IC/93/123

FINITE-DIMENSIONAL REPRESENTATIONS

OF THE QUANTUM SUPERALGEBRA Uq[gl(3/2)]

T.D. Palev

and

N.1. Stoilova IC/93/123

International Atomic Energy Agency and United Nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

FINITE-DIMENSIONAL REPRESENTATIONS

OF THE QUANTUM SUPERALGEBRA Uq [gl(3/2)]

T.D. Palev1 International Centre for Theoretical Physics, Trieste, ItaJy and N.I. Stoilova Applied Mathematics and Computer Science, University of Gent, Gent, B-9000 Belgium.

ABSTRACT For generic q we give expressions for the transformations of all essentially typical finite­ dimensional modules of the Hopf superalgebra Uq [gl(3/2)]. The latter is a deformation of the universal enveloping algebra of the Lie superalgebra gl(3/2). The basis within each module is similar to the Gel'fand-Zetlin basis for gl(5). \Ve write down expressions for the transformations of the basis under the action of the Chevalley generators.

MIRAMARE TRIESTE June 1993

IPermanent address: Institute for Nuclear Research and Nuclear Energy, 1784 Sofia, Bulgaria; e-mail: [email protected] This letter is devoted to the study of a subclass of representations of the quantized associative superalgebra Uq[gl(3/2)]' which we call essentially typical (see Definition 1). The motivation to consider this particular algebra stems, firstly, from the observation that it is relatively simple and nevertheless its representation theory remains so far undeveloped explicitly and, secondly, that it carries all main features of the general Uq[gl(n/m)] superalgebra. The hope is, therefore, that Uq[gl(3/2)] could serve as a generic example to tackle as a next step the representation theory of the deformed superalgebra gl(n/m). To begin with we recall that a quantum algebra Uq[G] associated with the Lie (super)algebra G is a deformation of the universal enveloping algebra U[G] of G, preserving its Hopf algebra structure. The first example of this kind was found for G = sl(2) [1] endowed with a Hopf structure given by 8klyanin [2]. An example of a quantum superalgebra, namely the L8 osp(1/2), was first considered by Kulish [3] and it was soon generalized to an arbitrary Kac-Moody superalgebra with a symmetrizable generalized Cartan matrix [4]; an independent approach for basic Lie superalgebras was given in [5-7].

Beginning with Uq[osp(1/2)] [8] the representation theory of the quantum superalgebras became also a field of growing interest. Various oscillator representations of all infinite series of the basic Lie superalgebras have been found (see, for instance, Refs. 5, 7, g, 10). Nevertheless the results in this respect are still modest, the main reason being that even in the nondeformed case the representation theory of the best known, namely the basic Lie superalgebras (L8's) is far from being complete. Even the problem of obtaining character formulae for all atypical modules is still not completely solved (see [11] and the references therein). Apart from the lowest rank L8's, explicit expressions for the transformations of all finite-dimensional modules have been given so far only for the class gl(n/1), n = 1,2, ... [12]. By explicit we understand what is usually meant in physics, namely introducing a basis in the representation space and writing down expressions for the transformations of the basis under the action of the generators ( or, which is the same, giving expressions for the matrix elements of the generators with respect to the selected basis). A class of finite-dimensional irreducible representations of gl(n / m), namely the essentially typ­ ical representations (ET irreps) were explicitly constructed in [13]. Our main hypothesis is that each ET irrep can be deformed to an irreducible representation of Uq[gl(n/m)] by an appropriate replacement of certain factors x in the matrix elements by q-factors [x] = q;~r~,x.This is the case for gl(n) [14] and gl(n/1) [15]. Here we show how it works for Uq[gl(3/2)]. Our initial idea was to solve the problem for any nand m and, in fact, we believe we know the solution. The proof, however, even for Uq[gl(3/2)] took enormous amount of computer time. At present we are not able to carry out similar proof for any nand m. Before going to the deformed case we recall that the Lie superalgebra (L8) gl(3/2) is an extension of the basic L8 sl(3/2) [16] by a one dimensional center with generator I. A convenient basis in

2 gl(3/2) is given by the Weyl generators eij, i, j = 1,2,3,4,5, which satisfy the relations

(1) where Bij = Bi + Bj ,Bi = 0 for i = 1,2,3, Bi = 1 for i = 3,4. The Z2-grading on gl(3/2) is imposed from the requirement that eij is an even (respectively an odd) generator, if Bij is an even (respectively odd) number. Then sl(3/2) is a factor algebra of gl(3/2) with respect to the ideal generated by I = en + e22+ e33+ e44+ e55. In the lowest 5 x 5 representation eij is a matrix with 1 on the cross of the ith row and the ph column and zeros elsewhere and I is the unit matrix. The universal enveloping algebra U[gl(3/2)] of gl(3/2) is the free associative algebra of all generators eij with relations (1). A representation of the L8 gl(3/2) is by definition a representation of U[gl(3/2)] viewed as a Z2 (= {O, 1})-graded associative algebra, i.e., as associative superalgebra. The irreducible finite-dimensional modules (=representation spaces) of any basic L8 are either typical or atypical [17]. It is convenient to consider any irreducible 81(3/2) module W as an irre­ ducible gl(3/2) module, setting I proportional to the identity operator in W. In this sense one can speak about typical or atypical gl(3/2) modules. More precisely, by a typical (resp. atypical) gl(3/2) module we understand a gl(3/2) module W, which remains typical (resp. atypical) with respect to sl(3/2) and for which I is proportional to the identity operator in W. A subclass of the typical modules will be relevant for us. Definition 1.[13] A typical gl(3/2) module W is said to be essentially typical, if it decomposes into a direct sum of only typical gl(3/1) modules. In the present letter we show that for generic values of q (=q is not root of unity) any essentially typical gl(3/2) module W can be deformed to an irreducible Uq [gl(3/2)] module, which we call also essentially typical. We proceed to describe in some more detail a basis within each typical module (and, more generally, - within a larger class of finite-dimensional induced modules, namely the Kac modules [18]), since the same basis will be used also in the deformed case.

Let ell, en, e33, e44, e55 be a basis in the Cartan subalgebra Hand e1, e2, e3, e4, e5 be the basis dual to it in H*. Denote by W([m]s) the Kac module with highest weight

(2)

and let

li5=mi5-i+4fori=I,2,3 and li5=-mi5+i-3fori=4,5. (3)

Proposition 1 [13]. The Kac modules W([m]s) are in one to one correspondence with the set of all complex coordinates of the highest weight, which satisfy the conditions:

3 m15 - m25, m25 - m35, m45 - m55 E Z+ [= all nonnegative integers]. (4)

The module W([m]s) is typical if and only if all 115 , 125 , 135 , 145 , 155 are different. The typical module

W([m]s) is essentially typical if and only if 115 , 125 , 135 =F145 ,145 + 1,145 + 2, ... ,155 , Let (m) be a pattern consisting of 15 complex numbers mij, i :sj = 1,2,3,4,5, ordered as in the usual Gel'fand-Zetlin basis for gl(5) [19], i. e.,

m25 m35 m45 m14 m24 m34 m44 m,,) (m) == m13 m23 m33 (5) m12 m22 C"mll Proposition 2 [20]. The set of all patterns (5), whose entries satisfy the conditions

(1) m15, m25, m35, m45, m55 are fixed and the same for all patterns,

(2) For each i = 1,2,3 and p = 4,5 mip - mi,p-l ==Bi,p-l E Z2,

(3) For each i = 1,2 and p = 4,5 mip - mi+l,p E Z+, (6)

(4) For each i:S j = 1,2 mi,Hl - mij, mij - mi+l,j+l E Z+,

(5) m45 - m44, m44 - m55 E Z+, constitute a basis in the Kac module W([mh), which we refer to as a GZ-basis.

The quantum superalgebra Uq [gl(3/2)] (we consider here), is a one parameter Hopf deformation of the universal enveloping algebra of g[(3/2) with a deformation parameter q = en. As usually, the limit q -+ 1 ( Ii -+ 0) corresponds to the nondeformed case. More precisely, Uq [gl(3/2)] is a free associative algebra with unity, generated by ei, Ii and kj ==qhjf2, i = 1,2,3,4, j = 1,2,3,4,5, which satisfy (unless otherwise stated the indices below run over all possible values) 1. The Cartan relations

(7)

(8)

ei /j - h ei = 0 i =Fj; (9)

-1)-1(k2k2 k- 2 k- 2 ). e3f 3 + f 3 e3 = (q - q. 3 4 - 3 4 ' (10)

2. The Serre relations for the positive simple root vectors (E-Serre relations)

(11)

4 i = 1,2; (12)

(13)

(14)

3. The relations obtained from eqs.(1l-14) by replacing everywhere ei by fi (F-Serre relations).

The eq.(14) is the extra Serre relation, discovered recently [21-23] and reported by Tolstoy, Scheunert and Vinet on the II International Wigner Symposium (July 1991, Goslar, Germany). The Z2-grading

on Uq [gl(3/2)] is defined from the requirement that the only odd generators are e3 and 13. Uq [gl(3/'2)] is a Hopf algebra with a counity c, a comultiplication ~ and an antipode S defined as

c(e;) = c(J;) = c(kd = 0, (15)

uA(k·)2- k·z \:yMI f\,2;•.

.6.(e;) = ei ® kiki.(:l + ki1ki+l ® ei, i =j::.3, ~(e3)= e3 ® k3k4 + k;;lk;l ® e3, (16) 1k .6.(f;) = Ii ® kiki.(:l + ki i+l ® Ii, i =j::.3, ~(13)= 13® k 3 k 4 + k;;lk;l ® 13,

Now we are ready to state our main result. To this end denote by (m)±ij a pattern obtained from the GZ pattern (m) after a replacement of mij with mij ± 1 and let

Iij = mij - i + 4 for i = 1,2,3 and Iij = -mij + i - 3 for i = 4,5, (18)

Proposition 3. For generic values of q every essentially typical gl(3/2) module W([mh) can be

deformed to an irreducible Uq [gl(3/2)] module. The transformations of the basis under the action of the algebra generators read

1 ('\'i . ,\,i-1 ) k (1n) qO L..j=l mj,- L..j=l mj,i-l (m), i = i = 1,2,3,4,5, (19)

k = 1,2, (20)

5 k = 1,2, (21)

(22)

(23)

(~:4)

3 14 f4(m) = ([144 -1 45 + 1][155 - 144 - 1])1/2(m)_44 + L Bi3(1- Bi4)(_1/ ++8i-1,4+8i+1,3++833 i=l

(25)

As we have mentioned, the proof is lengthy, It goes on a case-by-case ground. In order to carry out the proof we have checked that all Cartan and Serre relations hold as operator equations in W([m]s). Eqs.(20)-(21) are the same as in [14] and (22)-(23) - as in [13] for n = 3. The irreducibility follows from the results of Zhang [24]. It is evident also from the observation that for generic q a deformed matrix element in the GZ basis is zero only if the corresponding nondeformed matrix element vanishes [see (19)-(25)]. Remark. If a vector from the rhs of eqs. (19)-(25) does not belong to the module under consideration, than it is assumed that the corresponding term is zero even if the coefficient in front of it is undefined. If an equal number of factors in a numerator and a denominator are simultaneously equal to zero, then one has to cancel them out. The operators ei-1, /;-1, ki- 1, ki' Z = 1,2, ... ,n generate a chain of Hopf subalgebras

Uq [gl(l)] for n = 1, Uq [gl(2)] for n = 2, Uq [gl(3)] for n = 3, Uq[gl(3/1)] for n = 4, Uq [gl(3/2)] for n = 5,

Uq [gl(l)] C Uq [gl(2)] C Uq [gl(3)] C Uq [gl(3/1)] C Uq [gl(3/2)]. (26)

6 The row [m]n ==(mIn, . ., mnn) in a GZ pattern (m) indicates that (m) belongs to an irreducible submodule W([m]n) C W([m]s) of the subalgebra with the corresponding n. Therefore the decom­ position of any essentially typical module W([m]s) along the chain (26) is evident and the GZ basis is reduced with respect to this chain. More precisely (as this is the case also with the GZ basis of gl(5)), each basis vector (m) is an element from one and only one chain of submodules,

(m) E W([m]d C W([mh) C W([mh) C W([m]4) C W([m]s). (27)

Unfortunately we do not know at present how to extend the above results beyond the class of the essentially typical representations, although it is known that each irreducible finite-dimensional rep­

resentation can be deformed to a Uq [gl(3j2)] representation [24]. We hope, however, that the present

approach could at least be generalized for the essentially typical representations of Uq [gl(nj m)], al­ though technically this could be quite difficult. It remains still an open question whether one can deform the Kac modules even for generic q.

Note. Our attention was drawn to a preprint to appear [25] on the quantization of Uq [gl(2j2)]

in a gl(2) ffi gl(2) basis. The comparison with our results shows how convenient the GZ basis is. The analog of the transformation relations (19)-(25) in written in [25] on more than 10 pages. The results describe, however all typical representation.

Acknowledgments

The authors are grateful to Prof. Van den Berghe for the possibility to work at the Department of Applied Mathematics and Computer Science, Uni­ versity of Gent. It is a pleasure to thank Dr. J. Van der Jeugt for stimulating discussions and introduction to l\1axima, which was of great help. One of the authors (T.D.P.) would like to thank Professor Abdus Salam, the International Atomic Energy Agency and UNESCO for hospitality at the International Cen­ tre for Theoretical Physics, Trieste. This work was supported by the European Community, contract No. ERB-CIPA-CT-92-2011 (Cooperation in Science and Technology with Central and Eastern European Countries) and grant <1>-215of the Committee of Science in Bulgaria.

References

[1] Kulish PP and Reshetikhin N Yu 1980 Zapiski nauch. sernin. LOMI101 112

[2] Sklyanin EK 1985 Usp. Math. Nauk 40 214

[3] Kulish PP 1988 Zapiski nauch. sernin. LOMI 169 95

[4] Tolstoy VN 1990 Lect. Notes in Physics 370 118 Berlin, Heidelberg, New York: Springer

[5] Chaichian M and Kulish PP 1990 Phys. Lett B 23472

[6] Bracken A J, Gould MD and Zhang RB 1990 Mod. Phys. Lett. A 5 831

7 [7] Floreanini R, Spiridonov V and Vinet L 1991 Commun Math. Phys. 137 149

[8] Kulish PP and Reshetikhin N Yu 1989 Lett. Math. Phys 18 143

[9] Chaichian M, Kulish PP and Lukierski J 1990 Phys. Lett B 237 401 [10] Floreanini R, Spiridonov V and Vinet L Phys. Lett B 242 383 [11] Van del' Jeugt J, Hughes JW B, King RC and Thierry-Mieg J 1990 Journ. Math. Phys. 312278

Hughes JW B, King RC and Van del' Jeugt .J 1992 Journ. Math. Phys. 33470 [12] Paley TD 1987 Funkt. Anal. Prilozh. 21 No3 85 (in Russian); Funct. Anal. Appl. 21 245 (English translation) Paley TD 1989 Journ. Math. Phys. 30 1433 [13] Paley TD 1989 Funkt. Anal. Prilozh. 23 No2 69 (in Russian); Funct. Anal. Appl. 23 141 (English translation) [14] Jimbo M 1986 Lect. Notes in Physics 246334 Berlin, Heidelberg, New York: Springer [15] Paley TD and Tolstoy V. N. 1991 Commun Math. Phys. 141 549 [16] Kac VG 1977 Adv. Math. 26 8 [17] Kac VG 1979 Lect. Notes in Mathematics 626 597. Berlin, Heidelberg, New York: Springer [18] Van del' Jeugt J, Hughes J W B, King RC and Thierry-Mieg J 1990 Commun. Algebra 18 3453 [19] Gel'fand IM and Zetlin ML 1950 DAN SSSR 71 825-838 (in Russian) [20] Paley TD (unpublished). Some of the results have been announced in [17] [21] Khoroshkin SM and Tolstoy VN 1991 Commun Math. Phys. 141 599 [22] Scheunert 1992 Lett. Math. Phys 24 173 [23] Floreanini R, Leites DA and Vinet L 1991 Lett. Math. Phys 23 127 [24] Zhang RB 1993 Jount. Math. Phys. 341236

[25] Ky NA 1993 Finite-dimensional representations of the quantum superalgebra Uq [gl(2/2)]: 1. Typical representations.

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