CUBO 6, 13-18 ( 1990) 

R11cilbldo14qo•lo 1989 

ON NUCLEAR BERNSTEIH ALGEBRAS 

by 

Rodolfo Baeza V. 12 

O. -Abstract. In [ J] P. Holgate preved that the core of any 

orthogonal Bernstein algebra is a special train algebra and 

consequently a genetic algebra . Let A be a Bernstein 

algebra. Then A is a nuclear algebra if and only if the ocre 

of A is A. The present paper preves that the core of a 

Bernstein algebra is a special train algebra. 

1.-Prelimlnaries. In the following let K be an infinita 

commutati ve f ield whose characteristic is nei ther 2 nor 3. 

Let A be a commutative nonassociative K algebra. For every 

sequence 

principal 

8 1 •ª2, • • • 'ªk 

product a 1 

of k elemente of A define the 

of all finita sume of products 

with a 1= a 1 . g'k is the set 

ak of k elemente in BU. g'k 

is called principal power of B. We say that aeA is nilpotent 

if there existe ne~ such that a"""º, A is nilpotent if there 

existe teli euch that At• O , If all elements of A are 

1 J FOMotCYT 227-89; 01\Ffll 710-881 CCINT-USP-BRAZIL 

;ll l m APPPA !JI L, • .A . A . 



PAC.14 C U B O 

nilpotents we say that A is a nilalgebra. A la called a 

Jordan algebra if xy=yx and x 2 (xy)=x(x2y) Vx,yEA. A.A.Albert 

preved that all finite dimensional Jordan nilalgebram or 

characteristic ~ 2 are nilpotent. 

Lemma l. Let A be a commutative algebra ,char(A)•2 and x3•o 

't'xEA. Then A is a Jordan algebra. 

Proof. The identity x 3 "" (x+y) 3 • (x-y) 3 • O impliea that 

o ""' (x+y) 3 -(x-y) 3 = 2(2x(xy)+x2y). , But char(A) • 2, hence 

x 2y=-2x(xy). Replacing y by xy we obtain x 2 (xy)•-2x(x(xy)) • 

x(-2x(xy)) "" x(x2y), Le. A is a Jordan algebra. 

Let (A,w) be an (n+l) dimensional commutative non· 

associative baric K-algebra where w:A ---+ X is a waight 

function. (A,w) is called Bernstein algebra iff 

(x 2 ) 2 ""W(x) 2 x 2 , VxeA. In any Bernstein (A,w) algebra th• 

nontrivial homomorphism w is uni(¡uely determined, and A 

possesses at least one non tri vial idempotent element ec.I, 

( see { 5] ) • The e-canonical decomposi tion of A is Xe•U•V whan 

U• { yeker { w) : ey• 'iy) and v-{ yEker ( w) : ey-o ) . The subspacea U and 

V satisfy the fundamental relations u2~, uv~, v2~, vv2•o, 

uY-o and the fundamental identities u~-o, u1(u 1v 1 )•O, 

u 1 (u 2 u 3 )+u 2 (u 3 u 1 }+u 3 (u 1 u 2 ) • o (Jacobi's identity) 1 vu 1tU, 

v 1EV, and (xy)(zt)+(xz)(yt)+(xt)(yz) •O Vx,y,z,teN • ker(w) 

• U• V. 

Lemma 2. Let (A,w) be a Bernstein algebra. Then N•Xer(w) and 

its princ ipa l powers are ideals of A. 

Proof . N is an ideal of A, since the kernel! of a 

h omomo rph i sm is an ideal. P.Holgate preved [3,p 615), that 

all yE.H ea t isf y e( y 1 y2 ) ... <iy 1 )y2-(2ey 1)(ey2 ).Thua if y 1.Y1 



e u. e o PAC.15 

and y 2EY2 where Y 1 and 'l2 are L8 - invariant suba paces of N, 

then the product 'l 1Y2 is L8 -invariant. For all (ae+u+v)EA we 

have (ae+u+v)~·'. ae(NNtc) + (u+v) (NNtc). For the induction 

hypothesis, let ua suppose that Ntc is an ideal of A . But we 

know that N is aleo an ideal of A, hence they are 

L8 -invariant subapac es of A. Finally, it is concluded that 

(cxe+u+v)Nk•t, Nh1 +(u+v)Ntc • Nk•t. 

2. - Bernatein Special-t.rain algebras. Let { A,w) be a baric 

K-algebra . (A,w) is callad a Special-train algebra when the 

principal po\lfers of ker(w) are ideals of A and ker{w) is 

nilpotent . 

Proposi tion l. Let {A ,w) be a Bernstein algebra. If x3•0 , 

'YXEN then A is a special train algebra. 

Proof. N- ker(w) is a subalgebra of B , hence N is a finita 

dimensional commutative algebra with x 3•0 'YxEN. By lemma 1, 

N is a Jordan algebra, and x 3• 0. Thus N is a finita 

dimensional Jordan nilalgebra and by Albert'& Theorem quoted 

on p. 96 of ( 4] 1 N ia nilpotent. Whan Cbar(IC) • O wa can 

use the result of Geratanhaber " All nilalgebra of finita 

nilindex ie nilpotant" (p 53, 5), for eetabliehing the tact 

that H ia nilpotent. By lamma 2 the principal powere of N 

are ideal• of A, hence A i• a apecial train algebra . 

Corollary If A ia a Bernatein-Jordan algebra than it ia 

a special train algebra. 

Proof. It will be proved that x 3•0 Vx•N. Let x•u+v be in 

Then By the 



l'AG . 16 CUBO 

fundamental relations and identi ti es in Bernatein algebraa 

we can vrite x 3- u 2 v+v3 +2v(vu). In [l] we proved that A b a 

Bernatein-Jordan algebra iff v2- o and v(vU)•O VV•V. H•nce 

x 3• 0, VxEN and A is a special train algebra. 

Lea.a J . Let C• Keeuau2 be a Bernstein algebra, N•Ueu2 .. o and 

D • Ann {N) - {XEC: x N""O), then C/D is a Bernstein algebra. 

Proor , It x-ae+u+u1 u 2 be in D, then xU•O impliH 

aeU+uU+(u 1u 2 )U• O Hence uU • o, being the only tenr. on th• 

lett that lies in U2 • By Jacobi ' s identity and uU•O we have 

utf• o ; thua UED. Consequently ae + u 1 u 2.0, and ueu3+ 

(u 1u 2) u3•0 for u3E u . But icxu3• -u3(u1u 2) impliH 

iau~·-u3 ( u3 { u1 u2 )) • o. If a .. O then u ~-o for u 3EU . Thua O 

• (U 1 + u 2 )
2• u: + 2u 1u 2+u: and u 1 u 2• o. Hence ae • D. But • 

ti D becauae U .. o , a nd eu • i u for u E U. That is, a • O and 

x•u+u 1u 3E N with u, u 1u 2 eD . Having, x • u+u1u2eD, y-ae+ntC 1 

n• I, we obtain xy• (u+u 1u 2 )ae • aeu • iau. But u•D ; bance 

xyd> and D is a n ideal of c . That ie , (C/D , W') h 1 

Berna te i n algebra, aince it is a homomorphic image ot c . 

Naturall y w' {c+D) • w(c) defines the weight function in C/D. 

IAt ua no te that x• c +D • ker(w') if f w(c) • O iff c • ker(w) I 

thua ke .r (1..1 ' ) • {c+D:cEN ) • ii • ü eV. 

Pr opo.t tion 2 . Let A • KeeU• V be a Berna te i n a lgebra, and 

C• IC tMU• tf•JC be i t s core . Then e is a apec i a l tra in algebra . 

Proo f . Pi rst , shall prove tha t C/ D i s a s pec i al train 

a lgebra . By lemma 3 we kn o w that C/ D i s a Bernstein algebra . 

It only rema in s to prove tha t x 3 • O Vx Eke r (w ' ) and then UH 

Propoai t lon 1. Let x • n+D be an el e ment ot ke r {w') , thua n • 

u-tu , u;t . Uaing the f undamental relat i o na and i dentitie• in 



OH HUCl.EAn .. e u B o PAG.17 

the Bernstein algebra e, 

2(u 1 u 2 )[u(u 1 u 2 ))EU. V
2s;:U, V2U=O and Jacobi's identity imply 

v 2 s;:o and, consequently u 2 (u1 u 2 )eD . Furthermore the second 

and the last fundamental identities imply that 

2u3 ((u 1 u 2 )[u(u 1 u 2 ))) ""-2(u(u 1 u 2 )][u 3 (u 1 u 2 )] 

= (uu 3 )((u 1 u 2 )(u 1 u 2 ))E(uu 3 )v2-o vu3EU, 

because v 2g>. Then we have (u 1 u 2 )(u{u 1 u 2 )) E Uf\Ann(U), and 

by Jacobi's identity it is in D. '.1'hat is x 3=n3+D=-D and C/D 

is a special train algebra. This shows that r E N existe 

such that Q. (ker(w 1 ) )r 

Nr+1=NNrs;: ND .. O. By lemma 2 we know that N 1 is an ideal of 

e far all integer i>O. But N is nilpotent, hence e is a 

special train algebra. 

Remark l. It was preved in (2} that the orthogonality is not 

a necessary condition to be a special train algebra. The 

present work provee that the orthogonal hypothesis can be 

removed from Proposi tion 4 of [3) • 

Remark 2. It is preved in (5] that the core e of a Bernstein 

algebra A satisfy e = A2 • Let A = KeeU•V be a Bernstein 

algebra, the.n A 2 mA implies V=if ,and the meaning of our 

proposition 2 is that "Every nuclear Bernstein algebra is a 

special train algebra". 

References . 

(1] M.T.A.lcalde, R.Baeza, e.Burgueño, Aucour des algebres de 

Bernstein, Arch . Math., Vol. 53,134-140 (1989). 

( 2] Baeza R. ,A non orChogonal BernsCein algebra vhich 

1.~ R SnJ':r:1R1 trR1n Rlrtt":hrR. A.tris dri X F.scnlri dA AlaAbrA