International Centre for Theoretical Physics
XA9745154
IC/97/92
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
HIGHEST WEIGHT IRREDUCIBLE REPRESENTATIONS OF THE QUANTUM ALGEBRA Uh(Aoc>)
T.D. Palev and N.I. Stoilova
INTERNATIONAL ATOMIC ENERGY AGENCY
UNITED NATIONS EDUCATIONAL. SCIENTIFIC AND CULTURAL ORGANIZATION
MIRAMARE-TRIESTE IC/97/92
United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
HIGHEST WEIGHT IRREDUCIBLE REPRESENTATIONS OF THE QUANTUM ALGEBRA
T.D. Palev Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria* and International Centre for Theoretical Physics, Trieste, Italy
and
N.I. Stoilova Department of Mathematics, University of Queensland, Brisbane Qld 4072, Australia and Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria.*
ABSTRACT
A class of highest weight irreducible representations of the algebra Uh(Aoo), the quan- tum analogue of the completion and central extension A^ of the Lie algebra gloo, is con- structed. It is considerably larger than the known, so far, representations. Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators are explicitly written. The verification of the quantum algebra relations to be satisfied is shown to reduce to a set of nontrivial g-number identities. All our representations are restricted in the terminology of S. Levendorskii and Y. Soibelman (Commun. Math. Phys. 140, 399-414 (1991)).
MIRAMARE - TRIESTE July
Permanent address. E-mail addresses: [email protected]; [email protected] 1. INTRODUCTION
We construct a class of highest weight irreducible representations (irreps) of the algebra Uh(Aoo) [8], the quantum analogue of the completion and central extension Aoo of the Lie algebra gloo [1], [6]. Our interest in the subject stems from the observation that certain representations of gl(n) (including n = oo) [9] and of Aoo (see Example 2 in Ref. [10] and the references therein) are related to a new quantum statistics, the .A-statistics. The latter, as it is clear now, is a particular case of the Haldane exclusion statistics [5], a subject of considerable interest in condensed matter physics (see Sect. 4 in Ref. [13] for more discussions on the subject). It turns out that some of the representations of the deformed algebra {/h(A») satisfy also the requirements of the Haldane statistics. More precisely, they lead to new solutions for the microscopic statistics of Karabali and Nair [7] directly in the case of infinitely many degrees of freedom. These results will be published elsewhere. We mention them here only in order to justify our personal motivation for the work we are going to present. One may expect certainly that similar as for A^ [2], [4] the representations of Uh{Aoo) may prove useful also in other branches of physics and mathematics. The quantum analogues of gl^ and A^ in the sense of Drinfeld [3], namely Uh.(gloo) and C//I(J4OO)> were worked out by Levendorskii and Soibelman [8]. These authors have constructed a class of highest weight irreducible representations, writing down explicit expressions for the transformations of the basis under the action of the algebra generators. The (/^(ylooj-modules, which we study, are labeled by all possible sequences (see the end of the intro- duction for the notation) {M} = {Mi}i€z € C[/i]°°, subject to the conditions:
(a) There exists m < n € Z, such that Mm = Mm-k and Mn = Mn+k for all k € N; (b) Mi - Mj € Z+ for all t < j € Z. Representations, corresponding to two different sequences, {M1} ^ {M2}, are inequivalent. The E//i(-<4oo)- modules of Levendorskii and Soibelman [8] consist of all those sequences {M^}, for which s = m = n € Z s) and M\ = 1, if i < s and M\S) = 0 for t > s. In Refs. [11] and [12] a class of highest weight irreps, called finite signature representations, of the Lie algebra .Aoo was constructed. The corresponding modules are labeled by the set of all complex sequences
{M} = {Mj}i6z € C°°, which satisfy the conditions (a) and (b). The name "finite signature" comes to indicate that, due to (a), each signature {M} is characterized by a finite number of different coordinates or, more precisely, by no more than n - m + 1 different complex numbers. FVom our results it follows that each .Aoo-module with a signature {M} can be deformed to an ^(-Aoo)-module with the same signature. The class of the finite signature representations of ^(^oo) is however larger, which is due to the fact that the coordinates of the l/h(-<4oo) signatures take values in C[/i). All representations we obtain are restricted in the sense of Ref. [8], Definition 4.1. In Section 2 we construct a class of highest weight irreps of the subalgebra Uhla^) of U^Aoo). Some of these representations, namely the finite signature representations (Definition 2), are extended to repre- sentations of Uh{Aoo) in Section 3. Throughout the paper we use the notation (most of them standard): N - all positive integers;
Z+ - all non-negative integers; Z - all integers; Q - all rational numbers; C - all complex numbers; C[/i] - the ring of all polynomials in h over C; C[[/i]] - the ring of all formal power series in h over C; n X - {(01,02,...,On)! a{ € X}, (including n = 00); [a,b] = {x| a < x < b,x € Z}, (a,b) = {x\ a < x < b,x € Z}; F({M}) - the C-basis of a module with a signature {M}; I, fori>0 fori<0;
N = [x, y]q =xy- qyx.
2. REPRESENTATIONS OF THE ALGEBRA
The algebra U^idoo) was denned in Ref. [8]. The authors denote it as f/h(g'(j4oo))/- It is a Hopf al- gebra, which is a topologically free module over C[[h}] (complete in ft-adic topology), with generators {ei,fi,hi,c}iez, and 1. Cartan relations:
[c,o]=0, ae {hi,ei,fi
[hi,hJ]=0,
2. e-Serre relations:
eiej =eiei, if \i - j\ £ 1, l e?ei+1 - {q + q- )eiei+1ei + ei+ief = 0, (2) e e 1 i+i t - (q + 9~ )ei+ieiei+1 + e^^ = 0. 3. /-Serre relations:
fifi^fjfi, if l*-j|/l, l fffi+i -(Q + q- )fifi+ifi + fi+iff = 0, (3) fhJi - (
We do not write the other Hopf algebra maps (A, e, 5) [8], since we will not use them. They are certainly also a part of the definition of Uk(a,oo)- Replacing throughout in the above relations {ei,fi,hi,c}i€z with {Ei}Fi, Hi,0}i&z, one obtains the definition of Uh{gloo)-
In terms of an equivalent set of generating elements {&i, fi, hi, c}i^z, with
* = «,<*••-*>/*, fi = hq{hi-h^)l\ (4) one writes the quantum analogues of the Weyl generators {eij}(i,j
eij = [ei,[ei+i,[...,[ij-2,ej-i]q...}q]q]q, i + l
©jZ+aj. Denote by C//i(n+) (respectively C//,(n_) ) the unital subalgebra in Uh(aoo) generated by {ej}i6z
(respectively {/i}iez )• Then
Uh{n±) = ®a€Q'+Uh(n±)±a, (7) where
Uh(n±)±a = {x€ Uh(n±)\ [h',x] = ±a(h')x, W € H) (8) for a ^ 0, and £//,(n±)o = C[[/i]]. Any element u e (/^(aoo) can be represented as
Yl Y, :Fo,,k,i,tY[hl(k'l)l£0,k,i,t, finite sums over a, 0, 7, (9)
fc=0 (=0 a,/3€Q'+ 7(fc,i)€Z~ t=l where Fa,k,i,t 6 f^/i("-)-o, £p,k,i,t € C/ft(n+)+^ and finitely many exponents 7(fc, i)i are different from zero. The words "finite sums over a, (3, 7" have been added in order to indicate that for a fixed fc only finitely many summands in (9) are different from zero.
The set Uhi^oo), consisting of all C[[/i]]-polynomials of the Chevalley generators {ei,fi,hi,c}i&z, is dense in Uh(aoo) with a basis (6). In particular J"Q,fc,;,t, £0,k,l,t and fltgz h] are in ^(aoo). Then according to (9) any element u € ^(aoo) is of the form
Kk) t{a,0) Ufc = Ec' E £ E *"«.*.'.' II hlik'l)i£0.kU e ^(aoo). (10) fc=0 (=0 a,0€Q'+ 7(fc,()6Z~ t==l 16Z
4 We pass to construct a class of highest weight irreps of t/h(ooo). To this end we define the ^(a
;6>,£i), labelled by &, fi € C[h] and by a sequence {M} = {Mi}i€Z e C[/i]°° such that
Mi-Mj€ Z+, Vi < j e Z, with a basis F({M}). The latter, called a central basis (C-bases), is independent on £o.fi- It is formally the same as the one introduced in Refs. [11] and [12] for a description of representations of OOQ. F({M}) consists of all C- patterns
M_i, Mo, Mi,
, (11) '-1,3, M13
Moi where k € N, 0 = 0,1. Each such pattern is an ordered collection of formal polynomials in h
Mii2k+e-i£C\h), V&eN, 0 = 0,1, i€[-0-* + l,*-l], (12) which satisfy the conditions:
(i) there exists a positive, depending on (M), integer N^M) > 1> such that (13a) Mii2k+e-i = Mi, V 2k + 8 - 1 > N(M), 8 = 0,1, i e [1 - 9 - k,k - 1];
(ii) for each Jfc € N, 6 = 0,1 and i € [1 - 6 - k, k - 1]
be its Denote by ^({M^fo.fr) the free C[[/i]]-module with generators T({M}) and let V({M};£0,£i) completion in the /i-adic topology. V({M};^Oi^i) is a topologically free C[[/i]]-module with a (topological)
basis T({M}) and ^({MIJ^OI^I) >S dense in it (in the /i-adic topology). V({M};£0,£i) consists of all formal power series in h with coefficients in
(14) >=o
If a is a C[[/i]]—linear map in V({M};£o,£i), a € End V({M};£o,£i), we extend it to a continuous linear
map on V({M};^0,^i) setting
av = t=0
Therefore the transformation of V({M}; ^0, £1) under the action of a is completely defined, if a is defined on tne We proceed to turn V({M};fo>£i) into a ^(a,*,) module. Denote by (M)±^?p} and (^)±ijp\ patterns obtained from the C-pattern (M) in (11) after the replacements
Mjp->Mjp±l and Mlq - Mlq ± 1, Mjp - Mjp ± correspondingly, and let
S(jJ;v)= (16) -1 for j >
Set moreover
e° = /„ e\ = et, i € Z. (17) Let {p{ei),p(fi),p(hi),p(c)}i£z be a collection of C[[/i]]-endomorphisms of V({M};£o,£i)> defined on any C-pattern (M) € T({M}), as follows (see also (50), (51), (54) and (55)):
= 0,1 , (18)
t-i 1+ E
(
(£1 ~ , i € Z, (20) -\i\
(21)
? Above and throughout [x] = ? t e C[[/i]]. If a pattern from the right-hand side of (19) does not belong to F({M}), i.e., it is not a C-pattern, then the corresponding term has to be deleted. (The coefficients in front of all such patterns are undefined, they contain zero multiples in the denominators. Therefore an equivalent statement is that all terms with zeros in the denominators have to be removed.) With this convention all coefficients in front of the C-patterns in r.h.sof (18)-(21) are well defined as elements from C[[/i]].
Proposition 1. The endomorphisms {p(ei)1p(fi),p(hi),p(c)}i&z satisfy Eqs. (l)-(3) with pfa), p(fi), p(hi) and p(c) substituted for ei, fi, hi, and c, respectively. Proof. The proof is based on a direct verification of the relations (l)-(3). This verification is lengthy and nontrivial. The most difficult to check are the last Cartan relations in (1), corresponding to i = j. In order to show they hold, one has to prove as an intermediate step that the following identities are fulfilled:
1 fc-1 fc-1 *-\\ ' ' ' • L—l V 5=0 J = l-fc Z=-fc t,2fc-l - ij,2fc-l k-l ~ Ljt2k-l
~ U,2k
k-l fc-2 fc-1 53 Lj,2k-1 ~ VfceN, (22) j=-k j=-k
1 fc-1 fc -3
s=0 j = -k ( = -*
fc-i fc-i , Vfc € N. (23) _i = -fc-l j=-, 5Z j=-k We proceed to verify (22) and (23). Let l fc-i fc-i rr*-1 rr : s=0 i=l-fc (=-* »,2Jfe-l - Lj,2k-l
~ U ,2k i^fc - ^i,2fe + S][^i,2*: - Ufik + S - 1]
fe-1 *-2 fc L 2fc fc-i Gk(h) = - 53 J- -+E j=-k+l i=-k i=-k Fk(h) and Gk(h) are formal power series,
Fk(h) = ^Ckih', Gk{h) = i=0 i=0 In order to prove that (22) holds one has to show that Cki = dk% for any fc and i. To this end replace h in (22) by a complex variable x € C, so that x takes values on any curve 7 C C and x £ itrQ. Then Fk(x) and
Gk(x) are well defined for any a; € 7 and
i=0 t=0 Therefore Cki = dki if Fk(x) = Gfc(x). Since h appears in (22) only through q = eh^2, we have to show that (22) is an identity for q being a number, which is not a root of 1. The same arguments hold also for (23). 2 2L 2L 2Li Let q € C be not a root of 1. Setting q ^"-" = Ai, g -" = Bit q >«+» = d, q <"-* = 2L Du 2k = n in (22) and qW-w-i) = A^ q2L,,3k+l = Bu q2{LiM+*-i) = Cu q ',^-^ = Du 2k + 1 = n in (23) and relabelling the indices in an appropriate way, one reduces both identities to the following one:
which can be written also in the form
(25)
Consider the complex functions:
{ }
z 2 The function fi(z) (resp. f2( )) is holomorphic over C except in its simple poles q Aj, B\,...,Bn, 2 2 2 2 2 q~ B\,... ,q~ Bn (resp. q~ Bi, Ai,...,An-i, q A\,...,q An-i.) Let C be a closed curve whose in- terior contains all poles of /i(z) (resp. f2(z)-) The Residue theorem of complex analysis implies that
§cfj(z)dz = 2m^2Res{fj{z)), j = 1,2. On the other hand,
I fj(z)dz = -2mResfj{oo) j = 1,2. (27) Jc Applying the Residue Theorem for both functions (26) and inserting the results in (25) one obtains:
U A nr;/(^ M) nr(^ 4^) h B n^(^ B> nr^c^ allt l 1 = i-g n-- 1 '"tr • (28)
In order to prove this last identity consider the function:
f(z) = ™i=1 , q Di' ^i=1 ^Z ~ q Ci>. (29)
Applying again the Residue Theorem on this function one obtains (28) which proves formulas (22) and (23). In addition to the identities (22) and (23) the last Cartan relation (1) is valid if:
n
which is proved in a similar way. The verification of all other relations (l)-(3) is based on (30) or/and on some of the following identities:
,,. _, _ll k _ > _ „ - |21|o _, _ 11|c _ „ + „ _H| e _ m (p^g, p^j (31) [q - 1][6 - 1] - [2][a][b - 1] + [a][b] [a - l)[b - 1] - [2][a - 1][6] + [a][6] _ [a-6 + 1] + [a-b-l] -°' (32)
[a-l]-[2][a] + [o + l]=0. (33)
This completes the proof. [] Remark. We may have constructed the endomorphisms {p(ei),p(fi),p(hi),p(c)}izz from the results on the representations of Uh(gloo), announced in Ref. [14] and, more precisely, from the endomorphisms
{p(Ei),p(Fi)>p(Hl)}i€z of the C[[h]]-module V({M}), which satisfy the Cartan and the Serre relations for Uh(gloo) (Ref- [14], Eqs. (16)-(18).) This possibility is based on the observation that the C[[h]]-linear map fi, and extended by associativity is an (algebra) isomorphism of Uh{gloo) ® C[[/i]]c onto Uh(aoo)- Then the endomorphisms {p(ei),p(fi),p(hi), p(c)}i6z defined according to (34) (with a(£o — £i) = ^i) as p(ei)=p(Ei), p(fi)=p(Fl), p(c)=G,-£i, ((&C)e(0 6) obey Eqs. (18)-(21). Here we have given a direct proof of Proposition 1, since the corresponding Proposition 1 in Ref. [14] was only stated. Its proof would have been based again on the identities (22) and (23). Consider p as a C[[/ij]-linear operator from Uh{a pb)=ap(a)+0p(b), p(ab) = p(a)p(b), o,6€%(aoo), ct,0eC[[h]]. (36) As we know from (10), any element u € U^doo) can be represented as a sum u = X^ou»'l*> u« Then for an arbitrary v € V({M};£o.fi). writing it as in (14), we have oo/n w n ^ = E E ?(«»-»)«« /* e V({M};£0,6), (37) \j ) n=0 \m=0 / since m=0 Using (37), we extend p on p{u) = Y,P(ui)hi € ^"^ ^(W^o.fi) V u € I/fcCooo). (38) •=o Hence p is a well-defined map from Uh(a,oo) into End P : Uh(aoo) - End V({Af};&,6). (39) Proposition 2. T/ie mop ^55,), acting on the C-basis according to Eqs. (18)-(21), defines a highest weight irreducible representation ofUh(a.oo) in V({A/};^0i^i))- Proof. According to Proposition 1, (36), (38), p is a C[[/i]]-homomorphism of Uh{a.oo) in End V({M};£o,£i)- It is continuous in the /i-adic topology. Indeed, let u € Uhla^). Then any neighbourhood W(p(u)) of p(u) contains a basic neighbourhood p(u) + hnEnd V({M};£o,£i) C W{p{u)). Evidently n n p(u + h Uh(aoo)) C p(u) + h End V({M};&,6) C W{p{u)) (40) a and therefore p is continuous in u for any u € E^,(aoo)- Hence V({M};^OiCi) is £//i(a Mi,2k+e-i = Mu VfeeN, 0 = 0,1, ie[-0-fc + l,fc-l]. (41) Moreover, V({M};£0,£i) = p(C//l(aoo))(M). Since V({M};^o,Ci) contains no other singular vectors, vec- tors annihilated from p(Uh(n+)), each V({M};£o,£i) is an irreducible C//,(aoo)-module. The proof of the latter follows from the results in Ref. [12] and the observation that each (deformed) matrix element in the transformation relations (18)-(21) is zero only if the corresponding nondeformed matrix element vanishes. [] 10 3. REPRESENTATIONS OF THE ALGEBRA In this section we show that some of the U^doo)-modules V({M};£o,£i) can be turned into irreducible highest weight [/^(.AooJ-modules. First, following Ref. [8], we recall the definition of the quantum algebra Uh(Aoo) (denoted by the authors as C^i(loo))) only within the algebra sector (for the other Hopf algebra maps see Ref. [8]). Uh(Aoo) consists of all elements S(a) = {i I m, ± 0}, 5(7) = {x I 7i ^ 0}, S(a,7) = S(a) U S(7). (43) Connecting i and j if \i - j\ = 1, one can view S(a, 7) as a graph. Denote by T{a, 7) the collection of its connected components. For any u = £)fclo hkUk as given in (42) consider the series t(k) t(a,p) ^ = Ec' E E E^«.^nfc7(*'l)i^.w4, (44) 1=0 a,^€Q'+ 7(fc,0€Z~ « = 1 t€Z and let jF(U,fc)=UJF(a,7), (45) where the union is taken over all a and 7, which appear in the non-zero summands of (44). For i € Z and keZ+ set Int(u, k, t) = {/ € T{u, k)\ie I). (46) For r € N define the series u(r), corresponding to u, by substituting 0 for all hi (i < -r or i > r) and for all e4l /< (|t| > r). Definition 1. [8] 77ie series u 0/ t/ie /orm (42) is said to belong to f//,(^4oo), provided (i) for any k € Z+ and any i e Z the set Int(u, k, i) is finite; (ii) u(r) £ Uhiaoo) for oi/reN. We turn to construct a class of representations of [^(^oo)- The idea is to show that within certain };£0,£i) the domain of the definition of the operator p : C//i(aoo) —> End V({M}\£o,£i) (see (39)) can be extended from Uh(aoo) to Uh(Aoo), so that p is a representation of UhiAoo) in V({M};£o,£i). The construction is a natural one. We assume that p(u) is a continuous C[[/i]]—linear operator in for any u € U^A^). Then for any v = £°lot;^ € p(u)Vj)^. (47) The above series is well defined, if p(u)vj 6 V({M};£Oi£i)- Since Vj is a (finite) C[[/i]]-linear combination of C-basis vectors the latter holds, and hence p(u) is defined as an operator in V({M};£o>£i)i if p{u)(M) 6 11 ;£0>£i) for any C—pattern (M). Thus, the first step is to clarify which are the C//i( Definition 2. We say that V({Af};£o,£i) w of a finite-signature or, more precisely, of(m,n) signature and unite {M} = {M}m We now proceed to show that each finite signature Uh(aoo) module V({M}m,n; Mm, Mn) can be turned into a Uh{Aoo) module. To this end we prove first a few preliminary propositions. Denote by V({M};fo,6)w, 1 < JV 6 N the subspace of ^({M};fo,6)> which is a C[[h\]-linear envelope of all C—basis vectors (M), for which Mi<2k+e-i = Mi V 2k + 6 - 1 > N, 6 = 0,1, i € [1 - 0 - k, k - 1]; (48) Note that VX{M};£O.£I)AT >S a finite dimensional subspace. Proposition 3. 1 p(efc)V({M};£0,6)N=0, if k i {- -{N + 1), ±(JV - 2)). (49) Proo/. Prom (19) one obtains: k k p(efc)(M) = - j=-k l=-k-l 1/2 (50) 1/2 fe-2 fc-1 1/2 (51) 1/2 i,2fc-2 ~ U,2k-\ {l-2k~1} k If(M)€ if k> -(N - 2)), then Li:2k+3 = Li<2k+2 = L4 = Af4 - i, (52) if k < --(AT + 1), then Li%2k-i = Li}2k = Li = Mi — i. 12 In both cases the r.h.s. of (50) and (51) contain zero multiples (Lj - L{) and therefore vanish. [] Proposition 4. p(fk)V({M}m,n;Mm,Mn)N=0, if (53) Proof. Let (M) € V({M}min;Mm, Mn)/y. The relations that follow from (19) in this case are k k p(fk)(M) = - j=-ki=-k-\ rfc-l 1/2 (54) 1/2 fe-2 Jfc-1 j = -fc+l/=-*+! 1/2 (55) -2 1/2 {3,2k-2 If k > iV/2, then I/i,2fc+3 = ^i,2fc+2 = •i't^fc+i = Li,2k = A = A^t — i and therefore (54) reads: 1/2 1/2 - u+1] n?w..»[i( - ^+ 13 Above only the term with j = k survives: k 1/2 p(fk)(M) = - T S(k,l-0) 1/2 {l'2k+2) 1/2 = -S(k,k;0) '{k'2k+2} 1/2 {''2A:+2} In the last term I ^ k. Hence k fc-l [Li-Lk\ = [Lk-Lk] and therefore it vanishes. Then 1/2 l,1/2 i.e., 1/2 k 2 p(fk)(M) = - \[Mk+1 - Mk}\ (M)Z\ kfkl \. (56) If k > n then Mk+\ = Mk and p(fk)(M) = 0. In a similar way one derives from (55) that p{fk){M) = 0 if k < min{-^(N + 3),m - 1}, which proves (53). [] Proposition 5. p{hk)V{{M}m>n;Mm,Mn)N=Q, if k £ (min{~(N + l),m},max{^N,n}). (57) The proof follows easily from (20). Set = max{-(N + 3), 1 - m,n}. (58) From the last three propositions one concludes. Corollary 1. If k £ (-rjv,r^), then p(hk)V({M}m>n; Mm, Mn)N = p(ek)V({M}m,n; Mm, Mn)N = p(fk)V({M}m Proposition 6. p(Uh(n+)0)V({M};^o,^i)N C V({M};^0^I)N for any N 6 N. More precisely, 7 then If S(P) C (-^-p^, ^Y^) = ^' p(UH(n+)+0)V({M};to,Si)N C ^({M};^,^)^. (60) // , then p(Uh(n+)0)V({M};£o,Zi)N =0. (61) 14 Proof. From (19) (or directly from (50) and (51)) one concludes that p(ej)(M) e V({M };&,&)*, V j € (-^y^, ^^) ™d V (M) 6 V({M}^0,^)N. (62) Hence (60) holds. Assume S(0) <£. IN and let Pp be a monomial of the generators e<, i 6 S((3). Pp can be represented as Pp = Q'enQ, where Q depends only on the generators e, with j € S((3) n 7^ and n ^ IN- Then (62) yields that p(Q)(M) e V({M};£O,ZI)N and therefore (Proposition 3) p(Pp)(M) = p(Q')p{en)p{Q)(M) = 0. 0 Proposition 7. Let u 6e any element from Uh(Aoo), represented as in (4%)- Then oo l(k) t(a,0) 7< 1 )< S^E^)' E E E /K^«.W,t)n^) * ' ^/».*.'.*)(Af)6V({M}m,n;Afm,Afn) (63) fc=0 /=0 a,0€Q'+ 7(fc,O€Z« t=l i6Z for any (M) from the basis T({M}) of V({M}min;Mm,Mn). For a fixed k the number of the non-zero summands in (63) is finite. Proof. In order to prove the proposition it suffices to show that t(<*,0) {k l)i E E E P^a^,t)\[p{hiV ' p{£0 Let for definiteness (M) € V({M}m>n\Mm,Mn)N and let /3o be the weight of (M). Since £p,k,i,t € Uh(n+)p, p(£p,k,i,t)(M) € V({M}m non-zero vectors p(£ptk,i,t)(M), corresponding to different /3 € Q^, are linearly independent (over C[[h]]). Since V({M}mn;Mm,Mn)jv is a finite dimensional subspace, p(£piicjtt)(M) ^ 0 only for a finite number of (3 e Q'+. Hence the sum over /3 in (64) is finite. Setting p(£0,k,i,t)(M) — v0ik,i,t € V({M}m t(a,0) 7(fci ) E E E^«.*.'.«)n^') ' '«/»lW,,. (65) t=l i€Z We proceed to show that the sum over a and 7 in (65) is finite too. Without loss of generality we assume that every Fa,k,i,t Iliez ^< is a monomial of {fit hi}iez- Consider any term from (65), corresponding to a particular pair (0,7): {k l) p(Fa*u) I] P(hir ' 'vp,ku- (66) i€Z + Let J-(a,-y) be the connected components of S(a, 7): jF(a,7) = {/ai>(,, = [ai)6i]|i=l,2,...,n}, 04 < b{ € Z, [a* - bj\ > 1, t/ i ^ j. (67) In (67) we do not distinguish between a connected component Iai,bi (with a beginning in a* and end in 6j) and the corresponding to it finite integer interval [aj,6i]. Then t€Z+ 15 where Ti is a monomial of fj,hj, j € [ai,&j]. FYom (1) and (3) it follows that the multiples ?i in (68) commute. If \a^b^ C (-oo,-r/v] or [a.i,bi] C (r^,oo), then Corollary 1 yields that p(Fi)v0tk,i,t=Q- Hence also p{Ta,k,i,t) riiez PihiV^'^'vpMj. — 0. Thus, the sum in (64) is over such pairs (a, 7), for which all connected components of S(a,7), namely the elements from ^"(0,7), have non-zero intersections with (-rjv.nv). There is only a finite number of pairs (0,7) with this property, for which Int(u,k,i) is finite. The conclusion is that the l.h.s. of the series (64) contains a finite number of non-zero summands. Since the generators of UhiA^) (see (18)-(21)) transform V{{M}m 00 l(k) t(a,0) C (k l) p{u) = E ^ E ^ )' E E E rt*"».W.«) II P{hiV ' 'p{S0,kU) 6 EndV({M}m,n; Mm, Mn) k=0 1=0 a,0€Q'+ -r(fc,/)eZ~ t=l i€Z (69) The map p is a homomorphism of U^A^) in End V({M}m>n; Mm,Mn). It is continuous in the /i-adic topology. Hence p defines a representation of Uh(Aoo) in V({M}m>n; Mm, Mn), which is a highest weight irreducible representation (since it is a highest weight irreps with respect to the subalgebra Uh.(a.oo))- k Let v = X)fcez+ h vk, vk e V({M}m Since each vk is a finite linear combination of C-vectors, for any k € Z+ there exists an integer Nk > 1 such that vk € V{{M}m,n\Mm,Mn)Nk. Then from (59) and (61) one concludes that the following properties hold: 1. p(Uh(n+)0)vk = 0, if S(/J) £ (-rNk,rNk), (70a) 2. p(Uh{n-)-a)vk =0, if S{a) C(-oo,-rwJ or S(a) C [rNk,00), (706) 3. p(hi)vk=0, if \i\>rNk. (70c) Therefore each finite signature representation of UhiAoo) in V({M}mjn; Mm, Mn) is a restricted representa- tion (see Definition 4-1 in Ref- [8]). Let us mention in conclusion that the {/^(aoo) modules V({M};£o,£i), which are not of a finite (m, n) signature and for which £0 ¥" Mm, £1 ^ Mn, cannot be turned into U^iAoo) modules. In order to see this consider the transformation of the highest weight vector (M) under the action of the operator I = ^2i€Z hi e ). Prom (20) and (41) one obtains: The r.h.s. of (71) is not divergent only in the finite-signature modules V({M}m 1 [8]. In that case the representations we have obtained remain highest weight irreps of C//lc(A00) realized in infinite-dimensional complex linear spaces. 16 Acknowledgments. We are grateful to Prof. Randjbar-Daemi for the kind hospitality at the High Energy Section of ICTP. One of us (N.I.S.) is grateful to Prof. M.D. Gould for the invitation to work in his group at the Department of Mathematics in University of Queensland. The work was supported by the Australian Research Council and by the Grant $ - 416 of the Bulgarian Foundation for Scientific Research. References 1. Date, E., Jimbo, M., Kashiwara M., Miwa, T.: Operator Approach to the Kadomtsev-Petviashvili Equation - Transformation Groups for Soliton Equations III. J. Phys. Soc. Japan 50, 3806-3818 (1981). 2. Date, E., Jimbo, M., Kashiwara, M., Miwa, T.: Transformation groups for soliton equations - Euclidean Lie algebras and reduction of the KP hierarchy. Publ. RIMS Kyoto Univ. 18, 1077-1110 (1982). 3. Drinfeld, V.: Quantum groups. ICM proceedings, Berkeley 798-820 (1986). 4. Goddard, P., Olive,D.: Kac-Moody and Virasoro Algebras in Relation to Quantum Physics. Int. J. Mod. Phys. 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