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Construction of vertex operator algebras fromcommutative associative algebrasChing Hung Lam aa Department of Mathematics , The Ohio State University , Columbus, Ohio, 43210Published online: 27 Jun 2007.

To cite this article: Ching Hung Lam (1996) Construction of vertex operator algebras from commutative associativealgebras, Communications in Algebra, 24:14, 4339-4360, DOI: 10.1080/00927879608825819

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COMMUNICATIONS IN ALGEBRA, 24(14), 4339-4360 (1996)

CONSTRUCTION OF VERTEX OPERATOR ALGEBRAS FROM COMMUTATIVE ASSOCIATIVE

ALGEBRAS

Ching Hung Lam

Department of Mathematics The Ohio State University

Columbus, Ohio 43210

Abstract

Given a commutative associative algebra A with an associative form ( , ), we construct a vertex operator algebra V with the weight two space V2 2 A. If in addition the form ( . ) is nondegenerate, we show that there is a simple vertex operator algebra with VL 2 A. We also show that if A is seniisin~ple, then the vertex operator algebra constructed is the tensor products of a certain number of Virasoro vertex operator algebras.

1 Introduction

The theory of vertex operator algebra has undergone many developments in the last few years. Some new esamples have been discovered. One impor- tant example is Frenkel-Zhu's construction [6] of vet tes operator algebras from the highest weight representations of affine Lie algebra. In their paper 161, Frenkel and Zhu also glve a similar construction for Virasoro algebras. Following Frenkel a n d Zhu. Lian [lo] constructed another class of vertex op- erator algebras associated with Lie algebras having nor~degenerate associative

4339

Copyright 6 1996 by Marcel Dekker, Inc.

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forms. In fact. i t is well known (cf. [I] and [5]) tha t if the weight zero space of a vertex operator algebra is one dimensional , i.e. 1.b = @ . 1 with the vacuum elerrie~it I , then the weight one space 1; has a Lie algebra structure with t,he Lie bracket given by

and an associative form ( , ) given by

(a , b ) . 1 = ~ ~ 6 , for a,b E \; (2)

Using this fact, Lian [lo] actually classified a class of vertex operat,or algebras which are generated by the weight one space. On the other tiand, if' 1.I = 0, it is shown by Frenkel, Lcpo\vsky and I I e u m a n [5] tha t tlie weight t,wo space 11; is a commutative (rlonassociative) algebra with t,he prodi~ct given by

Here. an algebra is simply a vector space 1' equipped with a bilinear m a p 1. x 1.' i 1.. Moreover, we can define an associative f o r n ~ or1 1.; with

( u , b) . 1 = a 3 b , for u , O E I/;, (4)

Therefore, it is natural t o ask if the converse of the above will also holtl, i.e.: If a cornrnutative algebra A wit,h an associativr f'orm ( . ) is given, does there exist a vertex operator algebra I/' such that I.', = 0 and t,he weight two space V2 is isomorphic t o A? The most interesting example is t,he moonshine module, which has I/; = 0 and the weight t,wo space I.; isomorphic t o the Griess algebra - the 196884 di~nerisional algebra constructed by Griess [7] for realizing tile hfonster simple group.

In this paper, we shall construct a vertex operator algebra from a com- mutative associative algebra A. Our construction is very similar to Frenkel- Zhu's construct,ion for Virasoro vertex operalor algebras. In rase t,hat A is also semisirnple, we show tha t the algebra constructed is, in fact, the tensor product of certain number of 1'irasol.o vertex operator algebras.

We shall begin with tlie main definition of' vertex operator algebra and its representatio~is. Some important consequences will also be mentioned. An algebra .4 (an affiriizatio~i of A) will then be constructed from a commutative algebra A. We shall show that '4 is a Lie algebra if and only if A is a n associative algebra. We shall next construct a vertex operator algebra V (A) using the highest weight representations of .j. If the ass~c ia t~ ive form on A is ~ionclegenerate. we shall fillti an irreducible (or simple) quotient of V ( A ) which has the n ~ i g l i t two space isomorphic to A. Finally, we shall show tha t tlie vertex operator algebra constructed is the tensor product of certain ~ i u ~ n h e r of 1'irasoro vertex operator algebras if .4 is also semisirnple.

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VERTEXOPERATORALGEBRAS

2 Definitions and Basic Properties

2.1 Definitions

We shall s ta r t by reviewing tlic definition of \.cSrttJx operator i d ~ e l m arid some of' its important consequences. Most of the results call br found In 1.11 and [j]. For more details, readers should r e f r t o these two books.

Definition 1. A oertez operator algebra rs a Z-gr.aded vector spctce

.such that din1 T ;, < x for. r r E Z. (6)

\ ;, = O fbr. 11 sqjjic,~errtly s111al1 , ( 7 )

und wrth t,wo rtz.strnguzshec1 horrroye~wous vectors 1. u E I . , .utrtr.qf:,yn~i the followzrq conditzor~s for 11, E I .:

w h e ~ e J(z) = CnEZ zn and the e q r e s s z o n s s w h as d ((zl - i2) /zo) o m to be expanded as formal powe7 serzes zn positzve powers o f the seco~rd ~ l r c i i i l e , z2 f07' h' ( (z i - 22) / 7 " ) . W h e n ench ezpressron zn (18) rs tr.pplrf:d to tr11.y eleir~,erlt of I - , t he coef iczer~t of each 7nonorr~rcd z n tire f o r ~ n n l vc1ma.blc.s r.s o ,firrrte .SU,IIL.

Moreover,

for nl, 71 E Z, where

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d -1. ( u . :) = I* ( L (-1) I , \ 2 ) . ds (17)

T h i s completes t he de,finitzon a n d the serzes I r ( . c . 2 ) z.s culled the ,c~ertez oper- utor assoczc~ted wzth u .

Definition 2. A yr.cl,ded vector. s p c e AII = CIIEG AI,, zs (1 \ 7-'rr~od~i1e tf there rs 11 linear m0.p

1 z ) : I + I [[z. z-I]]

Cleaxl?;. I,. is a I.-niodule itself. Kc call th is t h e atl,jo~nt niotlule of I:

Definition 3. A v e ~ , t e z operutor ulqebru I . z s .szrryle $1. (us otljo,irrt v m l u l e j has onhj 0 and I,' us rts submodules.

2.2 Basic Consequences

X P S ~ . we shall w n s i t l r r thv ,Jarobi itlent,ity (12). B?; taking rrsi~lues wi th irlspcc.t t o zo m(i z,. n.r shall obtair~ the f'ollo\~ing f'o~rnulas:

( I ) (Coxn~~lut,ator Formula)

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VERTEX OPERATOR ALGEBRAS

and (11) (Associator Formula)

Moreover, with the L (-1)-property (17), we car1 show that (111) (Skew-symmetry)

Y (a , 2 ) b = r Z L ( - ' ) l r ( b , - 2 ) a ( 2 0 )

1 i + , l + l a,b = (-1)"" b,,a + - (-1) L (-1)' (b,,+,a) , for 71 E Z (21) z > I 2!

3 Vertex Operator Algebras with Vl = 0

Now, suppose t h a t V is a vertex operator algebra with

Then by formula (21), we have

for all a . b E 1; since bl+,a C 'I 1-, for 1 > 1 and L (-1) 1 = 0 Thus. 1; forms a commutative algebra with its product g1wn b\

On the other hand. s i n c ~ we have

wla = alw = L ( 0 ) a = 2n, for n E l;,

we see t h a t w/2 is a n ideritity element in V2 Aloreover, we can clefirie a form on LT2 by

(ci,b). 1 = n3b for a , b E 1;. (23)

By (19), we can show the form ( . ) is also associative, i.e

(ab, c) = (a . bc) for a . 6 E 1,;

Therefore, I' is a commutative algebra with an associative form as s tated ir i

151 .

Lemma 2. Let 1)' be a vertex operator algebra such thud

V, = 0 for n < 0, I/O = @ . 1, and 1; = 0. (24)

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where 1

I".% ker L ( 1 ) n 13 (~nd tu,' = - ( q , b - boa) . 2

Proof: It is well known (cf. [3] and [ 8 ] ) that i f 1. sntisfies coliditior~ ( 2 4 ) .

1 , = k e r L ( ) , I L ( - I ) , for all 11 # 1.

In particular. we have r r o b = t + L ( - 1 ) r

for some t E ker L (-1) and r. E since nob E 1 >

Applying L (1) to both sdes of'eq~~at,ion ( 2 5 ) . we have

L (1) nab = L ( 1 ) L ( - 1 ) r .

Since 1'; = 0. wc h w e L ( 1 ) I , = 0. So,

[ L ( 1 ) : ao] l ) = [ L ( 1 ) . L ( - l ) ] ~ .

;\loreover. I)? the co~nniutator formula:

we have

[ L ( 1 ) .(LO] = ( L ( - 1 ) a ) > + 2 ( L (0) a ) , + (L ( 1 ) a)"

= -2u , + 2 ( 2 4 )

= 2a ,

Note that L ( 7 1 1 ) = w,,,, and L (1) n = 0. Tlius, we haw

2ulb = 2 L (0) r = 4r.

and so 1 1

1. = -a,b = - ( a b ) . 2 2

Thr1.efor.e.

oob = t + :L (-1) (ab) 2

011 thc other hand. by the skew-sym~netric property (21) , we have

bo(t = -ni,b+ L ( - l ) ( n l b ) 1

= - f - ; L ( - l ) ( n b ) + L ( - l ) ( n b ) 2 1

= - t + ; L ( - l ) ( n b ) . 2

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Herice, 1

t = - ((lob - boo) 2

as required. I

Proposition 3. Let I . be (I, ver tex opelator algebra wzth the p,i.oprv.tres that

1,; = O for71 < 0 , I /a=c. l , a n d \ ; = 0 .

T h e n , for every a , b E 1.5, we have

where Fb = (oob - boa) .

P m o f . Because of the commutator formula (18), we know t,liat for. evcry (1, b E /5> me have

In atidition, we know that

1 [a,,,, b,,] = ( ( ~ 0 ~ ) , , + ~ ~ + ~ 7 ~ ( ~ b ) , , , + , , - ~ + - I n ( ~ r z - 1) (7rz - 2) ( a , b ) L + , , - 2 0 . (26)

G

1 = (tf13% 5L ( - 1 ) ( a b ) ) rrl+ri + 7 1 ~ ( d ) , , ~ + , ~ - ~

1 1 = (ta",,,+, + I (tn - 7 1 ) (ub),,,+,,-, + i l t (in - 1 ) (tit, - 2) ( ( 1 , b) 6,,,+11-2,0

6 as in the proposition.

In general, it is difficult t o determine the propert,! of' f : ~ic~~ertli ' less. nrc have the following result.

Proposition 4. Let 1' be a ver tex operator algebrtr .mch th,(~.t

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4346 LAM

2.( , . Pb = 0 Jbr d l ( L O E 1,;

t h e n 1 rs assocrutrve us a.n algebru wzth the prod,u,ct pverl by/ (16 = a l b and ( ( 1 . b) . 1 = nab.

Now let us assume 771 = n . B y ( 2 8 ) ; we have

Therefore, if k # irr or 1.

ilrld SO

b ( c a ) = u (OC) = (bc)u

as u- , . 1 = v for E 1.-. Theref'orc, 1.5 is associative. I

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VERTEXOPERATORALGEBRAS 4347

4 Affinizat ion of Commutative Associative Al- gebras

U~lless otherwise stated, we shall assume A is a c:ommutitt~vc associativc algebra with an identity e and an associative fbrln ( , ) .

Remark 2. Since .-I zs com7nutatzvc a,ud hu,s a , ~ zdentit?), oll c~ssocri~.tz~ue fbr.rrl on A are symmetrzc.

Definition 4. Let A be a c o n m ~ ~ t a t z u e ( riot necessi~rily assocrntrvr) ulgeb,ru

T h e ufinzzutzon of A zs the ulgebru.

with the bracket gzven by

Proof. Because of Proposition 4 , the "onlj. i f " part IS (,as>. to sl~on. by rcmar11- ing u@t rn = I L , ~ ~ + ~ . Note that in the proof of Proposition 4, no other relations of thc \ w t ~ x olwrator is l w t l est:clpt for ( 27 ) . Kcst , n.e shall h11on. tlw ..if"' pa r t , i.e. .4 is a Lie algebra if .-1 is associativc. Since .4 is commutative and the form ( . ) is sgmmrtric, wt. can easily show that

[:I.. y] = - [y. .I.] for all .r, !, € -4. Therefore, the bracket [ . ] is ar~t isy~rimetr ic . Next. wc s l~al l show the .Jat:obi identity fhr Lie algebra. Ob~ious lg ,

Therefore, it. suffices t o show

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But . by forniula (29) .n.e liavc,

(n 8 tr". (6 4 1". c 8 t ' ] ]

1 1 = [ ( I @ t"'. 5 ( 7 1 - k ) 6 r 8 t 7+% + (1,' - r ~ ) h , ,+k ,o ( D ) r ) c ] -

1 1 = - ( 1 1 , - X.) { - ( r 7 1 - 1 1 - A : ) (l(bC) @ t"'+"+k

3 2 1 +i ( rn i - m) b ,n+."~ ,~ ( ( I . 6c ) c )

1 + - ( 1 7 , - i i ) ( k : . ' - X , ) t i ,,,,,, + A , o ( 6 , c n ) c

12 ;\a .4 is assoclativr and co~lm~rl tat ive.

( L ( L C . ) = 6 ( c n ) = c ( ( L O ) .

Bcsides, the form ( . ) is associative. So, we also h a w

( n , 6c) = (6. ca) = ( c , ( ~ 6 ) .

Therefore:

[ o m tT7' , [ O B t n . i g t * ] ] + [ ~ g t 7 1 , [car*, n c ~ ~ t r ~ ~ ] ]

+ [ ( 4 I * . [o Q IT". b 8 t 7 ' ] ]

1 + ( n 2 - m 2 + - I & ) } n (hc) 8 tTrL+"+' + - { ( n - k ) (711" ~ r * ) 12

+ ( k - 7n) (n3 - n) + ( m - n ) ( k 3 - k))bm+, ,+k .O ( a , Dc) c 1

= -{(I) - k ) ( 1 n 3 - m ) + ( k - ,rn)(n3 - , rr ) 12 + ( m - 17,) ( k ' - k:) }h , , ,+ , ,+k ,~ (a , Dc) c .

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VERTEXOPERATORALGEBRAS 4349

Now. it re~nairis to show

If m+ i t -t k f 0 then 6,n+n+kx0 = 0 arid hence S = 0. Now suppose n + 11 + k = 0. Then m = - rz - k and we have

1 = -{(n - k ) ((-n - k ) 3 - ( - I ? - k ) ) + ( k - [ - I L - k)) ( t t . ' - I L )

12 +( ( -n - k ) - n ) ( k 3 - k ) )

- - 1

- { ( - I / ' ' - 3r: 'k + 2711.' + A,.' + 11' - k 2 ) + (2krt:' - + - 1 2

wi th

a n d '-iO = '463 to $ @ C .

Lr t b = .-I+ @ .30

Tlleii, I\ (A) is a g r i l d ~ d . i - m o t l u l ~ with thc grading i ~ ~ t l u c c d by .-I. \'I:' define a ( n ) = (1, @ t" on 111 (.A) and define the weight of' an clemc~it to bc the negative degree,

By Poiiicare-Birkhoff-Witt Tlieorem, we Itriow that

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Lemma 6 . n ( - 1 ) . 1 LS ti hlghrst ~r~rzght uector oJ I11 (.4) for (111 (1 E .-I

Proof \Ye need to prove that

6 ( 1 1 ) n ( - 1 ) . 1 = O for all 6 E .-I, n > 0

But

b ( 7 1 ) ( 1 ( - 1 ) . 1 = [ l I ( l l ) , n . ( - l ) ] ~ l + ~ ~ ( - l ) b ( r l ) ~ l 1

= - ( l l + 1 ) ( h 1 ) ( 7 ~ - 1 ) 1 2

S ~ n w rl > O, we have 11 - 1 >_ O So,

and thus a ( - 1 ) . 1 is a highest weight vect,or.

Suppose tha t I is the , ~ - ~ n o c l , ~ l e generated by { a ( - 1 ) . 11 n. E . A ) . B y Lenima 6, I does not contain 1 arld so it is a proper subniod~lle of 2 1 1 ( A ) . 111 atlditio~i, the PBW theorem implies that

It is clear t h a t I is also graded sutxrlodule, again with the grading induced by 2.

Define a quotient module t)y

Then. we have = 0

and I ' (.A), = ~ p u n { u ( - 2 ) . 1 / a E A} S A as a linear space. (34)

If there is no ambiguity, wr shall usually write C' ( A ) as I;..

5 Vertex Operator Algebra Structure of V ( A )

In this section, we shall define a vertex operator structure on 1. ( '4). O u r main idea comes from Frenkel- Zhu [6]. 111 fact, some results are essentially proved in (61 but we modified them to suit our cases. In order t o simply some of the proofs. wr shall also ernploy several telminology of H.S.Li [9].

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5.1 Vertex Operators

W e shall start with a tlefinition clue t o Li.

Definition 5. A f o ~ m a l serzes (L ( 2 ) = ( I . , ~ Z - " - ' E ( E I ~ '11) [ [ z . z p ' ] ] rs culled a weak vertex operator o n a v e c t o ~ , spocr: r, f for. ~ , 1 1 , ? / '11 E dd. U , ~ U = 0 for 72 suJlfj~rentl?j larqe.

Following Frenkel and Zhu [ G I . a I~ullet upcrat1o11 1s (1di1itd as f'ollon.~:

Definition 6. Suppose 6 (2) = X,,,, 6,!i.-"-1 E E l 1 0 T .[[:. z - l ] ] rs oi cutwl~ vertex o p e m t o ~ and a ( 1 1 ) , 7 1 E Z. 2s an .A. D ~ f i l l e

u ( n ) 6 ( 2 ) = ( ( 1 ( 7 2 ) L ( Z ) ) , , ~ z - ~ - ' rnEZ

= ResW ( a ( w ) 6 ( 2 ) 1 (u - z ) " ~ ' - 6 ( z ) ( I (11.) I :,,,. ( t r - z ) "+ ' )

Proposition 7 (Frenkel and Zhu). For ( I E -4.

ruhe,re 1 zs the ~den t r t y e r ~ r l o r r ~ o ~ p i ~ r ~ r r ~ of' 1 (.A)

Proof. B y Definition 6 ,

!L+ 1 , l + l If n > -1, 1 ,,., (tu - s) = I , 1 - ) . Tlicl.clfi)r(,. ( t ( 1 1 ) 1 = 0

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where 6 ( - " - L ) ( 2 ) is the ( - 1 2 - 2)-th der~vat ive of b ( z ) Hence,

as required. I

Now. \ye can define the vertex operators uslng the bullet oprrittions

Definition 7. T h e verte.c operators on 1. zs (1 luteor rrlup

.su.rh thot l ' ( l . 2 ) = I

and 1- (a i ( - T L , ) . . (1, ( - i t , ) . 1 . Z ) = nl ( - 7 1 1 ) . . . ti, ( - 7 1 , ) 1

Remark 4. By P r o p o s h o n 7. w e have

f o r every n E .A and n 2 2

Tlle~c. d~ t s a n algebra z son~orphz sm w l w h preserves the assoczatzue f o r m ( , ) .

Proof'. \,Ye already kliow by (31) that 4 is a linear isomorphisni. So. we only lirctl t o show q~ is also an algebra Iiomo~riorphism. However, b ~ , Proposition 7. Kc hit\.?

1. ( ( L ( - 2 ) . 1 , z ) = (L ( z ) = c a (7%) z - " - ~ for u E '4. nEZ

Therefor?, u ( r n ) = ( n (-2) . I),,,,, and we have

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Thus, is an algebra I lorno~noqhisrr~. Uorrover.

(@(a) .d(D)) . 1 = (a(-2) . . 1

= n (2 ) 6 ( -2) , 1 = [ ( I , ( 2 ) . 6 ( -2)) . 1

= ( c i . h ) . l .

Hence, I$ preserves the assoc~ativc f'orrri. ;~lso. I

Proposition 9. For 1, E 1; ,I:,, . 1 = O f i r . 71 > 0 : c.., . 1 = 1 , ( I , I L ( ! wt o,, = 1vt 1. - 11 - 1.

Proof. \IT? shall prove tlicl 1)~oposition t ~ y intluctioli on tlic wc$$t of' I:. Clearly, tlic proposition holds f'or l i = 1 . Solv supposr the p~ul)ositioli holds fbr h E 1,' arid let o = rr ( - I L ) 11 ~ i t h 71 > 1, (L E .4. Then.

-

( 3 7 ) Sirice a ( 1 ) . 1 = 0 for L 3 -1 itntl 6, . 1 = 0 f'or ~1 2 0 . Tlirwf 'or~. n.r Iiave on, . 1 = 0 for all 111 > 0. hloreover, if 7n = - 1 . w havr I * - ] . 1 = n ( - 1 1 ) b I . 1 = ci ( - 7 1 ) 6 = v . Colisidcr t h e right halid sitlr of' (37). wt) 1.2111 s rc that both (1 ( - 1 2 - i) b ,,,, , and 0 , , , - , , - h + , ( L ( 2 - 1) l iaw w i g h t I I + I I ~ b - 111 - 1 which is mdepertdcnt of 1 . Theref'ore.

cl't c,,, = I I + 1c.t 6 - r r ? - 1 = cot a - 111 - 1.

,with c = 8 ( 6 . . e) . Moreover, u~r : h a u e

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[ L ( 1 7 1 ) . L ( l l ) ] = [2? ( 1 7 1 ) . 2c ( I ! ) ]

1 1 = 4{5 ( 7 7 ! , - I ! ) r ( 1 7 7 , + t , ) + - ( 1 1 1 , J - 7 7 1 ) ( c . P ) d,l,+,,,(,]

6

In addition. nrc Iiaw

Proposition 11. For eorry 1: E 1 ..

Prnoj. Again, \IT shall use induction on w t 7; It is trivial i f c = 1. .-\ssnme for so~rir. 6 E 1 -.

[ L ( - 1 ) . / I , , , ] = - I U 6,,1- I for n! E Z.

Consider 1: = ( 1 ( - r ~ ) 6 ; a E .-I, 17 < -1. Then.

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So, by change of indices, we obtain tha t

Similarly, we have

Thus.

as required.

Corollary 12. For, every u E 1 - ,

5.2 Jacobi Identity

Sow. we are ready t o prove the .Jacobi idmt,ity of' I (.-I). \\'r sl~wll I~asically fhllow the approach of Li in [9]. For con~pleteness, some results of Li will also he mentioned here.

The following fact is first observrtl by Dong and Lcponsky [2].

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Proposition 13. I n the rlcfinztzon of uerte:c operator algebrcr, the Jacobz identity can be equzz:dently s~~bstzt lrted by the follo~uzng corn,mutatzr~rty coridz- tion: For all 1 1 . o E 1; there 2s nl E Z+ s l~cl i tliut

By the proposition above, we lleetl to show the colrin~utativity condition (40) holds in 1,- (.-I) .

Definition 8. Let a ( 2 ) and b ( 2 ) E E n d :\1[[2, z - ' ] ] be two weak zlerter oper- ntors. a (2) arid 1) ( i ) u,re s t rd to be I I ~ U ~ I L Q ~ ~ : I ~ ~ O L . C L ~ (01. a ( ~ ) 1s !!(JC(II! (~:,it/i / ) ( z ) ) if t h u t rs 1 1 > O sr~clr t lmt

(21 - i 2 ) l i [ ( I ( q ) (1) (41 = 0 (-11)

Definition 9. Let n ( 2 ) untl b ( 2 ) be weak ver tex opertrtors ori 111. Define

Remark 5. A s i n R e n ~ u r k 3, we ran s h o u ~ that ( L ( z ) , , b ( z ) IS (L ,weak z;erte.t. operutor.

S e s t . we shall recall two crucial lenlmas d u e to Li and to Li a i d Dong

PI. Lemma 14. Su.ppose ( L ( z ) trnd b ( z ) are tluo ~ r ~ , ~ ~ t r i n l l y local ,weak ver tex op- erators o n A l . T h e n . a ( 2 ) untl $ b ( z ) w e rnutlrtrlly loco1 nlso.

Lemma 15 (Li and Dong). Lrt ( I ( 2 ) . 11 ( 2 ) clrld (. ( 2 ) lie rumk uerte.7: opera- tors on d l . Suppose both (1 ( 2 ) iiud b ( 2 ) rlrp l ocd ,wrth c ( 2 ) . TILPIL , (1, 1) ( 2 )

is local w i th c ( 2 ) for all 7~ E Z.

By the two lemmas above. we have the following result

Proposition 16. T h e c o ~ r ~ ~ ~ ~ ~ ~ t u t ~ v z t y cotidztron h.olds zn \'(.-I), z e. , for every 1 1 , L ' € \'(.I),

( z l - z ~ ) " ' [I. (11. z l ) . 1. (L ' . z L ) ] = 0 for home m > O

Proof. L\.I' shall first show that the set {a ( z ) = C,lEZ~ ( t i ) Z - " - ~ ] (L E '4) is a niutually local set. Csing the colnmutatol. fol.rnula:

1 1 [a (ni) , b ( T I ) ] = 5 ( t n - n ) ( n b ) ( m + t l ) + - ( R L ~ - m,) ( a , b) hm+n,o ,

6

we can show that for every n . b E .-I. Dow

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5.3 Simple Vertex Operator Algebras

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Therefore. J is also p r o p c ~ Then, by t , h ~ clefillition, J is clcarly thc 11nique niininial subniotlule of \ : 1

Since .I is a sub~notlule,

h ( 2 ) r ~ ( - 2 ) . 1 E .I for d l 6 E .4

011 the other hand, wr havr

Thrwfhre, ( 0 . I L ) = 0 as thc ~ ~ a c u u r n e lemwt 1 $ J. T h a t means ( b . ( 1 ) = 0 for all 0 E .4. It c~oritratlicts t,hat ( . ) is ~ i o n d e g e ~ ~ e r a t c . I

Theorem 19. L (.-I) I.? (1, sxrr~plr, c:er.tez operntor rrl,qab7u ,ui,~,th 1; = @. 1, I; = 0 ( L T I , ~ \ .L S .4 ( ILY 011 ( I , ~ ~ P / ) , I . ( L .

Pr.oyf. Clear from the above discussiori. I

5.4 Semisimple Case

where e is the itleritit,y element of .-I and e,'s are mutually orthogorial primitive iderripot,ents, i.e. r:f = e, and e,e, = 0 i f ' z # J . Note also that (e , , e,) = (e , e , , e ) = 0 if I # J .

S o w let 1 . 1 ~ (c ' , ) bc, the Lie algebra generated by e, ill '4.

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Tl ie~ i , by (29), we have

Therefore, each T . 2 1 . (e,) is is om or phi^ to n \'isasor<> algebri~ and

So. t l l ~ gene~alizctl \'crrrla rrlodulr

\\-llcl~.e ,If (c,. 0) is the \ ' m n a rnodlilt: of' 1 . L I , (c , ) \vith c.cx!it,ral (.Ililrg(' (., illld higlicst wriglit 0.

Kow. lve can easily show tl1c1 f'ollon.mg:

References

121 C.Dong ant! .J.Lrpo\vsky. Gcncritlizetl 1.c1.tc.s a l g r l m s i111il r('lati\.(' vortt's opcwtors, Progress in mitt 11. ,Birkllausrr, Boston.1003.

[4] I.B.Frer~kcl. Y.Huang arltl .J.Leponzskj-, On asiorllatic. alq)roachr~s t o 1.w

tes operator algcbrws a r d ~nodules , hie111 ..A II I c.r...\liltl~.So(~. 104. No.-19-1 (1993).

[5] 1.B.Frerikel. . J .L<'1)o~~liy. imd . - \ . l l t v r ~ ~ l a n . \ 7 t ~ r t ( ~ ~ 01)t~riltor algrbrizs il110 tht, .\fonst,er. A c i ~ d e ~ n i c Press, S e w Yor1<(1988).

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[GI I.B. Frerikel imd I . . Z h u . \ 'rrtcs opvrator a lge lnx associatc~tl to rt~plxw~n- tat ions of' aff in~ iilltl \ ' i~i lso~.o algtabra. Duke llat11. .I 6 6 . So. 1 (1992). ppl23-168.

[7] R.L.G~icss, Thr Friendly Gian t , 11nmt . l la th . 69 (1982). 1q).1-102

[8] Ii.lii~rittla. and C.H.Lam.. '.Inverses" of' \'il.asol.o opC,riItols. Co~nri~uni- c.wtio~is in A1gcl)r.a 23. So.12(1995), pp4-103-4413.

Received: January 1995

Revised: June 1996

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