Cqq + £ *0 0 ^ 0 1 “ (^’/2 )c00?r01 + ^01 ’ Thus

0 “ H(yoi,yoi,aoi^ * ^y01,a0 1 ^ + 2 (y0ia01 ^ 0 1 }

- <* (e,ao1 > + * (zoo'aoi) + 2 X(e>yoi) + 2(c00'y01} - - « **01 + * *00*01 - 2 * yoi + 2Xyoi

- * (*00a01 * a01* and since oL / 0 ,

(1 01) *00*01 " *01 * Recalling that vas an arbitrary element of ve have, by (90), (2 2 ), (1 01), and (9 6 ),

<102> z00c00 " W W o i “ (x/2 )((®ooaoi*yoi^00 + ^ 00^0 1 ^ 0 1 W - (l/2 )((a01y01)00 + (y0ia0i)00 )

“ c0 0 * On the other hand, by (92), (90), (22), and (100),

(103) *00C00 “ * (y0 1 ^00°00 “ <* (y0 1*°0 0 ^ “ 1 (c00y01 *y01^00 » *. ^ O ^ O O

■ D< U ■ V Zq q * Comparing (102) and (103) ve get 55 ( “ * ) coo * W o o - * *00- Next, toy (95), (96), (10k), and (101),

0 * "(•m . ' V ’W - '-OI'V + ^ W o i ' V ■ + ^oo,yoi^ + +

* W o i - />yoi- Thus

<1Q5> W ’ t yo r Finally, toy (90), (95), (22), and (101),

(1 0 6 ) zo0b00 “ *00*01 “ * *00a0l’*0l\>0 “ (*01^00 * b00J but toy (9 0 ), (9 2 ), (22), and (105),

(107) z(X)b00 “ * ^yO l W bOO “ * yoi‘boo

“ ** ^b0(/01*y01^00 " ** 1/3 (y0l\)0

- * *oo “ /^oo* By comparing (106) and (107) ve see that

b00 - fi *00* Thus we have shown that for every a ^ In A^,

•oi * /*(« + W - Hence toy our earlier consent,

^Di “ £ (e + zoO^* 2 Since we have already shown that ( ^ J q q “ a ^q ** conclude that

*00 " ^ *00 is one-dimensional. Moreover our work shows that Zqq is an ldempotent and for every y ^ in Aq^,

W o o ' 0 and 56

W o i " yo r To complete the proof of Theorem 2 let { * * * ,yg ] be a basis of Aq 1 over $ . Suppose

V j ■ *v<* ♦*> where z « Zqq. Since y ^ « y ^ j the matrix ( °< jj) Is symmetric.

Suppose ( «*C jj) is singular. Then there is a non-zero vector

( * V ‘ * *'r>s) in its null-space. But then the element

a ■ f , 1=1 has the property that ea ■ 0, ae ■ a, za » a, az = 0, and e ay^ = ^ ^ i lj “ 0 = yja * Hence $ a is an ideal of A. This is impossible since A is simple and # » S + A. Therefore the matrix ( «< ^ ) is non-singular and A has precisely the properties of the example G. k. EXAMPLES

In this chapter ve present more examples of simple Lie- admissible paver-associative algebras*

Ve remarked in Chapter 1 that not every simple Lie-admissible pover-associative has an idempotent, not the identity, giving a

Peirce decomposition* It vill be convenient to look at one such class of algebras here.

Albert [l, 2) characterizes a reduced central simple Jordan algebra of degree two over a field $ of characteristic not two as an algebra J with basis • • *,u ^ , n 1, over

$ where the basis elements multiply according to the table

e . . *uy . . z - / e \ /' e UJ 0 L/ 2 e • • ui Ui *ij( e + z ) ui 2 e 2 • #

z 0 z 2 with the matrix ( ^ ) symmetric and non-singular.

57 58

The (for us) salient facts about such an algebra J are that

J is coomutatlve (hence Lie-admissible), has an identity element e + z, 1s simple and quadratic. The first two properties are evident from the multiplication table. To see that J is quadratic, let n x « £e + Yi K- + * be an arbltraxy element of J. Then

2 n x ■ E ex + F" M-4u «x + J a J-l J 0

* £(€e + f l (l/2),A'iUi )

+ r ^ i ^ / s H U i + (JLjXu(e + *) + (1/2)11^)

+ s ^Uj + 1 8 ) ' + ’,2'+ 7E ^ n

-u+r)* + [r £?](« +i).

Since J is quadratic it is easy to see that J is power- associative. Indeed j for every x, the subspace $*x + *(e + z) is a subalgebra, is associative, and contains J^.

To prove that J is slnple we first note that if an ideal of J contains either e or z then it contains every ut ■ 2eu^ = 2sui and 59

therefore contains every ig ^(e + z). Since the matrix ( ^ Is non-singular not every ^ ^ can he zero, and consequently any Ideal

containing e or z contains e + z, the identity element of J. Thus

the only such ideal Is J Itself.

Let n X“ £e+Z +J* J-l J J he an element of an ideal I of J. Then I also contains

n ex = £e + (1/2)£ K i V d=l 1 3 3 and at « (1/2)2^ K-JUJ + 1 *•

Therefore I contains

ex - zx = £ e - f z,

e(ex - zx) = £e,

and

- z(ex - zx) ■ Jz.

Hence unless t = 0, I contains e and by the above, I * j.

Similarly unless 'S * 0, 1 « J. Assuming that I / J we find that

I can contain only elements of the form n x Z M - i V i»l r 1 1 But then I contains, for every j,

“ j - z)- 60

Since I J, e + z^ I and, for every J, we must have

This says that the vector ' *, H-n) is in the null space of the matrix ( X Since ( « y > la non-singular

B (0>* * *>0). This says that x ■ 0. Hence if I / J then I - 0.

Therefore J ia simple.

J does not have a Peirce decomposition with respect to the idempotent e since

(eje,u^) * ( 1 / M ^ i 0.

Indeed if a commutative algebra A has a Peirce decomposition with respect to an idempotent e then

*10 * eAio * V 5 ■ 0 and

*01 “ *01® " eAoi “ °* Moreover, it is implicit in (l6) that A ^ and axe suhalgehras when A is commutative. But then

A " * 1 1 * * 0 0 and, if A is simple, ve can only have A « A ^, in which case e is an identity element for A.

Kosler [l6] recently studied algebras which satisfy the so-called anti-flexible identity

(x,y,z) - (z,y,x). Assuming power-assoclatlvlty he proves that such an algebra Is

Lie-admissible and that every idempotent yields a Peirce decomposition. Be proves that simple algebras of this type are associative If + A ^ / 0 for ease idempotent e, and introduces examples of new simple algebras having A^Q + A ^ ■ 0. These are defined as the supplementary sum of n orthogonal subspaces

i ■ 1,* • *,n where has a basis { | 8ucl» that ei 2 2 is the unity of A,, u, ■ v, « 0, and u.v. ■ - v4u. ■ e, +• • •+ e i ll' ii ill n the unity element of the algebra. Hosier also uses such an algebra to construct another having nil radical (maxiawl nil ideal) zero but which is not a direct sum of simple algebras. This example is

Lie-admissible, power-associative, and has a Peirce decomposition.

It has no unity element.

Any set { e^, • * *'en t °* mutually orthogonal ldempotents in an algebra A is linearly Independent since if ot ^e^ +• • •+ e

■ 0 then e.( of ,e +• • •+ e ) * ,e. « 0 Implies <*. * 0 for x x i n n x x x i m i,» • • ,n. A finite-dimensional algebra therefore has a maximum

□Timber of mutually orthogonal ldempotents.

If an algebra A has no identity element then it is said to be of degree zero. The three-dimensional algebra B in Theorem 1 is of degree zero. If an algebra A has an identity element u then the degree of A is the maximum number of mutually orthogonal ldempotents whose sum is u. By the preceding paragraph we see that every 62

(finite-dimensional) algebra A baa a (finite) degree. The algebra

J is of degree two. Kosier1s examples may have arbitrary positive degree.

Ve proceed now to present more examples of simple Lie-admissible power-associative algebras having Peirce decomposition. These appear to be new.

Let ^ be a field with characteristic different from two.

Let A^ be the vector space over with basis

( v w * *'Vyi'* * *'yt} s,t 1, and for i = 2, • • • let be the vector space over with basis

n^ 1. Define the algebra A to be the orthogonal sum

A ■ A. +• * •+ A X B of the (vector) subspaces A^, where the multiplication in A^ is defined by the table

• • • Xjg see *1 * * * yk * * * *1 el el *k 0 0

0 XJ A Jk^el+zl^ V i tf» v (1/S|£ (v*n) XJ

0 yJ yJ

0 0 *1 *1 63 and Multiplication in for 1 > 1 li defined by the table

41 e e stf^ e • • Z1

0 •i ei (1/2>(Vik'n:5i( V ‘“)) 'e e (l/ 2 )(w1J- Z f(V zn )) *u l^)(V l (V ‘.)) e •

(l/^tv^te^ )) zi 0 Xi

The Greek letters in the tables are scalars which satisfy the following conditions:

(I) For all i and j,

*ij “ *Ji'

M-ij * ^ i '

°*i3 + *ij " /®ij + f i i y (H) If ( \^j) and are zero matrices then or 1J ij' and ^ij' or *’* not a *er0 Batrlx* (III) The a by (s + Ut) composite matrix

((X u X * ij )(<* )( )(/6ij )) has linearly independent (with respect to $ ) rows.

(IV) The (t + Us) by t composite matrix

((p-1(J)(•< ±j )T( * ^ )T( £ )T( fl±i )T f , where T denotes transpose, has linearly Independent rows. 6k

(V) For each k - 2,* • •,«, the matrix + Xj^) *■ noo-slngular.

For our demonstration of the properties of A ve shall take a simplified case; namely where o< ^ m “ 0 and

A j j ■ “ X [j ^ ■ <5"^ the Kronecker delta. These assuaptlons are stronger than conditions (I) through (v) hut are much easier to deal with In our calculations.

The algebra A has an attached algebra A ^ which Is the same vector space as A but with a multiplication (*) defined by

x * y - (l/2)(xy + yx) where Juxtaposition denotes the product in A. Examining the multiplication tables for A ve see that is the direct sum of subalgebras A^+ ^, i « 1 / • • ,m, each of which is a reduced central simple Jordan algebra of degree two. The identity element of A ^ is

ui " ei + *1 and this is also an identity element for A^. The element

u - u^ +• • *+ u^ is the identity element for A and for A ^ .

Every element of A may be written uniquely in the form

a ,ai +' • *+ \ with in A^. Noting that squares in A agree with squares in (+) AN ' ve have by orthogonality of the Aj, 2 2 2 a ■ a^ +• • •+ a^

- + r(ai)*ui) +* * ,+ + Thus we see that the subspace 65

m ^ ^ • a 1 +

Is a subalgebra, is associative, and contains A . By definition,

A is power-associative.

Observe that

and, for 1 > 1

u ' '"11 + ‘l’u * eiwil 6 Ai- 2 Thus u € A± for i » 1,• • .,m.

If I is an ideal of A then I is also an ideal of the semi-

simple Jordan algebra A ^ and must therefore be a sum of certain

of the A^. But if A^ C. I then € I and since u ^ for

i = I,* * ‘,m, it follows that the only ideals of A are 0 and A.

If i / j then by orthogonality

(A±,AJ ) = 0.

Moreover from the tables ve see

(Ai,Ai ) <$-un

so that ((A^,A^),Aj) = 0 and therefore

(1 0 8 ) ((a',a"),a-) + ((a",a'"),a*) + ((a"’,a' ),a") - 0

holds in A if and only if it holds when the elements a 1, a”, a"'

are all in the seme A^. For i > 1,

< V Al) £ ^ < & un

80 66

» 0 and (106) bolds. Thus ve fix our attention on the Identity (108) for elements of A^.

Let

a'- f.1+ £ li*t + with a similar notation for the elements a" and a'" of A^. Then

- ( i * ) £ - n j H K o a ~ • - \)

+ 11 < £ ‘ - fJlJ - U <£" • * n)]xi

t ♦ L |(V - £ ’)*\3 - nj(i" - f")]y,, J*1 and therefore

((a1,a"),a'" ) ■ /t(a',a",am ) £ u + £ L ( a ' , a ”,a",)x. n-2 n i-1 1 1 t + I n (a,,aH,a”*)y j-1 J J where

(>-(.• ,a",a"' ) . ( 1 / 2 ) ^ Z { [ ( £ ' - J '))l- ti

-[<*’ - £ - ^(-j" - f">] \ "i) i

JJa-.a-.a"') . - [( 6> - , •) J J - jjU " - ■$")](£"• - t h and

^j(a',a",af") - - [( 1' - £ ')nj * *\ j(j" - £")]( % - £'" ).

A short calculation reveals that

IT (a',a",a"' ) + tr (a",a,,,,a') + IT (a,M « 0 67 when IT it toy one of fju, or This says that (108) hold*

In A^. our earlier remarks this suffices to show that A Is

Lie-admissible.

We have demonstrated that A is single power-associative and

He-admissible. It is evident that A has Peirce decomposition with respect to the idempotent e^ and in fact with respect to every idempotent of the form

e » e^ + +• • •+ X n where is a subset of [2,* • *,m}. Indeed in the decomposition with respect to e we have

s

V*> - J*1£ ‘S-v

t ^ ( e ) - I U A j * l J

n V ( e ) * + Z. A, , A > 1 and Agg(e) is the sum of $ z.^ and those sub spaces A^ whose indices run over the complement of the set * *>^n \ 10 l2>* * *>a\*

There is an interesting special case Imbedded in the preceding.

The algebra

C ■ Ag +• • •+ A^> m >,"$» as vector space, but with the multiplication table redefined for each A^ so as to delete e^ + as an entry, is easily seen from 66

the proofs given for A to have the properties of simplicity, pover-assoclativity, and Lie-admissihility. La fact C has the

interesting property

((C,C),C) - 0.

Moreover relative to every idenpotent e ■ £ u. C has a Peirce 3 3 decomposition

c - cii + coo where C-. « J A. . 3 XJ The algebra G arising in Theorem 2 la another special case

of the algebras presented here; namely the case m ■ 1. When m ■ 1

one may take either s ■ 0 or t a 0 (but not both) and still have

a simple algebra. BIBLIOGRAPHY

1. Albert, A. A., On Jordan algebras of linear transformations, Trans. Amer. Hath. Soc. vol7 59 (1944) pp. 524-555.

2. ______, A structure theory for Jordan algebras, Ann. of Math. vol. to (19^7 ) pp. 546-547.

3»______> On the pover-associatlvlty of rings. Sunma Braslllensls Mathematical vol.2 (1948) pp. 21-32.

4. . Power-associative rings. Trane. Amer. Math. Soc. vol. 64 (1948) pp. 552-593.

5* » Almost alternative algebras, Portugal!ae Math. vol. 8 ( W ) PP. 23-36.

6. , On right alternative algebras, Ann. of Math. vol. 50 f W ) PP. 318-328.

7* ______t Structure of right alternative algebras. Ann. of Math. vol. 59 (1954) pp. 407-417.

8. Artin, E., Hesbitt, C., and Thrall, R., Rings With Minimum Condition, Ufciversity of Michigan, 1944.

9* KLeinfeld, E., Simple alternative rings, Ann. of Math. vol. 58 (1953) PP. 544-547T

10. ______, Rings of (X ) type, Portugal Math. vol. 18 (1959) pp. 107-110.

11. , Simple algebras of type (1,1) are associative, Can ad. J. of Math. vol. 13 (1961) pp. 149-158.

12. ______, Assoclator dependent rings, Archlv der Mathematik vol. 13 (19^2 ) pp. 203-21 2 .

13. ______> and Kbkoris, L. A., Flexible algebras of degree one. Proc. Amer. Math. Soc. vol. 13 (19 6 2 ) pp. 891 -8 9 3 .

14. , Rosier, F., Osborn, J. M., and Rodabaugh, D., The structure of associator dependent rings, to appear in Trans. Amer. Math. Soc.

69 70 15- Kokoris, L. A*. A class of almost alternative algebras, Canad. J. Math. vol. 8 pp. £$0 -2 5 5 .

16- ^ *”• *”• ^ 3°e‘

17- . Flexible nllatahle algebras. Proc. Amer. Math. Soc. vol. 13 (l$G2j pp.

18. Kosler. F., On a class of non-flexible algebras. Trans. Amer. Math. Soc. voT:“162 (1$65)' pp. 2 ^ - 3 1 8 . -----

19* Laufer, P. J., and Tomber. M. L., Some Lis admissible algebras. Caaad. J. Math. vol. 14 (1 9 6 2 ) pp. 287-292. ----

20. Manerlrl, C., Simple (-1.1) rings irtth an idempotent. Proc. Amer. Math. Soc. vol. 14 (19 0 3 ; pp. U O - 11 7 .

21. Oebake. R. H.. On flexible algebras. Ann. of Math. vol. 68 (1958) pp. 221-255:------

22. Weiner. L. M.. Lie admissible algebras. Univ. Nac. Tucuman Rev. Ser. A. voTTT T a ^ f J'pp';' ?6 ^ . AUTOBIOGRAPHY

I, William Eugene Coppage, was Born in Geary, Oklahoma,

November 2k, 193^. I received my secondary-school education in Galveston, Texas, graduating from high school in that city in 1951* From 1951 until 1956 I attended the Agricultural and

Mechanical College of Texas, receiving the B. A. degree in mathematics in 1955» and the M. S. degree in mathematics in 1956.

During the period 1956-1959 I served as a Lieutenant in the United

States Air Force; my duty assignment vas Instructor in Mathematics at the Air Force Institute of Technology, Wright-Patterson Air

Force Base, Ohio. From 1959 through 1962 I vas a graduate student at Ohio State University. During this period I vas an Assistant

Instructor in the Department of Mathematics for two years and a

National Science Foundation Cooperative Graduate Fellov for one year. I also spent tvo simmers on National Science Foundation summer fellowships. The sumner of 1962 I spent as a research mathematician for the Institute for Defense Analyses. I vas a part-time Instructor in Mathematics at Syracuse University during the 1962-1963 academic year.

I have accepted an invitation to participate in the 1963 summer research project at U.C.L.A. sponsored By the Institute for Defense Analyses. I have also accepted an Associate

Professorship in Mathematics at Indiana State College for the

1963-196^ academic year. 71