Available at: http://www.ictp.trieste.it/ pub_off IC/99/178

United Nations Educational Scientific and Cultural Organizationand

International Atomic Energy Agency

THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

QUANTUM GROUP Uq(D^) SINGULAR VECTORSIN POINCARE-BIRKHOFF-WITT BASIS

V.K. Dobreva

Arnold Sommerfeld Institute for Mathematical Physics, Technical University Clausthal,Leibnizstr. 10, 38678 Clausthal-Zellerf'eld, Germany,h

Bulgarian Academy of Sciences, Institute of Nuclear Research and Nuclear Energy,72 Tsarigradsko Chaussee, 1784 Sofia, Bulgaria0

andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy

and

M. EL Falakid

Laboratoire de Physique Theorique, Faculte des Sciences,Universite Mohammed V, BP. 1014, Rabat, Morocco

andThe Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.

Abstract

We give explicit expressions for the singular vectors of Uq(Di) in terms of thePoincare-Birkhoff-Witt (PBW) basis. We relate these expressions with those in terms ofthe simple root vectors.

MIRAMARE - TRIESTE

December 1999

a E-mail: dobrev@inrne.bas.bg

Current address. E-mail: asvd@rz.tu-clausthal.dec Permanent address.

E-mail: mantrach@fsr.ac.ma

1. Introduction

Singular vectors of Verma modules appeared in the representation theory of semisimpleLie algebras and groups, cf. [1], [2], [3], [4]. After that singular vectors have also madea great impact in physics and not only from the point of view of applications. In fact,the generalization of singular vectors to other symmetry objects was done primarily in the(mathematical) physics literature starting with the paper [5], where singular vectors ofthe Virasoro algebra were crucially used. More explicit examples in this case were givenin [6], [7], [8], [9], [10]. Further, singular vectors were given for Kac-Moody algebras [11],[12], the conformal superalgebra su(21 2/n) [13], the N = 1 super-Virasoro algebras [14],[15], quantum groups [16], [17], [18], ^-algebras [19], [20], [21], the N = 2 super-Virasoroalgebras [22], [23], [24], the N = 4 super-Virasoro algebras [25], Kac-Moody superalgebras[26], [27].

In the present paper we consider singular vectors on Verma modules over the Drinfeld-Jimbo quantum groups [28], [29]. These are g-deformations Uq(Q) of the universalenveloping algebras U(Q) of simple Lie algebras Q and are called quantum groups[28], quantum universal enveloping algebras [30], [31], or just quantum algebras. Thepresent paper may be viewed as a natural continuation of the paper [17], where explicitformulae were given for the singular vectors of Verma modules over Uq{Q) for arbitraryQ corresponding to a class of positive roots of £/, which were called straight roots, and someexamples corresponding to arbitrary positive roots. Note that these results are completeonly for Q = Ag since in that case all positive roots are straight. The singular vectorswere given only through the simple root vectors as in earlier work in the case q = 1, cf.[32], [33]. (This basis turned out to be part of a more general basis introduced later inthe context of quantum groups, though for other reasons, by Lusztig [34].) On the otherhand, there were examples, both in the undeformed case [35] and the q-deformed case[36], [37], [38], when it was convenient to use singular vectors in the Poincare-Birkhoff-Witt(PBW) basis. In principle, the paper [18] generalizes the results of [11], (from where PBWsingular vectors may be extracted), to the quantum group case, however, the formulae arenot so explicit as it is necessary for the applications.

Thus, the first result of the present paper gives explicit expressions for the singularvectors of Uq(Di) in terms of the PBW basis. The second result relates these expressionswith those in terms of the simple root vectors. This we also do for the non-straight roots,which were omitted in [17]. The second result is not known also for q = 1.

2. Preliminaries

Let Q be a complex simple Lie algebra with Chevalley generators Xf , Hi , i = 1, ...,£ =rank Q. Then the quantum algebra Uq(Q) is the g-deformation of the universal envelopingalgebra U(Q) defined as the associative algebra over W with generators Xf , Ki = qi

l ,Kr1 = q~H- , and with relations [29] :

^K% = 1 , K% Xf Kr f

\ , (16)

i , Kj\ = 0, KiKr1 = K^K% = 1 , K% Xf Kr1 = qfa^Xf , (la)

Efc=0

where qi = q^ai >ai)/2, (a^-) = (2(pn , aj)/(ai, a^)) is the Cartan matrix of Q, (•, •) isthe scalar product of the roots normalized so that for the short roots a we have (a, a) = 2,n = 1 — a,ij ,

4Further we may omit the subscript q in [m]q if no confusion could arise.

The above definition is valid also when Q is an affine Kac-Moody algebra

We use the standard decompositions into direct sums of vector subspaces Q = % ©Qp = Q+ © % © Q-, Q± = © gp , where % is the Cartan subalgebra spanned by

3 A ±

the elements Hi, A = A+ U A~ is the root system of £?, A+, A~, the sets of positive,negative, roots, respectively; A5 will denote the set of simple roots of A. We recall thatHj correspond to the simple roots OLJ of g, and if /3V = ^2- rij a j , /3V = 2/3/(/3, /3), thenf3 corresponds to Hp = ^ . n^ i^-.

For the PBW basis of Uq(g) besides Xf , X ^ 1 we also need the Cart an-Weyl(CW) generators X ^ corresponding to the non-simple roots (3 G A + . Naturally, we shalluse a uniform notation, so that X^. = Xf-. The CW generators X"t are normalized sothat [29], [16], [39] :

We shall not use the fact that the algebra Uq(g) is a Hopf algebra and consequentlywe shall not introduce the corresponding structure.

The highest weight modules V over Uq(g) are given by their highest weight A GH* and highest weight vector VQ G V such that:

KiV0 = q^v0, X+v0 = 0 , i = l,...,£, A, = (A, a,v) (3)

We start with the Verma modules VA such that VA = Uq(g~)<S>vo . We recall severalfacts from [16]. The Verma module V is reducible if there exists a root (3 G A"1" andm G IN such that

[(A + p , / ? v ) - m ] g , = 0 , (4)

holds, where p = \ S a eA+ ot . li q is not a root of unity then (4) is also a necessarycondition for reducibility and then it may be rewritten as 2(A + p , (3) = m(/3, (3). (Inthat case it is the generalization of the (necessary and sufficient) reducibility conditions forVerma modules over finite-dimensional g [1] and affine Lie algebras [40].) For uniformitywe shall write the reducibility condition in the general form (4). If (4) holds then thereexists a vector vs G VA, called a singular vector, such that vs ^ @ vo , and:

KiV8 = q t > - m i P ' a r ) vs , i = l,...,£, (5a)

X+vs = 0, i = !,...,£, (56)

The space Uq(Q )vs is a proper submodule of VA isomorphic to the Verma moduleyA-m,p _ Uq(C/~)<S)v'o where v'o is the highest weight vector of VA~rn^; the isomorphismbeing realized by vs i—>• 1 (g) v'o. The singular vector is given by [32], [33], [16]:

vs = vp'm = Vi®vG (6)

where V^ is a homogeneous polynomial of weight m(3. The polynomial V^ is uniqueup to a non-zero multiplicative constant. The Verma module VA contains a uniqueproper maximal submodule IA. Among the HWM with highest weight A there is aunique irreducible one, denoted by LA, i.e., LA = VA/IA . If VA is irreducible thenLA = VA. Thus we discuss further LA for which VA is reducible. If VA is reduciblewith respect to (w.r.t.) to every simple root (and thus w.r.t. to all positive roots), thenLA is a finite-dimensional highest weight module over Uq(Q) [41]. The representationsof Uq(Q) are deformations of the representations of U(Q), and the latter are obtainedfrom the former for q —> 1 [41].

In [17] the singular vectors were given only through the simple root vectors, namely:

, . . . , A 7 ) ® W o , (7)

so that V^ is a homogeneous polynomial in its variables of degrees mrii, where rii E%+ come from f3 = ^ niai •

The aim of the present paper is to give expressions for the singular vectors in termsof the PBW basis and to relate these expressions with those of [17].

3. Singular vectors for the straight roots

3.1. Singular vectors in PBW basis

In this paper we consider Uq(Q) when the deformation parameter q is not a nontrivialroot of unity. This generic case is very important for two reasons. First, for q = 1 allformulae are valid also for the undeformed case and formulae for the relation with [17] arealso new for q = 1. Second, the formulae for the case when q is a root of unity use theformulae for generic q as important input as is explained in [17].

Let Q = Di, £ > 4. Let «j , i = 1 , . . . ,£ be the simple roots, so that(«i , «j) = —1 if either \i—j\ = 1 , i,j ^ £ or ij = £(£ — 2) and (pn , a.j) = 28 ij inother cases.

Then all positive roots are given as follows

otij = Ui + ai+1 + ... + oij , 1 < i < j < £ - 2 ,

J3j = otj + aj+i + . . . + Oit-2 + oit-i, 1 < j < £ - 2 ,

Po = ai_2 + Oii-i + ai,

7j = aj + aj+i + ••• + otg-2 + a£-! + a£, 1 < j < ^ - 3 ,

lij = oii + ai+i + . . . + 2 (ay + . . . + at-?) + Oil-i + Oil, 1 <i < j <

We recall that the roots a^-, (3j , f3j , (3$ , are positive roots of various An subalgebras.Thus, we have to consider only the roots 7̂ and 7^ . We recall from [17] that 7̂ arestraight, while 7^ are not straight.

In this section we deal with the straight roots 7^. Now we recall that every root 7̂is the highest straight root of a Di-j+\ subalgebra of Dp . This means that it is enoughto give the formula for the singular vector corresponding to the highest straight root 71 .

Further we shall need the explicit expressions for the non-simple-root Cartan-Weyl(CW) generators of Uq{Q). Let XJ , Y± , Yp, Y± , Z± and Zf^ be the Cartan-Weyl generators corresponding respectively to the roots ±«ij , ±/% , ±/3j , ±/?o ,and

The CW generators corresponding to the nonsimple roots are given as follows:

X± _ _|_ ^T 1 / 2 (Q1/2 X±X± — a~1l2 X^ X^\ = (9a)

- « / i - \ / - i / i - \ i i - i / t - \ i i \

= ±

= ± «T1/2 (<11/2 X±Y±+1 - q~^ Y±+1X±) , 1 < i < £ - 2

= (9c)

= ± ?T 1 / 2 (?1 / 2 X^+1 - q-1/2Y±+1X±) , 1 < 3 < t - 2

y± _ 1 OTI/2 (ai/2 x± Y± - a~1/2 Y± X± ) -

1/2 Y£_2 q i 2 i )

112 tf^ = (9e)

± q*1/2 (q1/2 XUY? - q-^Ytxt,) , 1 < 3 < I - 3

± 1T1/2 {i1/2 ZfXftl_2 - q-1/2 X^_2Zf) , l<i<j<l-2 (9/)

Now the PBW basis of Uq(Q ) is given by the following monomials:

V ( 7~ \St~3,l-2 ( V~ \Sl-4,l-2 ( 7~ \Sl,l-2 (V— \il-3(V— \tl-3-A \Zj£-3,£-2/ \^£-4:,£-2/ • • • \Zjl,£-2) K1 £-3/ \J£-3/

X I ZJ n A n o ' . . . [ ZJ-I o o I ' . . . M i ) M i ) U n A

(10)

~\ai

These monomials are in the so-called normal order [39]. Namely, we put the simple rootvectors XJ in the order X^_2 , XJ , X^_1, X^_3, ... , X2 , X]~. Then we put a root

vector Ea corresponding to the nonsimple root a between the root vectors Eg andE~ if a = /5+7, a, /3,7 G A+. This order is not complete but this is not a problem, sincewhen two roots are not ordered this means that the corresponding root vectors commute,e.g., [Xr , Xr_kt.+k] =0, and [Yr , Y~] = 0, 1 < i < £ - 2.

Let us have condition (4) fulfilled for 71, but not for any of its subroots % , i > 1 :

p,7iV) -m]q = 0 , me IN, (lla)

e W . (116)

(The necessity of the condition (116) was explained in [17].) Let us denote the singularvector corresponding to (lla) by

vJi,m _ ST^ T~,li,m 1 y - \ae-2(v- \te-3,e-2 (v~ \ti,i-2 (y- \ii-2 (y- \te-2 v

T

V (7~ \Sl-3,l-2( V~ \Sl-4,l-2 (7- \S\,L-I(y— \i,_-3(y- \ti-3 vx {^e-3,1-2) \Ai-A,i-2) • • • lZ/i,^-2/1 \xe.-z) \xt-z) x

v ( 7~ \si-4,i-3 ( 7— \s\,i-z (V~\t\ (y—\t^ (V~\t vx {^1-4,1-3) ••• \^x,i-z) ••• \ Y \ ) \ Y \ ) {yo ) x

v ( 7~ \si-3 ( 7~\Sl ( Y~\a>- ( Y~ \a>--i-( Y~ \ai~3 \/x {^£-3) ••• l z i ) \^i ) K^i-i) \ A i - z ) x

V IY~ \tl-4,l-3 (Y~ ^l,'-3fY- \ai-4^X {^1-4^-3) ••• l A i , £ _ 3 J ^1-4) X

x . . . (x2-)a2(xr2) i-(x1-)a i®Vo ,(12)

where T denotes the set of summation variables ai,Uj,Sij,ti,ti,Si,t, all of which arenonnegative integers. Next we impose the conditions (5) with f3 —> 71. (Inequalities (116)mean that no other conditions need to be imposed.) Conditions (5a) restrict the linearcombination to terms of weight 77171 . In our parametrization these are the following£ conditions:

p 1-2 1-2

ap = m-^2 {(fi + ii + Si) + ^2 fij + ^2 sij + 2 ^2 Sii)' 1 - p - (j ~ 3

i=l j=p j=p+l

1-2 £-3

2

1-2 1-3 ^ '

g-i = m -

1-3 1-3

This eliminates the summation in aj in (12) and also restricts further the summationtij,Sij,ii,ti,Si,t so that the a^ in (13) would be all nonnegative.

Furthermore, conditions (56) fix the coefficients Dj,1'"1 completely and we have:

= De ( -v-« n^i

j=2 j = l

£ 2 ^ 3 £ 3

xf=[ r,(A+j + l)rg(A£-i + 1 - a£-i)rg(Ai + 1 - a£) £

r ( A + 2 ) r ( A + 2) ' ^

Ar := (A, /3r), with

where

1-2

and the factor A is given by:

- 4 1-4 1-3

Ep=0 p=0

E1-3

\ fz—[(P — O\ \ / . . _i_ /"/ _i_ 1 ^ \ e. . _i_ / \ G. \rn

e-3

E- 2 i = l

+ Ep=l 3 =

V

E *

I i<i<j<e-

/ •< %i

1-3

,( E -

-2 i=l

t-3

£ - 3

£ — p)

£ - 3

fc=l

E1-3 1-2

1-3where t6 := 5] tbfc .

k=j+i

Finally, we explicate how to obtain the singular vectors for the roots 7J , i > 1 fromthe above formulae. For this one has to replace £ —>• ^ — z + 1, and then to shift theenumeration of the roots, namely, to replace 1,..., £ — i + 1 by i,..., £.

3.2. Relation between the two expressions for the singular vectors

Here we would like to present the relation between the expressions for the singular vectorsin the PBW basis given in (14) and in the simple root vectors basis given in [17]. Thelatter formula is (cf. f-la (16):

2^ " f c i - " f c £ - i l A i ) •••(^1-3) \ x i - i ) x

fci=0 ki-1=O

x ( X " ) m - ^ - 2 (X-_2)m(X-)^-2(X£ l1)^-1x

x (X"_3)^-3 . . . (Xr)fcl<S^o , (16a)

Xq

^ ^ x[(A + p, 0i) - fei], • • • [(A + p,

[(A + p, a£) - k£-2]q [(A + p, ae-i) - ke-i]q

X

(166)

where As = (A, as)1

The D—coefficients are given in terms of the d—coefficients by the following formula:

[tv'msijWsj-iV-'mtjWiV- n t^-]'j=2 j = l

( ) gX X

[m-2t- J2(U + U) ~ 2 E % - 2

[m-fc£_i]! [m-fc<-2]! (fc<_ia<_1+fc<_2a<) , ,

where 0 < kp < api 0 < p < £ — 3, fe^_i < a^_i and fe^_2 < a^ .

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4. Singular vectors for the nonstraight roots

4.1. Singular vectors in the PBW basis

The nonstraight roots of Di are given in (8). We shall also write them as:

e~frp = ^2 rij aj , 1 < r < p < £ - 2

j=r

{ 1 for r < j < p2 for p < j < £ - 21 for j = £- 1,£

Like in the case of straight roots we could use the fact that every root j r p can be treatedas the root jip of a Di_r+i subalgebra of Di. This means that it would be enough togive the formula for the singular vector corresponding to the roots 7ip . However, we shallnot do this for these roots, since anyway it is not reduced to single root.

Let us have condition (4) fulfilled for ̂ rp , but not for any of its subroots. The singularvectors corresponding to these roots are given by:

yjrp,in __ \ "* j-yJrP,in (j£— \2m — bt-2(j£— \ti —3,1-2 (X~ \tr,i-2 (Y~ \tt-2(y~— \ti-2-^

T

l-4l-2 I 7~ \Sr,l-2 (V— \• • • \Zjr,e-2) \Yi-3)

Sl-4,l-3 X

X . . . yJ£r_i_i) \ r r+1) r'r \X-r ) r ® VQ (19)

where the coefficients D^rp'm are given by:

i-r [mnB-bB]\Z_d

Sii A A [mB s -6 s ] !nJr-P,m j^ns < 1\r.^,- s=r+l

[t\i n [srMsj-iV-nfoWiV- nj=r+l j=r

[2m -2t-^2(ti + k) - 2 £ Sij - 2 E ̂ " Ei=r r<i<j<£—2 i=r i=r

E'~\ Jmrn-biA'i) Tq(A'J ~ mUj + bj + ^J-l.j) „

rg(A£_i + 1 - m + 6£-i)rg(A^ + 1 - m + bj)r ( A

, DnS ̂ 0 (20)

10

where we have set for r < p < £ — 3 :

V i-2

1-2 1-2

i=r j=p j=p+l

1-2 1-3

i=r i=r1-2 es

i=r i=r1-3 1-3

—2

4.2. Singular vectors in the simple roots basis

T h e s i n g u l a r v e c t o r s c o r r e s p o n d i n g t o t h e n o n s t r a i g h t r o o t s , j r p , l<r<p<£ — 2 , i nt h e s i m p l e r o o t b a s i s a r e g i v e n b y

kr=0kr+1=0 ke-!=0

(v- \2rn-ke-3 ( y— \m-ki-X ( y-\m-ki-2

x (XiLj"'-! (Xl_3)k^ ... (X-)*- ® Vo (22)

Now the coefficients d are given by the following expression

l y O y-> \ K y-> TTL O p "I '^ h. p "I

( j j ^ ^ j m n _ hM

I ^ —[mrij — kj]\[mrij — bj]l[kj — mrij + bj]\

_ [mri

[m - bi]\[ki-2 - m - bg]\[m - &£_2]! [m - bt-i]\[kt-i -m

1-2 1-3 1-3i=r r<i<j'<^—2

1-2 1-2

[t]l I ] W![«i-i]!llfe]!fej=r+l j=r

more explicitly the d—coefficients are given by

) q

ff

[(A + p, p r ' r ) - kr\q [(A+ p,fjr>1 6) - ki-3\q

[(A + p, «£) - &£_2]g [(A + p , «£_l) - fe£-

[At + 1 - fe£_2]q [A£_i + 1 - fc£_i]q 'i

i<*i> A ' J = (A, 13rd) , n* := £ n, . (24)

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