ideals of kac-moody algebras and realisations of w∞

Volume 245, number 1 PHYSICS LETTERS B 2 August 1990

Ideals of Kac-Moody algebras and realisations of

C.N. Pope a, L.J. R o m a n s b,~ a n d X. Shen a

Center for Theoretical Physics, Texas A&M University, College Station, TX 77843-4242, USA b Department QfPhysics, University of Southern California, Los Angeles, CA 90089-0484, USA

Received 23 March 1990

We have recently constructed two new higher-spin extensions of the Virasoro algebra, denoted W~ +~ and W~, with generators of all conformal spins s >/1 and s >/2 respectively, which admit central terms for all spins. In this paper, we show how these algebras may, respectively, be realised as enveloping algebras of the U( 1 ) Kac-Moody algebra and the Virasoro algebra, factored by certain ideals. The algebra W~ + ~ may be viewed as the algebra of all smooth differential operators on the circle.

The Virasoro algebra of conformal-spin-2 fields admits generalisations to WN algebras describing fields with all conformal spins 2<~s<~N [ 1-4]. For N>~ 3, WN is not a Lie algebra, owing to the occur- rence of non-l inear terms in the commuta t ion relations for higher-spin generators. It was recently ar- gued that by taking the l imit N - , ~ in an appropriate way, one obtains an algebra describing fields with all conformal spins s>~2 that is linear. This algebra, which we shall call w~, admits a central term only in the conformal-spin-2 sector. Recently, we addressed the question of whether there exists a different N - . ~ limit of Wx, in which all the higher-spin central terms of the WN algebras are retained, while the non-line- arities of the finite-N case disappear in the limit. Our approach was based on a direct study of the Jacobi identities for the N = ~ algebra, rather than investi- gating the details of the N- ,ov limit itself. We have shown that two N = ~ algebras of this type exist [5 - 7 ]. One, which we call W~, contains generators corresponding to conformal spins s>~2 and the other, called W~ +~, contains generators for conformal spins s>~ 1 ~. As we shall show below, one can make a non- trivial redefinit ion of the generators of W~ +~ in such a way that W~ can be obtained as a t runcat ion to the s/> 2 sectors.

Our original construction of the W~ algebra pro-

Supported in part by the US Department of Energy, under grant DE-FG03-84ER40168.

ceeded by means of the brute-force imposit ion of Jacobi identities [5]. Subsequently, we uncovered some of the underlying structure by discovering that W~ could be viewed as an extension of the enveloping algebra of SL(2, ~) tensor operators [6]. This construction also revealed the existence of the W~ +~ algebra described above [7]. There is also an intri- guing realisation of these algebras as the ant isymme- tric part of associative product algebras, which we call lone-star algebras [6,7]. The purpose of the present paper is to show that there are in fact very elegant realisations of WI +~ a,O W~ in terms of enveloping algebras of the U ( 1 ) Kac-Moody algebra and the Virasoro algebra respectively. In each case, the enveloping algebra is factored by an ideal whose form is motivated by the structure of the corresponding lone- star product. In the case of W1 +~, we show that this construction can also be viewed as a realisation in terms of the algebra of all smooth differential opera-

tors on the circle. The algebras that we shall be considering all take

the general form

~ To be more precise, the s~>4 generators in W~, and the s>~3 generators in WI+~ correspond to quasi-primary fields, that is, they transform covariantly only under the SL(2, 1~) subalgebra of the Virasoro algebra. In principle, we could make redefinitions so as to obtain true primary fields. However, this can be achieved only at the price of introducing non-lineari- ties into the algebra. For brevity, in the rest of the paper we shall refer to the generators as having conformal spin s.

72 0370-2693/90/$ 03.50 © 1990 - Elsevier Science Publishers B.V. ( North-Holland )


[ V ~ , V ~ l = g ~ ( m , • g+J n, it ) V ,,, + ~

+ q2gi~( m, n; It) - m+,Vi+J-2

+ q4g~( m, n; It) vi+j-4 + --re/÷ n ~ . . .

v~, ~ [ / ' i + j - - 2 r ,.4- + qZrgi~r( m .. . . It J -- ,,,+~ --...

+ q2~ci( m; it )OiJ3m+ ,, o . (1)

The generators V~ correspond to the mth Fourier mode o f a conformal spin ( i + 2 ) field; g ~ ( m , n ;# ) are the structure constants; ci(m; It) are the central terms and q is a parameter (which may in fact be shifted at will by rescaling the generators). The structure constants are given by [ 5,6]

~f,( i t ) gi~r(m , n ; i t ) - - N ~ , ( m , n) , (2)

2 ( 2 r + 1 )!

where the N ~ ( m , n) are given by

2r+ I l ) N ~ , ( m , n ) = Z ( - 1 ) k (2 r+

k=O \ k [ i + l + m ] 2 r + l _ k

× [ i + l - - m ] k [ j + l + n ] k [ j + l - - n ] 2 , + l _ k , (3)

and the 0~,(it) are polynomials in It of degree r. In (3), [ a ] , denotes the descending Pochhammer sym- bol [ a ] , - a ( a - 1 ) . . . ( a - n + l ) = a ! / ( a - n ) ! . Note that the functions N ~, (m, n ) do not depend upon It. I f we parametrise It in terms of a variable s according to

I t = s ( s + 1 ) , (4)

the ~ (it) can be expressed as

O~r(it) =4F3( - - ½ --2S, 3+2S, - - r - - ½, - - r ,

- - i - - ½, - - j - - ½, i + j - - 2 r + ~ ; 1) , (5)

where the right-hand side is a saalschiitzian 4F3 ( 1 ) generalised hypergeometric function [6 ].

For generic values of It, the sequence of terms on the right-hand side of ( 1 ) will continue indefinitely, so that generators of all conformal spins from - ~ to + ~ are present in the algebra. When I t= 0, the structure constants g~r(m, n; It) turn out to be zero when- ever i + j - 2 r is less than zero. This has the effect of terminating the sequence of terms at conformal spin

2, giving rise to the W~ algebra that we described ear- lier. Some of the zeros o fg~r (m, n; 0) occur because o f "obv ious" zeros o f N ~ r ( m , n) , whilst the remain- der occur because of non-trivial zeros of ~ , ( 0 ) . As discussed in ref. [6], the functions ~ ~r(it) have a formal representation in terms of Wigner 6-j symbols, whose non-trivial zeros are not understood in any systematic way. When I t= - ] , the functions ~ r (it) again have many non-trivial zeros. Together with the zeros of N~r(m, n), these imply that g~, (m, n; It) is zero w h e n e v e r j + j - 2r is less than - 1. Thus there is another terminating algebra in the case It = - ~; this corresponds to WI + ~ [ 7 ].

The central terms in ( 1 ) take the form

ci(m; It) - ( m - i - 1 ) ( m - i ) . . .

X ( r e + i ) ( m + i + 1 )c i ( i t ) , (6)

where in fact the Jacobi identities require that all the central charges G(it) vanish, unless either I t = 0 or It = - ~. In the case It = 0 we find [ 5,6 ]

22i -3 i ! ( i+2) ! G(0) = c, (7)

(2 i+ 1) ! ! (2 i+3)! !

where c is an arbitrary constant coinciding with the usual central charge in the Virasoro sector. When It = - ~, the central charges are given by [ 7 ]

Ci(-- ~ ) = 22i-2[ ( i + 1)!]2 ( 2 i + 1 ) ! ! ( 2 i + 3 ) ! i c" (8)

Both WI+~ and W~ can be viewed as extensions of the SL(2, ~) tensor algebras #-(it), where the parameter It is the value of the quadratic Casimir Q of SL(2, Y~). As discussed in ref. [6] , the generators of these tensor algebras may be associated with modes V~n of conformal fields, restricted to the "wedge" I m I ~< ( i+ 1 ). We showed in ref. [ 6 ] how the algebra J-(i t) for general It could be extended "beyond the wedge" to an algebra W~ (it), in general requiring the addition of fields of negative conformal spin. Owing to the presence of non-trivial zeros among the structure constants for the I t= - ¼ and I t = 0 algebras, the extensions W~( - ~ ) and W~(0) can be consistently truncated to closed algebras containing only those generators with conformal spins >/1 or 2 respectively. These are the algebras W~+~ [7] and W~ [ 5,6 ] respectively.

The tensor algebras .Y-(it) are constructed by start-

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ing from the enveloping algebra of SL (2, ~ ) [ i.e. the algebra of all polynomials in the generators of SL(2, ~ ) ] , and factoring out by the ideal generated by ( Q - # ) , where Q is the quadratic Casimir operator of SL (2, ~ ) and # is a freely-specifiable constant. This SL(2, ~) algebra is in fact the subalgebra of the Virasoro algebra generated by L_ l, Lo and L~. Rather than constructing W~+~ and W~ by extending the corresponding SL(2, ~) tensor algebras beyond the wedge, it is natural to enquire whether it is possible to construct them directly from an enveloping algebra of the Virasoro algebra (or some other infinite- dimensional algebra) that is already defined beyond the wedge. To show how this can be done, we begin by recalling some relevant properties of the lone-star algebra given in refs. [6,7]. There, we showed that the W~ +~ and W~ algebras could be realised by the antisymmetric parts of corresponding associative product algebras.

For the case of W~ +~, the relevant lone-star products that we need in this paper are [ 7 ]

J,,, *L =jm+n, (9)

1 i Lo* V',,, = VI + l - ~mV, , ,

- ( i + 1 ) 2 [ ( i + 1 ) 2 - m 2 ] Vim - l , (10) 4 1 4 ( i + 1 ) 2 - 1 ]

where we have set the parameter q appearing in equa- tion ( 1 ) and in ref. [ 7 ] equal to 1. For convenience, we are writing the conformal-spin- 1 and 2 generators V - ~ and V ° as j and L respectively. For the case of W~, the relevant lone-star products are [ 6 ]

L m * L ~ = V,,+,~ +~l(m--n)Lm+~ , (11)

- - v i + I ! i Lo * V ',, - - - m - ~ m V ,,,

- i ( i + 2 ) [ ( i + l ) 2 - m 2 ] VI7, ~ . (12) 414( i+ 1 )2 - 1]

With these lone-star products, our realisation of the W~ + ~ algebra now proceeds as follows. We begin with the U(1 ) Kac-Moody algebra, without central extension, together with the derivation generator d:

[jm,L] = 0 , [d, j m ] = - m j m . (13)

The enveloping algebra for ( 13 ) is then obtained by taking the tensor algebra ofjm and d, quotiented by the ideal generated by x ® y - y ® x = [x, y] for all generators x and y. In other words, it is the algebra of

all polynomials in the generators, modulo the use of the commutation relations ( 13 ). This enveloping algebra is much larger than Wl+~. To obtain W~+~ itself, we define an additional ideal J , motivated by (9)

J : j , , , j , , - - jm+, ,=O, (14)

for all m and n. One can easily verify that the com- mutator of (14) with j or d gives back expressions of the form (14), as must be the case if ~¢ is to be an ideal.

Using the relations (13) and (14), any polynomial in j and d can be re-expressed as a polynomial in dacting on a singlejm whose index is equal to the sum of those on the individual f s in the original product. In particular, it is easy to see that if we define

Lm = ( d + ~m) j , , , , (15)

then the L,, satisfy the Virasoro algebra (without central extension). Note that since Jo functions as the unit operator, Lo may be identified with d. Higher polynomials in d will give rise to higher-spin generators. To organise these appropriately, we can turn eq. ( 10 ) around, and view it as a recursive definition for V',n in terms of lower-spin generators:

V i i i - i i 2 ( i 2 - m 2 ) V i-2 (16) _ ( d + ~ m ) V , , , + 4(4 i2_1 )

This recursion relation can be solved iteratively, to give

VI, = Pi(d; m ) j m , (17)

where the P,(d; m ) are polynomials in d of degree ( i+ 1 ). The first few polynomials take the form

P _ ~ = I ,

P o = d + ½m ,

Pl = d 2 + m d + l ( 2 m 2 + 1 ) ,

P2 =d3 + 3rnd2 + ~ ( l Zm2 + 7 ) d + ~o( 2m3 + 7m ) ,

P3 = d 4 + 2 m d 3 + ~ ( 1 8 m 2+ 13) d2

+ 1 ( 4 m 3 + 1 3 m ) d + ~ ( 8 m 4 + 100m2+27) . (18)

The entire Wl +~ algebra, satisfying ( 1 ) with # = - ], follows from (13), (14) and (17), where the polynomials are defined by (16). Thus W~ +~ is the enveloping algebra of the U(1) Kac-Moody algebra

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with derivation d, quotiented by the ideal (14). In fact the entire lone-star algebra for ~t= - ~ follows from (14) and (17).

The algebra Wo~ may, in a similar fashion, be realised in terms of the enveloping algebra of the Virasoro algebra. Again, we must quotient by an additional ideal. In this case, motivated by ( 11 ), we in- troduce the ideal J :

J : L , , L , , - ( L o + m ) L m + , , = O , (19)

for all m and n. Using these relations, any polynomial in L can be re-expressed as a polynomial in Lo acting on a single Lm, whose index is the sum of those of the L's in the original polynomial. As in the case of W~ +~, these can be organised in a manner appropriate to the W~ algebra by turning ( 12 ) around to give the recursion relation

V ~ = ( L o + ½ r n ) V ~ - ~ + ( i 2 - - 1 ) ( i 2 - m 2 ) Vi_2,

4(4i2--1 )

(20)

and solving this iteratively to give

VI, = @(Lo; m ) L , , , (21)

where the Qg(Lo; m) are polynomials in Lo of degree i. The first few polynomials are

Qo = 1,

Ql = L o + ½ m ,

Q2 =L2 + mLo + ~ ( m 2 + 1 ) ,

Q 3 = L 3 3 2 1 2 + u n L o + i 3 ( 9 m + lO)Lo + l~(m3 + 5m) ,

Q4=L4 + 2 mL3 + ~( 4 m a + 5 ) L 2 + l ( m 3 + 5 m ) L o

- t - ~ 2 ( m 4 - t - 1 5 m 2 - 1 - 8 ) . (22)

The algebra Woo is realised as the enveloping algebra of the Virasoro algebra [Lm, L,] = ( m - n ) L m + , quotiented by the ideal (19). The entire lone-star algebra for/~ = 0 is given by ( 19 ) and (21 ).

The W~ +~ and W~ algebras can in fact be represented in terms of algebras of differential operators on the unit circle, parametrised by 0. For W~ +oo we do this by taking

0 j m = e x p ( i m O ) , d = i ~ . (23)

One can easily see that these operators satisfy (13)

and (14) identically. This has the consequence that W~ + ~ may be identified as the algebra of all polynomials in the operatorsjm and d; this is nothing but the algebra of all smooth differential operators on the circle. Note that these include all smooth functions (corresponding to the spin-I generators jm) as well as differential operators of arbitrary order. For Woo, we take

0 Lm =i exp(im0) ~-~. (24)

This satisfies ( 19 ), and the Virasoro algebra without central term. In this case the generators of W~ are all represented by differential operators of strictly positive degree. The subalgebras W(~ +~)/2 and W~/z obtained by the restriction to even spins (as discussed in refs. [6,7] ) correspond to the subalgebras of these operator algebras for which V~,~-V~_m under 0~ - 0 .

By comparing the above representations for W~ +~ and W~, one can show that Woo may in fact be obtained as a truncation of W~ +~, after an appropriate redefinition of the generators. If we denote the generators of W~ +~ by V~, and those of W~ by ~' Vm, then these redefinitions take the form

i

Vm= ~ ao(m)V~, , , ( 2 5 ) j = - i

where the constants ao(m) are polynomials in m of degree ( i - j ) and i takes the values 0, 1, 2, .... For example, we have

~ o = v,n° + ½ m V ; . ' - ~ca, . ,o ,

--1 V I ~_I o Vm = -- 1 ) Vm 1 , ~mV,~ + l~(rn 2

f ' ~ = V2 + l~mV,n' + ~ ( 2 m 2 - 3 ) V °

+T~6( 2m3 + 7m ) V m I --3@6c~m,o , (26)

The generators ~i Vm, i>~0, together with V2~ ~, yield the Wl+oo algebra in a new basis. After making the redefinitions (25), the conformal-spin-1 generator Vm I can be consistently truncated from the W~+~ algebra. The remaining generators yield Wo~. Note that, for convenience, we have included constant shifts in the definitions of I?~, which ensure that the "trivial" lower-order parts of the central terms have their canonical form. The resulting central charge g

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of the W~ algebra is related to the central charge c of W, + o~ by

~= - 2 c . (27)

As is well known, in order to have unitary representations of algebras of this kind (for example Kac- Moody and Virasoro algebras), one requires that the central charge be positive. Thus (27) suggests that unitary representations of W,+o~ may not straight- forwardly decompose into unitary representations of

In fact the above discussion can be viewed as a special case of a more general class of redefinitions. Con- sider the shifted Virasoro generators £m, defined in terms of the generators of W, +~ by

£ m = L m + ( S + ½ ) m j m , , 2 - ~ ( S + ~ ) C~,,,,o. (28)

These generators satisfy the Virasoro algebra with central charge ~ given by

? = [ 1 - 1 2 ( s + ½ ) 2 ] c = - 2 ( 6 s 2 + 6 s + l ) c . (29)

When c = 1, which can be realised with a single free boson, this construction corresponds to the introduc- tion of a background charge c% = s + ½. This scalar boson can also be viewed as the bosonisation of (b, c) ghost system with spins s + 1 and - s . Note that the shifted SL(2, ~) generators satisfy

(/20) 2- ½{1,, l ~ }=s(s+ l ) = ~ , (30)

therefore they can be used to build up the wedge algebra ,<(~t). It is straightforward to derive expressions analogous to (28) for the higher-spin generators, for example

m = V~,, + (S+ ½)mL, , + ~ ( s + ½)2(m~-- 1 ) j .... (31)

It is natural to consider the redefinitions (28), (31 ) etc. for general values o f m. This corresponds to an extension of.~-(/z) beyond the wedge. However, this extension does not coincide with W~ (~t). It is easy to see that one has simply written the 14"j +~ algebra in a different basis. Note that for every value of kt be- sides 0 and - ~, the basis is off-diagonal, in the sense that there are central terms between generators o f different conformal spins.

As we discussed previously, one can extend the tensor operator algebras .¢-(/z) beyond the wedge for any value ofl t , to obtain algebras H~( / t ) that con-

tain generators with all conformal spins f r o m - oo to oo. Only when/z is equal to 0 or - ¼ can one truncate out the negative conformal spins. Also, central terms can be introduced only at these two special values of /t [ 6,7 ]. It is natural to wonder whether the representations of W~ and W, +~ in terms of differential operators may be extended to representations of the full algebras W~(0 ) and W~( - ¼ ). As we shall see, each of these algebras contains a "penumbral" subalgebra including the entire set of positive-spin modes, plus all the negative-spin modes outside of a "shadow wedge" of generators. We shall show how to realize the penumbral subalgebra in terms of formal inverses of differential operators. The representation of the modes within the shadow wedge (defined to be those {V',,,} with i < - I and Im[ < I i l - 1 ) cannot be spec- ified by the methods at our disposal.

The extension of our differential-operator representations to negative spins will involve the intro- duction of formal inverses of differential operators. We begin the construction by considering the particular family of lone-star products

l ~ ' i + j + 1 [:t+(i+,)~VJ+(j+l ) v ±,+/+2) , (32)

valid for any value of/~. Considering the top choice of sign, we see that the set of generators { V~+, } closes under multiplication. For i>~ - 1 these generators are simply 1, L~, (Ll)2, (L~)3, .--, constituting the right- hand edge of the wedge subalgebra. The rest of the generators may be identified by taking i> 0 and not- ing that (32) implies that

V t t + l , V - ( l + 2 ) = v - ( i + 2 ) i - ( ,+ , ) - - ( i + , ) * V i + l = V o 1, (33)

or in other words

V-¢ ,+2 )_ , = ( L ~ ) (34) - ( , + l ) - ( V , + , ) - I - ( ,+1)

since the mode Vff J is the identity element of the algebra. Thus the generators {VI+~) function (under the lone-star product) as the set of all integral powers o f L~, as i ranges over all integers. Similarly, the generators { V i ~,+ ~) ) may be identified as the set of all integral powers of L_ ~; in the positive-conformal-spin sector they delineate the left-hand edge of the wedge subalgebra.

For definiteness we now concentrate on the case ~ = - ¼, for which L+, are represented as the first-order differential operators

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L+ ~ =exp( _+ i0) (d-T- ½ ) = (d_+ ½ )exp(+_i0)

=exp( +_ ½i0)d exp( +_ ½i0), (35)

and therefore

1 V+~ =exp(-T-i0) d+~½' (36)

where we adopt the convention that the 0-dependent exponentials should be moved to the left. Note that the expression (36) may in fact be consistently defined to act on the space of all smooth functions on the circle, since then the operators d+_ ½ have no zero- modes.

The other generators on the edge are found by taking higher powers of ( 36 ). For example,

1 1 VS~ = e x p ( - i 0 ) ~ e x p ( - i 0 ) d+--~!

1 = e x p ( - 2 i 0 ) (d+3) (d+ ½ )

= e x p ( - 2i0)(~-+ ½ d T ~ ) " (37)

Note that one does not actually encounter higher negative powers of differential operators, contrary to what one might have expected from the form of (34).

The remaining generators in the penumbral subalgebra may be determined by repeatedly acting upon the generators (34) with appropriate SL (2, ~) rais- ing and lowering operators L+ ~, taking advantage of the commutation relation

Vm] [ + - ( i + l ) - m ] i [L+_l, i = Vm+_l , (38)

in fact valid for any value of#. In this way one obtains

-2 1 V+m = -- exp(+_imO) m

l ) × +d_T_~--3+...+d_T_(m+½) (39)

for all positive integers m. Note that the mode Vff 2 cannot be reached in this fashion. Similarly, one finds

-3 6exp(+_im0) ~ m - 2 a + l V + ~ - m ( m 2 _ l ) ~=ld-Y-(a-½)'z~ (40)

where again the shadow-wedge modes Vff 3 and V ¥ 3 are excluded, since they never appear in a com- mutator of modes outside the shadow wedge.

Turning now to #=0 , we find that the analogous considerations give simply

1 (41) Vm2=exp(imO) d - m

for all integers m # 0. (Unlike the previous case, here there is no analytic problem at m = 0; one simply does not encounter that particular mode.) Furthermore,

-3 6 exp(+_irnO) V ± m - m ( m 2 - 1 )

( , m o) × - ½ r n ( m - 1 ) ~ + ~ l d ~ a (42)

where now we must restrict m > 1. It is interesting to note that it is possible to deter-

mine the "conformal-spine-one" modes VT~ ~ in the W~ (0) algebra. By demanding consistency with the recursion relation (20), one finds that

½m ~ (43) Vm l=exp( im0) l + d _ m ] .

As required, the mode Vff ~ is the identity operator, but the higher modes do not span the space of smooth functions, as was the case with W ~ ( - ¼ ) . On the other hand, there is a particular combination of the "spin-one" and "spin-zero" fields which can be identified with the spin-one currentjm of the Wl +~ algebra, as we see by comparing ( 41 ) and (43) with (23):

P~' - -½m~'~z=jm. (44)

In a sense, this correspondence provides the missing first line of eq. (26), leading to an embedding of W~+~ in the full W~(0) (of course, this particular construction can only be effected for c = 0).

In fact (before truncations and the inclusion of central terms) it may be that all of the algebras W~o (#) are equivalent, in the sense that one can make formal redefinitions of the generators, involving infinite number of terms,

i

V~,(# ' )= ~ a o ( m ; # , # ' ) V ~ ( p ) , (45) j = - -o~

to relate the generators of W~(#' ) to those of W~ (#). However, the remarkable thing is that it is possible to

77


t runca te out the nega t ive -confo rma l - sp in genera to rs

o f W ~ ( - ¼ ) or W ~ ( 0 ) to ob ta in Wl+oo or W~

respect ively.

O u r work has b rough t to l ight a c o n n e c t i o n be-

tween h igher -sp in algebras and the algebra o f all

s m o o t h d i f ferent ia l opera to r s on a circle. In par t icu-

lar we see that this a lgebra a d m i t s a non- t r iv ia l cen-

tral extens ion. The ana logous centra l ex tens ion [8]

for the subalgebra o f the f i r s t -order d i f ferent ia l op-

era tors (V i ra so ro a lgebra) has m a n y in te res t ing

physical consequences . It r ema ins to be seen whe the r

the same will be t rue for the W~ +~ and W~ algebras.

References

[ 1 ] A.B. Zamolodchikov, Teor. Mat. Fiz. 65 (1985) 347. [2] V.A. Fateev and S. Lykyanov, Intern. J. Mod. Phys. A 3

(1988) 507. [ 3 ] A. Bilal and J.-L. Gervais, Phys. Lett. B 206 ( 1988 ) 412. [4] F. Bais, P. Bouwkne~t, M. Surridge and K. Schoutens, Nucl.

Phys. B 304 (1988) 348, 371. [5] C.N. Pope, L.J. Romans and X. Shen, Phys. Lett. B 236

(1990) 173. [6] C.N. Pope, L.J. Romans and X. Shen, W~ and the Racah-

Wigner algebra, preprint CTP TAMU-72/89, USC-89/ HEP040.

[7] C.N. Pope, L.J. Romans and X. Shen, Phys. Lett. B 242 (1990) 401.

[8] I.M. Gelfand and D.B. Fuks, Funks. Anal. Pril. 2 (1968) 92.

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ideals of kac-moody algebras and realisations of w∞

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