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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 - (</ + q-l)fi+ififi+i + fifli = 0. 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<j, (5) eji = l/i. [/i+l. [• • • 1 [/j-2, fj-l\q • • }q}q)q, i + K j. The "commutation relations" between these generators follow from (l)-(3) and are given in Ref. [8]. The relevance of the generators {^ij}(i,j)ez3 stems from the observation that the set of ordered monomials (see Ref. [8] for the ordering) ci n <r (6) with finitely many non-zero exponents n^ € Z+, !eZ+ forms a (topological) basis in Uh(a<3o)- Set if = ©iC/ij, define linear functionals £j : H —> C by £i(/ij) = 5<j and set a< = ei - £j+i> Q'+ = ©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.
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