Three lectures on quantum groups: representations, duality, real forms

JOURNAL OF

JournalofGeometryandPhysics North-Holland

11(1993)367—396

GEOMETRYAND

PHYSICS

Three lectures on quantum groups: representations, duality, real forms V.K. Dobrev’ International Centrefor Theoretical Physics, P.O. Box 586, 34100 Trieste, Italy

In these lectures three topics are discussed: the representations of quantum groups, duality between quantum algebras and matrix quantum groups and q-deformations of real forms of quantum groups.

Keywords: quantum groups 1991 MSC: 17B37, 81R50

Contents Introduction 1. Preliminaries 1.1. Hopf algebras and quantum groups 1.2. Quantum algebras 1.3. Matrix quantum groups 2. Representations of Uq( ~I) 2.1. Verma modules and their irreducible subquotients 2.2. Singular vectors 3. Duality 3.1. Duality between Hopf algebras 3.2. Matrix quantum group GLpq(2, C) 3.3. Duality a Ia Sudbery for GL~,~(2, C) 4. Real forms 4.1. Overview of the procedure 4.2. q-deformation with the most non-compact Cartan subalgebra 4.3. q-deformations with arbitrary Cartan subalgebras 4.4. q-deformations for arbitrary parabolic subalgebras and reductive Lie (super-)algebras References

368 368 368 370 372 375 375 376 382 382 382 384 388 388 389 392 393 394

At ICTP until July 31, 1992; permanent address: Bulgarian Academy of Sciences, Institute of Nuclear Research and Nuclear Energy, 72 Tsarigradsko Chaussee, 1 784 Sofia, Bulgaria.

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VK. Dobrev / Representations, duality, real fbrrns

Introduction

Quantum groups appeared first as quantum algebras, i.e., as one-parameter deformations of the universal enveloping algebras of complex simple Lie algebras, in the study of the algebraic aspects of quantum integrable systems in the papers of Faddeev, Kulish, Reshetikhin and Sklyanin [F, KR 1, KS, KRS, Si, S2 1. For recent reviews we refer to refs. [FF2, FRT]. Then quantum algebras related to trigonometric solutions of the quantum Yang—Baxter equation were axiomatically introduced as (pseudo) quasi-triangular Hopf algebras independently by Drinfeld and Jimbo [Dl, Ji, J2, D2 1. The mathematics of these objects was studied in refs. [Ri, R2, R3, Li, DK] (for more references on the representation theory of quantum algebras cf. refs. [Dol—Do4]). Later, inspired by the Knizhnik—Zamolodchikov equations Drinfeld developed a theory of formal deformations and introduced a new notion of quasi-Hopf algebras [D3]. Other approaches to quantum groups, in which the objects may be called quantum matrix groups and are Hopf algebras in duality to the quantum algebras, are developed by Faddeev, Reshetikhin and Takhtajan [FRT], Manin [Ml, M2 I and Woronowicz. The first approach, called R-matrix approach, is based on the main relation ofthe quantum inverse scattering method. There the quantum group matrices M played the role of quantum monodromy matrices (with operatorvalued entries) of the auxiliary linear problem and the Yang—Baxter equation was a compatibility equation. The approach of Manin considers quantum groups which act as symmetries of non-commutative, or quantum, spaces. The resulting objects are the same as in the first approach. For the approach ofWoronowicz we refer to ref. [W]. For the connections between the different approaches we refer the reader to refs. [Ri, Mj, DHL I. We should mention also the development by Wess Zuminoand collaborators of differential calculus on quantum hyperplanes (cf., e.g., refs. [WZ, SWZ]). Because of the lack of space these notes represent only the lectures given by the author at the School. They can be viewed as an update oftwo earlier long reviews [Do2, Do3]. Nevertheless, these lectures are self-contained.

1. Preliminaries 1.1. HOPF ALGEBRAS AND QUANTUM GROUPS

Let F be a field of characteristic 0; in fact most of the time we shall work with F= C or F=P. An associative algebra cW over F with unity 1 ~, is called a bialgebra [A] ifthere exist two algebra homomorphisms called co-multiplication ö: ô:~1®~t’, ö(l5,)=l~®l5,,

(l.la)

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and co-unit

369

:

~:4?i—~F, f(l~)=1

.

(l.lb)

The co-multiplication 5 fulfills the axiom of co-associativity: (ô®id)~ö=(id®ö)oö,

(1.ic)

where both sides are maps ~‘li—~ ~® ‘W® oh; the two homomorphisms fulfil (idøc)oô=i1

,

(~®id)~ô=i2 ,

( l.ld)

as maps ohl—+F® °hand ohi—~‘W®F, respectively, where i1, ~2 are the maps identifying ohi with %®F and F® ohl, respectively. Next a bialgebra ~11is called a Hopf algebra [A] if there exists an algebra antihomomorphism y called antipode: y:QI—~°&’, y(l~,)=l5,,

(1.2a)

such that the following axiom is fulfilled: (l.2b)

mo(id®y)oô=io,

as maps ohi—~~W,where m is the usual product in the algebra: m ( Y® Z) = YZ, Y, Zr °11 and i is the natural embedding of F into ~W: i(c) = ci ~, ccF. 2The = id.antipode plays the role of an inverse although there is no requirement that y One needs also the opposite co-multiplication [J2, D2 ] ö’ = ir~ö, where ir is the permutation in ohl® °le.Ifthe antipode has an inverse then one uses also the notion of opposite antipode [J2, D2] : y’= y A Hopf algebra °liis called a quasi-triangular Hopf algebra or quantum group [D2] if there exists an invertible element R a ohI® °lI,called universal R-matrix [Dl, D2], which intertwines ô and ô’: R5(Y)=ô’(Y)R,

VYeohl,

(l.3a)

and obeys also the relations (ö®id)R=R 13R23, R=R.3,

(l.3b)

(id®ô)R=R13R12, R=R1.,

(l.3c)

where the indices indicate the embeddings ofR into °l1®cW® ohl. From the above it follows that (~®id)R=(id®)R=l,,, and using also (l.4a) one has

(l.4a)

1=R. (l.4b) (y®id)R=R~, (id®y)R The term quantum group is used [D2] also if R is not in °h®~1Ibut in some completion of it (cf. next subsection). From (1.3a) and one of (1.3b,c) follows the Yang—Baxter equation for R:

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V.K. Dobrev / Representations, duality, realforms

R12R13R23=R23R13R12

(1.5)

.

A quasi-triangular Hopf algebra is called a triangular Hopf algebra if also the following holds: nR~=R.

(1.6)

1.2. QUANTUM ALGEBRAS

From now on (unless specified otherwise) we set F=C. Let ~ be a complex simple Lie algebra; then the q-deformation Uq ( ~) of the universal enveloping algebras U( ~) is defined [Dl, Ji, J2, D2] as the associative algebra over C with generators X~, H,, i= 1, ..., 1= rank ~ and with relations [H,,H1]=0, H,/2

[H,,X]t]=±a,~X1~,

(1.7) 2

—

[X~, X1] =~~‘ q~ 1/2 —q q, 1

=ô,j[Hilq,,

X~~(X~y~k=0, i~j,

~ (_l)k(~) (X

(1.8)

q1q((~~.c~~)/

(1.9)

k0

where (a,1)=(2(a1, a1)/(a,, a,)) is the Cartan matrix of ~, (, ~) is the scalar product of the roots normalized so that for the short roots a we have (a, a) =2, n= l—a,~, [fl]q!

‘%.,k)q

[k]q! [n—k]q! 2...q_tn/2

[m]q_

—

qm/ qI/2_q_I/2

— —

[m]q!= [m]q[m— 1 ]~“ [1 ]q, sinh(mh/2) sin(irmr) sinh(h/2) sin(irr)

(l.lOa)

— —

(l.lOb)

q=ei~=e2~1t, h,TEC,

(l.l0c) Further we shall omit the subscript q in [m]q ifno confusion can arise. Note also that instead of q some authors use q’=q2. This definition is valid also when ~ is an affine Kac—Moody algebra [Dl]. The algebras Uq ( ~) were called quantum groups [Dl, D2], or quantum universal enveloping algebras [Re 1, KiR 1]. For shortness we shall call them quantum algebras as is becoming commonly accepted in the recent literature. For q—* 1 (h.-÷0), we recoverthe standard commutation relations from (1.7), (1.8) and Serre’s relations from (1.9) in terms of the Chevalley generators H 1, X~.The elements H, span the Cartan subalgebra of ~, while the elements X~generate the subalgebras We shall use the standard decompositions into .~°

~

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direct sums of vector subspaces

~

~±=

~

fl~4±

where A=A ± uA is the root system of ~, and A~and A are the sets of positive and negative roots, respectively; A~will denote the set of simple roots of A. We recall that II~correspond to the simple roots a~of ~, and if/3” = >~n~a~’ , /3 2/1/ (/1, /1), then to /3 there corresponds IIp= ~ n~H,.The elements of ~ which span ~ (dim ~= 1) are denoted by X4. These Cartan—Weyl generators H4, X4 [J1, J2, Dol, Tl] may be normalized so that [X4, X_4]

[H4, X÷4,]= ±(/3”, /1’)X+4,, 2. (1.11) fl,fl’eA~, q4~q(fl.fl)/ In some considerations it is necessary to use a subalgebra Uq( ~) of Uq( ~) generated by X~and =

[Hp]qp,

(1.12)

K=q~”4

then (1.7) and (1.8) are replaced by 4X~,

K

K,X~K=q~

1K~’=K~’K1=1, [K,,K~]=0, ~

(1.7’) (1.8’)

One may also use instead ofX/: the generators [R3] ~

F

4=X~K,. (1.13) 1=X~q~ In ref. [S2] for ~=sl(2, C) and in refs. [Dl, Ji, J2, D2] in general it was observed that the algebra Uq ( ~) is a Hopf algebra, the co-multiplication, co-unit, and antipode being defined on the generators of Uq ( ~) as follows: ~5(H~)=H

4+qiH1~/4®X~,(l.14a) 1®1+l®H1,

y(H~)=—H,.,

ö(X~)=X~ ®q~ y(X~)=—qq’2X/q’2=—q~”2X,t,

(l.14b)

(l.l4c)

where ~5c~ corresponds to P=~>aE4± a, A~is the set of positive roots, p= ~ + H,~. The action of ö, ~, y on the Cartan—Weyl generators is obtained easily from (1.14) since H 4 and X4 are given algebraically in terms of the Chevalley generators. [Note that if a~A~ the co-algebra operations ô, y look more complicated than (1.14).] The axioms in (1.1), (1.2) are fulfilled by the explicit definition (1.14). The opposite co-multiplication and antipode [J2, D2] introduced above define a Hopf algebra Uq ( ~)‘ which is related to Uq ( ~) by Uq(~)’Uq_i(~#).

(1.15)

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In terms ofthe generators K~~, E., F,. the above relations are rewritten as follows: 2®E,, 5(K1)=K1®K1, ô(E1)=E,®l+K1 ô(F 1)=F,®K~+l®F1, (i.l4a’) ~(E1)=(F1)=0,

,

2.

(l.14b’)

(l.14c’)

y(E1)= —K~E1, y(F,)= —F1Kr One may also rewrite the Serre relations (1.9) as [R3]: (adqE,Y’(Ej) =0= (ad~F 1)~(1~,), i#j,

(l.9’a)

where adq:Uq(~)—*End(Uq(~)), adq=(L®R)(Id®y)ô,

(l.9’b)

ad~:Uq(~)—*End(Uq(p—)) ,

(l.9’c)

ad’~=(L®R)(Id®y’)ô’,

and L (R) is the left (right) representation. Furthermore adq(E,) acts as a twisted derivation: for X, Ye Uq ( ~) homogeneous of degree 2Xadq(Ej)( fi, ye ~“ weY). have adq(Ej)(XY) =adq(Ej)(X)Y+q”~°’~ The action of ad~(F,)on X, Ye Uq( ~ is defined analogously. For ~ = sl (2, C) the universal R-matrix is given explicitly by [D2] R=q”®~4 ~

~l—

q

—1\n

q [n].

n(n.—1)/4

/

(qH/4x±)n®(q_H/4x_)n

(1.16)

where H=H 1, X± =X~,r= 1. 2, C)) ® U~(sl(2,C)), since it contains Note that this R-matrix is not in Uq(sl( power series involving the generators X ~, but in some completion of it [in the hadic topology used in refs. [Dl, D2] (q = e”)]. This is valid for the R-matrices of all Uq ( ~). Hopf algebras with such an R-matrix are called pseudo quasi-triangular Hopf algebras [D2] or essentially quasi-triangular Hopf algebras [Mj]. For ~= sl ( n, C) an explicit formula for R was given in ref. [R3]. Then explicit multiplicative formulas for R were given in refs. [KiR2, LS] for all complex simple Lie algebras ~ and in ref. [KT] for all finite-dimensional superalgebras with symmetrizable Cartan matrices. The centre of Uq( ~) and for generic q the centre of Uq( ~) is generated by qanalogues of the Casimir operators [S2, Jl, J2]. For ~=sl(2, C) one has (1.17) For Uq(sl (n + 1, C)) the (second-order) Casimir operator was given in ref. [MZ]. C

2+XX~.

2=[(H+1)/2]

t.3. MATRIX QUANTUM GROUPS

The quantum plane [Ml] Rq ( n 0), or rather the polynomial ring on it, is generated by coordinates x,, i= 1, ..., n, with commutation rules

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373

x.x3=qU2x~x1, fori
(1.l8a)

The Grassmannian quantum plane [Ml] Rq (01 n) is generated by coordinates i= 1, n, which satisfy 2~, fori
quantum matrices, which perform linear transformations of Rq ( n 0) and Rq (01 n): {x’i,...,x,}eRq(nlO), x=M~x 1, (l.19a) {~,...,~‘,,}eRq(0In),~

(l.19b)

where one assumes that the elements of M commute with all x,, Implementation of (1.19) gives the following restrictions upon the elements of M: ~,.

112M, M,1M,1=q

1M,1,

forj-’zl,

(l.20a)

fori
(l.20b)

for 1< k, 1
(1 .20c)

for i
(l.20d)

2Mk M,JMkJ=q~”

1M,~,

M,/MkJ = MkJM,t, 2—q”2)M [M,l,Mk/] = (q” 1/MkJ,

i,j=

Let us denote by Aq ( n) the bialgebra generated by the matrix elements M~, 1, n, with the following co-multiplication ô and co-unit ~: ...,

or o(M)=M~M, ~(Af,1) =ô~, or

(l.2la)

~(7i4) =1,,,

(l.21b)

where ~ denotes the tensor product of algebras and the usual product of matrices, I,, is the unit n X n matrix. Further, a quantum determinant is defined in the following way: 2)~1 (1.22) detqj=~ji4i,j,jij2...AIij~(~_qU ~

where l( i 1,

...,

i~)is the number ofinversions in the permutation (i1,

...~

in).

Note

that ö(detqM)=detqM®detqM, (detq Al)

=

~

f(.A~111,)~~(Is41 ~ ( _ql/

=

~

ô

2)

1,,,

~(

2)

(l.23a) 1(~i

/(,i,..,i~)= 1

.

( l.23b)

_ql/

It is easy to check that detqM commutes with the elements of M. Next let M’,

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M”eAq(n), and let all elements of M’ commute with all elements of M”. Then both products M’M”, M”M’eAq(n) and detqM’M”=detqM”M’=detqM’ detq M” = detq M” detq M’. If detq M 0 then one can find a matrix M’ which is both left and right inverse of M. However, M~ belongs to Aq_ (n) instead of Aq(n). Thus one can obtain the quantum groups GLq ( n, C) and SLq ( n, C), as the Hopf algebras generated by the matrix elements M,~,i, 1= 1, n, such that the condition detqM~0and detqM= 1, respectively, holds [FT2, Ml, W, CFFS]. The antipode is given by the formula (1.24) y(detqM)=(detqM)1 ...,

.

(Woronowicz [W} calls these objects also quantum pseudogroups.) The above notation is natural since for q= 1 and assuming that M~become complex numbers one obtains the standard commutative Hopf algebras of polynomial functions on the classical groups GL ( n, C) and SL ( n, C) with co-multiplication and co-unit given by (1.21) and the antipode given by (1.24) with q= 1. Ofcourse in the q= 1 case one works usually with the groups GL(n, C) and SL(n, C) themselves without reference to this related Hopf algebra (even when one considers tensor products ofgroup representations which are by default governed by the comultiplication structure). The quantum group SL~(n,C) is dual to the quantum algebra Uq(Sl(fl, C)). This duality is manifested in several forms. The first is through the R-matrices [cf. (1.12)]. The R-matrix of Uq(sl(n, C)) in the fundamental representation has the form [FRT] 2_q_1/2)~ EIJ~~EJI. (1.25)

R~=q~2~ E, 1

1~E11+1,1=1 ~ E,e~E1j+(ql~

‘,J~I 1>1

Now one may check that the following relation holds: R~M 1M2=M2M1R~,

(1.26)

where M1 =Mt~I~, M2=I~c~M. Conversely, one may start with relation (1.26) imposing it on an arbitrary n x n matrix M; then one would obtain relations (1.20). This characterizes the approach of Faddeev, Reshetikhin and Takhtajan [FRT}, for which the starting point is formula (1.26) and the Yang—Baxter equation (1.15). Their motivation comes from the original context of the quantum inverse scattering method [FST, FT 1, F], where the matrix M played the role of quantum monodromy matrix (with operator-valued entries) of the auxiliary linear problem and the Yang— Baxter equation was a compatibility equation for eq. (1.26). Following their approach, Faddeev, Reshetikhin and Takhtajan [FRT] have defined in a similar way the quantum groups SOq(n, C), SPq(fl, C).

V K. Dobrev / Representations, duality, realforms

375

2. Representations of Uq( ~) 2.1. VERMA MODULES AND THEIR IRREDUCIBLE SUBQUOTIENTS

The highest weight modules (HWM) V over Uq ( ~) [J 1] are given by their highest weight Ae~r and highest weight vector v0e V such that X~v0=0, i=l,...,l,

Hv0=)~(H)v0, HeW’.

(2.1)

We define a Verma module VA as5,the HWM from over Uq( with highest weight )~e~W” induced the ~) one-dimensional representaand highest weight vector v0e V tion V 0~Cv0of Uq(PI), where ~ ~ are Borel subalgebras of ~, such that Uq( ~Yjv0=0, Hv0=)~(H)vo,He,W’. (Note that the algebras Uq(~~) with generators H,, X~are Hopf subalgebras of Uq( ~) [Re 1].) Thus one has VA~Uq(~)®L1q(~i) Vo~Uq(~)®Vo.

The representation theory of VA parallels the theory of Verma modules V(A) over [V(A) is defined as the HWM over ~ induced from the one-dimensional representations of We recall several facts from ref. [Do 1]. The Verma module VA is reducible if there exists a root fleA~and meff~Jsuch that ~.

~.]

[~L+p,/J”)—m]qp= [~+p)(Hfl)—m]qfl=0,

f3”~2/3/(/3,/fl, (2.2)

holds. If q is not a root of unity then (2.2)2(2+p, is alsofl)=m(fl,fl). a necessary condition [In that for casereduit is cibility and then itofmay rewritten and as sufficient) reducibility conditions for the generalization thebe(necessary Vermamodules over finite-dimensional ~ and affine Lie algebras.] For uniformity we shall write the reducibility condition in the general form (2.2). If (2.2) holds then there exists a vector v~eVA, called a singular vector, such that v,#v 0,

X~v~=0, i=l,...,l,

Hv~=~(H)—mfl(H))v5, VHe,W’. The space Uq( ~ )v~is a proper submodule of VA isomorphic to the Verma module VA_mfl= Uq( )®v’o, where v’0 is the highest weight vector of VA_mfl, the isomorphism being v~’—~ 1 ®v’0. Thisthat situation will bedepicting denoted the by 5-”4, i.e., werealized use the by usual convention the arrows VA_~ V embedding maps point to the embedded HWM. The singular vector is given by [Dol] (2.3)

where ,.9 ~, is a homogeneous polynomial in its variables of degrees mn,, where

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VK. Dobrev /Representations, duality, realforms

n,e7L~come from /3=

~n

1a,, a, being the system of simple roots. The polynomial ~ is unique up to a non-zero multiplicative constant. If (2.2) holds for several pairs (m, /3) = (m,, fl~),i= 1, ..., k, there are other HWM modules VA_m~4~ all of which are isomorphic to submodules of VA. The Verma module VA contains a unique proper maximal submodule jA~Among the HWM with highest weight). there is a unique irreducible one, denoted by LA, i.e.,

LA=V~t/IA.

(2.4)

2 is irreducible then L,= VA. If Suppose V that q is not a root of 1. Then the representations of Uq( ~) are deformations of the representations of U( ~), and the latter are obtained from the former for q—~1 [R2, L 1]. Consider V2 reducible with respect to (w.r.t.) to every simple root (and thus w.r.t. to all positive roots): [~+p, a,”) —mj]q, = [).(H,) + 1 —mj]q, =0,

m•el\J,

i= 1,..., 1,

(2.5)

where we used p (a 7’ ) = 1. Then LA is a finite-dimensional highest weight module over Uq( ~) and all such modules may be obtained in this way [KiR2, KR1, R2, Ll]. If we restrict Uq( ~) to its compact real form Uq( ~k) then the set of all L 2 coincides with the set of all finite-dimensional unitary irreducible representations of U~(~‘k). Recently, De Concini and Kac [DK] have given a formula for the determinant of the contravariant form on the Verma modules VA. This result implies in the usual way the description of irreducible subquotients of V~’. In particular, this confirms results on the embeddings of the reducible modules VA [Do 1] summarized partially above. If the deformation parameter q is a root of unity the representation theory of Uq( ~) differs very much from the generic case (cf., e.g., refs. [L2, DK, Dol ]). For the lack of space we do not discuss this case and we refer to the contributions of Felder and of Cuerno, Sierra and Gomez in these Proceedings (cf. also ref. [Do2}). 2.2. SINGULAR VECTORS

The importance of the singular vectors was explained in the previous subsection. In this subsection we present, following ref. [Do4], the singular vectors corresponding to a class of positive roots which we call straight roots. For lack of space we shall restrict ourselves to the case when q is not a root of 1; otherwise we refer to ref. [Do4]. In order to introduce this class we shall need the Weyl group W of the simple complex Lie algebra ~. The group Wis generated by the Weylreflections Sa, acA, of ,W’*, defined by Sa(2) ~, a” )a. Actually W is generated by the simple reflections s, ~ ~i~
VK. Dobrev IRepresentations,

duality, realforms

377

ment we Wmay be written as the product of some simple reflections. Every such product which uses a minimal number of simple reflections is called a reduced expression or reducedform for w. The number of simple reflections in the reduced form is called the length of wand is denoted by 1(w). Note that there may exist many reduced forms for a fixed w. We shall also need the Weyldot reflections wj). defined by w).~w~+p)—p. Note that if (2.2) is fulfilled and q is not a root of 1 then).—m/1=s4~).. It is well known [B] that every root may be expressed as the result ofthe action of an element ofthe Weyl group Won some simple root. More explicitly, for any fleA~we have: ~

(2.6a)

,

and consequently ~

(2.6b)

,

where a~is a simple root, and the element we Wis written in a reduced form. The positive root/I is called a straight root if all numbers ~, P1, P2, ..., Pr in (2.6a) are different. Note that there may exist different forms of (2.6) involving other elements w’ and a,,,~however, this definition does not depend on the choice ofthese elements. Obviously, any simple root is a straight root. Other easy examples of straight roots are those which are sums of simple roots with coefficients not exceeding 1, i.e., /3= >~, na,, with n,= 1 or 0. All straight roots of the simply laced algebras A1, D1, E1 are of this form. In what follows we shall use also the following notion. A root y’eA~is called a subroot of y”eA~ if y”—y’#O may be expressed as a linear combination of simple roots with non-negative coefficients. For any ~ it is enough to consider roots for which nk #0 for 1 ~ k ~ 1. Any other root /3’ may be considered as a root of a complex simple Lie algebra isomorphic to a subalgebra of ~ of rank 1’ <1, so that /3’ = n a ~, and n . #0 for 1 ~ k ~1’ (a ~being the simple roots of p’). Thus in the case of the straight roots we shall consider always the case when u=l— 1, and {i1, ...~ i~,v} will be a permutation of{l, ..., l}. 2 over Uq( ~) as defined in the previous subsecLet us have a Verma module V tion. Letfl=n 1a1+n2a2+~~~+n,a1, where nkE7L±,be a straight root of the positive root system A~of ~, and m a positive integer. Let ).e ,W’* be such that (2.2) is fulfilled with this choice of/i and m, but is not2fulfilled for anyto subroot corresponding /3 and of/I. m is Then by: the singular vector of the Verma module V given

,

inn,,

in,,,,,

k,=O

~ ck,...k,,(X,~)”’~(X,,,)’~~ k,,=O

X

)“(X,T,

(X~

)Icu...(Xi )kl®vo,

~‘

(2.7a)

VK. Dobrev IRepresentalions, duality, realforms

378

k,±•••+k~

—

x

(mn , 11\~ \ “i Iq,,

[().+p)(iij,)]q,,

[().+p) (P1, )

(mn1~ \ ~u

(27b)

[~+p)(Rj,,)]q,,,

...

k1 Iq,~ [ ().+p) (R1,,) ‘“ku]q,,,

where the indices i1, i,,, v come from the representation (2.6), and ii’,, are linear combinations of the basis elements H1 of the Cartan subalgebra ,W’ of ~, which can be computed explicitly in all cases. This is presented below. 2ö,~ otherwise. Then every (1)fleA~ ~I=A1,(a1, a1)= for li—il = 1, (a,, a~)= root is given by—1fl=/3 1~=a1+a,~1+..+a1±~_1, where 1 ~
31,n /

.. .1?,

=s 1(/31±1~) =s1”~s1±,,_2(a1±,,_1 )=s1±~_1~”s1±1(a1) 5,±t+15i”~5j±t1 (a =si±nl

1±,) ,

0~t’~n— 1

where we have demonstrated different forms of (2.6) in this case. For A1 the highest root [31] is given byâ=a1+a2+~”+a1. Thus every root fleA~is the highest root of a subalgebra of A1 explicitly /3,, is the highest root of the subalgebra A,, with simple roots a, a1~1,..., a, + I. This means that it is enough to give the formula for the singular vector corresponding to the highest root. Thus in formula (2.7) with /3=à we have nk= 1, 1 ~ k~
(2.8) ~

JH~,

‘s_ ~ H~Hl+H7+-+Hk,

l~s~t,

t+l~
29 .

)

(2.10)

(2) ~=D1, l~4, (a1, a1) = —1 for li—il = 1, i,j#l and for ij=l(l—2), (a1, a~)= 2ô,~otherwise. First we note that if n12 + n~_1+ n, ~ 2 then the root/i is a positive root of a subalgebra of D1 of type A,,, n
fl=fl1 =a1 ~ =s1”~s2(a1) =s1s2~~~s,_3s1_ s1_2(a1)

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379

Thus in formula (2.7) with /3=flwe have nk=l, l~
{i1 ,

i1_1 v}={l, 2, ...,l—3, 1—1, 1; l—2}

...,

,~

“

JH~, ~

l~s~l—3, s=l—2,l—1.

211

)

(3) ~=E1, 1=6, 7, 8, (a1, a1~~)= —1, i= 1, (a3, a1) = 1, (a1, a1) 2ö,3 otherwise. First we note that if n2 + n4 + n1 ~ 2 then the root /1 is a positive root of a subalgebra of E1 of type A,,, n <1. Analogously, if n2 + n4 + n1= 3 and n1 + n5 ~ 1 the root/I is a positive root ofa subalgebra of E1 of type D,,, n<1. Thus it remains to consider the straight root ...,

1—2,

—

=

/3=a1 +...+a1=s1s2sts1_1...s4(a3) =

5~” ~s2( a1) = s1s2 s,_1~~s4s3 ( a1)

=s1s2s1s3.”s12(a1_1) Thus in formula (2.7) with /i=$’we have ~k= 1, 1 ~
(Ha,

s=l,2, s=3,

(Hi~~13~,

s=4,

(2.12) ...,

1—1

,

(2.13) (4) ~=B1, l~2, (a1, a1) = —2 if li—il = 1, (a1, a1) =2ô,3(2—ö11) otherwise. The straight roots are of two types: /31~=a1+a1~1+”~+a1±,,_1, 1 <~i<~l, ~ itive root of a subalgebra2). of B1 oftype <1 (with scaled by Thus we A,,, are nleft with the twoscalar typesproducts of straight roots 2/3, =and q replaced by q ~ = a, + a 1~1+ + a1, 1 s~i
+ ... + a1

+ 2a1

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380

We note that (fl’,fl’)=2,

if”==2a~”+.”+2aj~1+a7’,

(if’if’)4

+~“+a7’.

flv=~if=a~

Thus in formula (2.7) with /3=/we have nk= 1, 1 ~
i1_1 v}={1, ..., 1—1; l}, 2, s=l, ...,l—l ; .R1,=iv, q,,=q while for/3=fl” we have ~k= 1 +ökl, 1 ~
(2.14)

{i 1,

~ (5) ~

l>~3

...~ i1....1

v}={l,

...,

1—2,1;

l—l}

,

2°”’~

{w~ s=l,...,l—2,

(2.lSa) (2.15b)

q1~=q

(C 2~B2),(a,, a1) = —1 if

=

li—il

1 and i,j
—2

if ij=l(l— 1), (a1, a1) =2ö,1( 1 +c5,1) otherwise. The straight roots are of two types: /31,,=a1+a,~1+~”+a1~,,_1,

l~i~l,l~n’~l—i+l l~i
/3’=2a1+~”+2a,_1+a1,

If i+ n 1 <1 then /3,, is a positive root ofa subalgebra of C1 of type A,,, n <1. Thus we are left with two types of straight roots/31=/3,,1±1_1=a1+a,~1+ +a,, 1 ~ i<1, and /1 As above it is enough to account for the roots with i = 1. Thus we consider —

~

~‘.

=a1 ++a1=s1~s2(a1)[=s1”~st_2s~(a,_1)] if”=j3~’=2a1+“~+2a1_1+a/=sl”~sl_l(a,). We note that ($,/7)=2,

if”=fl=a1’+...+a~1+2a7’,

(fl”,if”)=4,

if”’=~if”=a~”+...+a7’.

Thus in formula (2.7) with/i=/I we have nk= 1, 1 ~
i1_1

v}={l,

2; l},

...,

q1~=ql±~~ , s=l,..., 1—1 ;

,

(2.16)

while for/l=$” we have nk=2—ôk1, 1 ~
i1_1

v}={l,

...,

1—1;

l},

q1~=q, s=l,...,l—l

.

(6) ~=F4, (a1, a1)=(a2, a2)=2(a3, a3)=2(a4, a4)=4

(2.17,) and (a1, a2)=

VK. Dobrev /Representations,

duality,

realforms

381

(a2, a3) =2(a3, a4)= —2 are the non-zero products between the simple roots. We have straight roots of type A2: a1+a2, a3+a4, of type B2: a2+a3, a2+2a3, of type B3: a1+a2+a3, a1+a2+2a3, of type C3: a2+a3+a4, a2+2a3+2a4. Thus we are left with the two roots

ff= a1 + a2 + a3 + a4 = s1 s2s4 ( a3) fl”=a1 +a2+2a3+2a4=s1s4s3(a2). We note that (fl~if)=2,

flv=if=2a~~+2a~~+a3v+a%,

(if~’if~’)4

fl~v~ifF~a~

+a2” +a~’+a%.

Thus in formula (2.7) with /3=/i we have nk= 1, 1 ~ {i1, ...,

v}

i3

~

={ 1, 2, 4; 3)

s=~2~

while for/l=jI’ we have nk= 1, k= 1, 2,

and

~k=2,

2_~~3; q1~=q k=3, 4, and

(2.18)

{i 1, ...,

v}={l, 4,3; 2)

i3

1~”. s—23

(2.19)

q1,=q

(7) ~‘=G 2,(a1, a1)=3(a2, a2)=—2(a1, a2)=6. The non-simple straight roots are the two roots fl’=a1 +a2=s1(a2)

,

if”=a~+3a2=s2(a1)

We note that (~fl)=2,

(if”, /1”) = 6,

if”=if=3a1’+a2’,

if”’ = ~if”= a + a ~‘

~‘

Thus in formula (2.7) with 13=/i we have n,,,= 1, k= 1, 2, and 3 {i1 v}={l; 2), .1?,, =H1 , q, =q while for /3=/i” we have n 1 = 1, n2= 3, and {i1 v) = {2; 1)

,

R,, =H2 ,

q,, =q .

(2.20)

(2.21)

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3. Duality 3.1. DUALITY BETWEEN HOPF ALGEBRAS

Two bialgebras d are said to be in duality [A] if there exists a doubly nondegenerate bilinear form ~‘,

>:°llXd—+C,

<

, >:(u,a)i—~,

UE°11,aEd,

(3.1)

such that, for u, ye ohs, a, bed: =<ô1,(u), a®b> ,

=,

(3.2a)

(u, l,~>=~,(u) .

(3.2b)

<1,,~,a> =~,,(a),

Two Hopf algebras °ul,d are said to be in duality [A] if they are in duality as bialgebras and if (3.2)

It is enough to define the pairing (3.1) between the generating elements of the two algebras. The pairing between any other elements of W, d follows then from relations (3.2) and the standard bilinear form inherited by the tensor product. For example, suppose ö(u) = >~, u®u~’,then one has (u, ab>

=

<5,,,(u), a®b> = ~ =

3.2. MATRIX QUANTUM GROUP

~



.

(3.3)

GLpq(2, C)

In this subsection we review the two-parameter deformation of GL (2, C) following ref. [DMMZ] (cf. also refs. [Ku, Su2, SWZ]). For more general multiparametric deformations we refer to refs. [M2, Re2, FZ, Si]. Let p, qe C \ {0}. Consider next 2 x 2 matrices M with non-commuting matrix elements which perform linear transformations of Rq (2 10) and R~(01 2), i.e., {x~,x}eRq(2I0), x=M,1x,,

(3.4a)

{~,~}eR~(0I2),~=M,4,

(3.4b)

assuming that the elements of M commute with all x1, ~ and summation over repeated indices is understood. Let us write the matrix M as M=(a

~).

(3.5)

VK. Dobrev / Representations,

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383

Then implementation of (3.4) gives that the matrix elements of M obey [DMMZ]: ac=q~2ca, ab=p1”2ba, bd=q”2db, cd=p~”2dc, q’~2bcp1~2cb, ad—da= (p~1”2—q “2)bc.

(3.6)

Let us denote byApq(2) the bialgebra generated by the matrix elements a, b, c, d with the following co-multiplication ö and co-unit [cf. also (1.21) for n = 2], ((a b’\\ ~k~c d))

=

(a®a+b®c a®b+b®d\ ~c®a+d®c c®b+d®d,)’

((a bV\ ~

(1 d))”~0

(3.7a)

o\ l)~

(3.7b)

Further, a quantum determinant detp,qMeAp,q (2) is defined as follows: det~,,~M= ad—p — C2bc = ad— q =da_pI~’2cb=da_qU2bc,

(3.8)

andthenwehave [cf. (1.23)] ö(~=~®~’,

f(~f)l.

(3.9)

The crucial difference with the one-parameter case, which is obtained for p = q (cf. subsection 1.3), is that the quantum determinant is not central but satisfies the following relations [DMMZ]:

[~, a]= [~,

p2~b=q112b~,

d]=0,

q”2~c=p”2c2~’. (3.10)

Further, if ~ #0 one extends the algebra by an element ~

obeying (3.lla)

from which follows [DMMZ]: [~1,a]=0,

[~1,d]=0,

q”2~’b=p”2b~’1,

p2~1c=qU2c~f1.

(3.llb)

Next one defines the left and right inverse matrix of M [DMMZ]: d

‘‘2b d M=~1 (_q_1/2c a (p_1/2c Suppose that the bialgebra operations are defined on

)=

a ~ —I

)~

(3.12) Then we have (3.13)

The quantum group GLpq (2, C) is defined as the Hopfalgebra obtained from

VK. Dobrev /Representations, duality, realforms

384

the bialgebra Apq (2) extended by the element 2i’ ~ and with antipode given by the formula (3.14) From the above definition we have 1,

y(~1)=~.

y(~)=~

(3.15)

For p=q one obtains from GLqq(2, C) the quantum groups GLq(2, C) and SLq(2, C), if the condition ~“#0, and ~= la,, respectively, holds. 3.3, DUALITY ALA SUDBERY FOR GLp,q(2,

C)

In this subsection we review ref. [DoS],where we have applied to Ap,q (2) the approach which Sudbery invented forAq(2) Aqq(2) [Sul]. For Apq(2) we use the basis given by all monomialsf=fklm,,=akdtbmcn, where k, 1, m, n e1±,andf 0000 = 1 ,~. We postulate the following pairings for f= a kd/blncn: (3.l6a) (3.16b)

(3.16c) (3.16d) and generated by A, B, C, D. Let uswe denote bialgebra to Apq( Later shallby see‘%~,,,, thatthe0/lpq has the dual structure of a Hopf algebra in duality with GLp,q(2, C). 2)

The following relations hold as consequences from (3.16): (A,(a (B,(a

~))=(~ ~),~ ~))=(~ ?)~ ~))=(~ ~), ~))=(? ~ (C,(a

(3.17a) (3.17b)

=0, kAlj,m

/1

~

Y=A,B,C,D,

(3.18)

~ cfl\...X /—V,,,0(J,,0.

We would like to find the commutation relations between the generators of 011p,q which ohlp,q.quadratic First we obtain the action onf=bya~~dlbmc~~ of the monomials in are in the that generators is given the following: k—I



ô,fl

1)/2+q J=0 (pq)U 0~5,,0~

1/2~51ö,,~,

(3.20a)

VK. Dobrev /Representations, duality, realforms i—I =o~

2+p112ö~ (pq)”

~

0o,,0

385

1ö,,1,

(3.20b)

1=0

(k+l)ômiôno(k+l)

(3.21a)

,

(3.2lb)

kömi~5no=k,



k~moöni=k

(3.22a)

,

(k+l)~moôni(k+l)

(3.22b)

,

(3.23a)

lômiönol ,

(3.23b)

= (1+ 1

)ômoöni =

(1+ 1)

(3.24a)

,

=lo~0ô,,1=l,

(3.24b)

==kMmoo,,okl.

Then we have:

(3.25)

2_p~/~2 qh/ =

I

(1—k)/2

pl/2_q_l/2

mO

nO=

<[A, C],f>=— ,

(3.27a)

<[D, C],f> =

(3.27b)



,

(3.27c)

<[A,D],f>=0.

Note that relations (3.20)—(3.25) depend on the elementf but the commutation relations (3.27) do not. This is also true for (3.26); in order to see this we need the following formulae: =k~ômoö,,o, sei”~J,

(3.28a)

~

sei”J,

(3.28b)

0,r=p, q,

(3.29a)

=r’ö~0ô,,0, r=p, q,

(3.29b)



=r”ô~~ô,,


where we use the formal power series 3’=l,, r

1+

k~I

Yk(lnr)k/k!

386

VK. Dobrev /Representations, duality, real forms

Thus we obtain that the commutation relations in the algebra 0&~p,q are given by:

(

)(A_D)/2

q112BC—p”2CB= [A,B]=B,

[D, B]

=

1/2

1

_~

(3.30)

[A, D] =0.

(3.31)

[A,C]=—C,

—B,

[D, C] =C,

Note that the generator K=A + D commutes with all other generators of ~ Let us denote by ~ the algebra spanned by K. Next we are looking for the analogue of the splitting Uq ( sl (2, C) ) ® U~ ( ~‘) which Sudbery [Su 1] obtained in the one-parameter case. We try a similar change of basis: H=A—D,

q’

~

I/4BqiH/4,

~

q~l/4~q~H/4,

(3.32)

q’_(pq)l/2,

and we get the generators H, ~, satisfy commutation relations (1.7) and (1.8) with 1=1, q 1 =q—i’q’, H1 =H, X7’redundant, =1~±. since factors q’—~”for arbitrary 4in (3.32) seem The factors q’±l/ veC will play the same role of the previous statement. Their significance becomes clear if we calculate the action of the new generators on akdlb~~1cn1, namely, ~‘

= (k—l)5ö~ 0ö,,0,


5örn

= (k+l)

~

0ôn0,


a”d’b”c”> 1b”c”> a”’d

q~kIo,noo,,o, ooo ,

qfk+Io

(3 33c)

a”dtb”c~>=qi(l_k)/4o,,,i~5,,o

=q~(l_k)/4o,noo,,i

(3.33a) (3.33b)

.

(3.33d)

Remark. Thus the two parameters are glued together in the commutation relations and the action ofthe new basis of the algebra ~hpq.This is in agreement with the general result ofDrinfeld [D2] stating that the q-deformation of U(sl(2, C)) is unique. However, we shall see below that in the Hopfalgebra relations the two parameters are not glued together, since in fact we are obtaining a deformation of U( (gl(2, C)). We turn now to the bialgebra structure of 0llpq. The co-multiplication in the algebra ~ is given by (3.34a) ö~(B)=B®p”~’2q”2+ 1 ,®B

(3.34b)

ô~(C)=C®q’412p’°~2 +1,

40C,

(3.34c) (3.34d)

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387

or in the new basis by (3.35a) (3.35b) ô~(X

~,(

) 5(± ® (P/q)K/4q~H/4+q~_H/4®~±, ) ~ ® (q/p)K/4q~H/4+q~H/4®~.

(3.35c) (3.35d)

For the proof we use the duality property (3.2a), namely, we should have = <ö±(Y),f 1®f2>,Y=A, B, C, D, for every splittingf=f1f2. The co-unit relations in %JJ.,~are given by ,11(Y)=0,

Y=A,B, ~

,

(3.36)

whichfollowsfrom(3.l8),(3.32)and=~(u) [cf.(3.2b)]. Let us assume now that °li~ is a Hopfalgebra in duality with GL1,,4(2, C). This assumption would be correct if we can define consistently the action of the generators of 0/1~on ~ and an antipode in %pq. We are even in a better situation since the action on ~ —1 and the antipode map in 0/i~~ are uniquely obtained as a consequence of the assumed duality. Namely, we have that the action of °l1,,,,,~,on ~ is given by
(3.37a)

, 0

337b)

-l)~

(.

To prove (3.37a) we use (3.2b) and (3.13): <1±,~>=.~(~‘~)=l. (3.37b) we use acorollary of (3.16): /(A B\ \ (1 \~C D)’~/~0

For

0’\ (3.37c)

1)’

and also (3.18), (3.37a). Next we obtain that the antipode map in ~/1p,qis given by 2 D/2 A B —A —B~ ~‘(C D) = (Cq~2p~2 Pq

)~

(3.38)

Finally we can state the main result of ref. [Do5]: The Hopf algebra °/4,.,~ dual to GL~ 4 (2, C) by relations (3.16) is isomorphic to U(pq)l/2(5l(2, C))®Up/q(.fi’) as a commutation algebra,Up/q(.ft) where ~% spanned by 4. The subalgebra is isa HopfsubalK, andofUr(~t) is spanned by K, r~’~ gebra ~ the commutation subalgebra generated by H, x±is not a Hopf subalgebra. For pq the dual algebra to GL 4(2, C) is U4(sl(2, C))®U(~”)as a tensor

388

VK. Dobrev IRepresentations, duality, real fbrms

product of Hopf subalgebras. For q = 1 the last statement reduces to the classical relation U(gl(2,C))=U(sl(2,C))®U(.~’).

4. Real forms 4.1. OVERVIEW OF THE PROCEDURE

In this section we review and explain a canonical procedure proposed in ref. [Do6] for the q-deformation ofthe real forms ~ of complex Lie (super-)algebras associated with (generalized) Cartan matrices. Besides several examples in the text we may refer the reader to the lectures of, e.g., Lukierski, Nowicki and Ruegg, and also to refs. [Do6, Do7]. Let ~ be a real simple Lie algebra (below we shall need to extend the construction to real reductive Lie algebras). We shall use the standard deformation from subsection 2.2 for the simple components of the complexification egC of ~ to obtain the deformation U4( ~) as a real form of Uq( (gC) As in the undeformed case this means that there exists an antilinear (anti) involution a of Uq ( ~) which preserves Uq ( ~). Unlike the undeformed case it is necessary to consider both involutions and anti-involutions, since there are two possibilities for the deformation parameter q, i.e., either I qI = 1 or qeP. For instance, U4 ( su (2)) has I q I = 1 when a is an involution and qe P when a is an anti-involution. Further, a is a co-algebra (anti)homomorphism, i.e., 5’ ô”a (aXa)o5, or ôoar (axa)° ~(a(X))=~(X) VXeU 4(~).

Then the relations for the antipode are aoy=yoa if a is an algebra involution and a co-algebra homomorphism or 2=id if it isotherwise an algebra[Sch]. anti-involution and a to co-algebra One approach the real antihomomorphism and (aoy) forms would be to try to classify directly the possible conjugations a. Our approach is more constructive and the conjugations a are obtained as a byproduct of the procedure proposed below. Though the procedure is described mostly in terms which are known from the undeformed case, we will stress which steps are necessitated by the q-deformation. The first basic ingredient of our approach relies on the fact that the real forms ~ of a complex simple Lie algebra ~ are in one-to-one correspondence with the Cartan automorphisms 0 of ~. This allows us to study the structure of the real forms and to find their explicit embeddings as real subalgebras of (gC invariant under 0, and consequently, using the same generators, to find U 4( ~). This ingredient is enough for the compact case though we still have to specify the range of q. The second basic ingredient of our procedure is related to the fact that a real

VK. Dobrev / Representations, duality, real fbrms

389

non-compact simple Lie algebra has in general (a finite number of) non-conjugate Cartan subalgebras. For each choice ofconjugacy class of Cartan subalgebras we get a different q-deformation. The third basic ingredient is constituted by the Bruhat decompositions ~=d~V, where d is a non-compact abelian sub-algebra, .11 (a reductive Lie algebra) is the centralizer of d in ~ (mod d), and uT” and ,~4”,are nilpotent subalgebras forming the positive and negative root spaces, respectively, of the root system (~,d). Consistently, the Cartan subalgebras of ~ have the decomposition ,W’ = d~ .W”, where ~W”is a Cartan subalgebra of ~#. A general property of the deformation U4( ~) obtained by our procedure is that U4(J1), Uq( ~),

Uq ( ~) are Hopf subalgebras of Uq ( ~), where ~ = d~t~4”, = are parabolic subalgebras of ~. (All notions are recalled below.) Our exposition is organized as follows. We fix a real simple Lie algebra ~ and its most non-compact Cartan subalgebra ,W~then we present the procedure since it is most simple in this case. Then we point out the modifications necessary in order to consider Cartan subalgebras .K of ~ which are non-conjugate to ~. Until this moment we consider only the so-called minimal parabolic subalgebras ~ (which are different for non-conjugate,çartan subalgebras). Next, for an arbitrary Cartan subalgebra, we extend the procedure for arbitrary parabolic subalgebras. Finally we note that we need to generalize the whole procedure to reductive Lie algebras, which is straightforward. 4.2. q-DEFORMATION

,

WITH THE MOST NON-COMPACT CARTAN SUBALGEBRA

First we recall some standard facts on real semisimple Lie algebras. The basic reference for that is ref. [B].Let ~ be a real semisimple Lie algebra and 1’ be the maximal compact subalgebra of ~. Then ~= ,W’tT~,9 is the Cartan decomposition of ~, the subspace ,fI’ is non-compact. Let 0 be the Cartan involution in ~ so that OX=X, Xe,W’, OX= —X, Xe,9. Let d0 be the maximal subspace of ~ which is an abelian subalgebra of ~ all non-compact abelian subalgebras of ~ with maximal dimension are conjugate to d~r0= dim d is the real (or split) rank of ~. For different ~ the real rank r0 may vary from 0 (then ~ is compact), up to 1= rank ~ [then ~ is called maximally split, e.g., sl(n, P), so(n, n), so(n+ 1, n), sp(n, OR)]. Let A°Rbe the root system of the pair (~,d,~),also called (d0-)restricted root system:

A~={).ed~I),#0,

~0},

~={Xe~I[Y,X]=).(Y)X,

VYed~}.

(4.1)

The elements of A°~ =A~7’uA~ are called (d0-)restricted roots; if).eA~,~ are

VK. Dobrev / Representations, duality, realforms

390

called (d0- ) restricted root spaces, dimR ~ ~ 1. Now we can introduce the subalgebras corresponding to the positive (40~7’ ) and negative (At) restricted roots: ~ A~

40±

~

~=x~=o7’~,

(4.2)

A~A ~‘

where .2’,~and ,X’~are the direct sum of ~ with dimR ~= 1 and dimR ~> l~ respectively, and analogously for .iV~= 0Yi~.Then we have the (Bruhat) decompositions which we shall use for our q-deformation: ~

(4.3)

where .A”0 is the centralizer of d0 in ~W’, i.e., .,t~={Xe.fI [X, Y] =0, V Yed0}. In general ~ is a compact reductive Lie algebra, and we shall write .A”~= ~1~’~ ~ where = ~ ,..#~]is the semisimple part of~/10,and ~ is the centre of~#0. Note that i~ ~ .~ ~d0 0E~.4’~ are subalgebras of ~, the socalled minimal parabolic subalgebras of ~ for that choice of Cartan subalgebra. Further let .W’~’be the Cartan subalgebra of Ji~0,i.e., .W’~’= .W’~~ .~J’,where ~ is the Cartan subalgebra of J1~.Then ~ ~*‘~‘ ~ d0 is a Cartan subalgebra of ~, the most non-compact one; dimR ~=dimR .W’~ls+ dime 2t~’+r0.We choose .*~ to be also the Cartan subalgebra of U4( ~). Let ,W’~be the complexification of .*~ (l=rank ~=dim~ ,W”)~then it is a Cartan subalgebra of (gC and U4( ~ It is important for our procedure to choose consistently the basis ofthe rest of ~ and and thus of U4( ~). For this we use the classification of the roots from A with respect to The set ~

~

,~.

A~

{ a cA I a l~g~= 0

4?

{aeAI a I ~ =0}

is called the set of real roots, the set of imaginary roots, and A~° ~A\ (A? uA?) the set of complex roots. Thus A=A? ~~A?uA?. Further, let aeA~,let ~ be the complex linear span ofHa, Xa, X_a, and let ~ n ~. Then dimR ,2~=3iffaeA? uA?. IfaeA? then XaE~1C and ,2~.is non-compact. Since the Cartan subalgebra is then XaE~W’Cand ,2~, is compact if acA?. The algebras 2~.are given by ,~,

(P

a

11 1.

V a,

a,

V

~ —aJ,

,40±

44a

r

~=rls{iHa,Xa—X.~a,i(Xa+Xa)}, ae4?~, (4.4b) where rls stands for real linear span. Note that there is a one-to-one correspondence between the real roots aeA? and the restrictedroots ).EA°Rwith dimR ~= 1 and naturally this correspondence is realized by the restriction ). = a I ~ Thus we take the elements in (4.4a), X~ for aeA?, also as elements of U4(’~). These generators obey (1.11) and if ae4?~nz1~also (1.14), and otherwise as explained after (1.14).

VK. Dobrev /Representations, duality, real forms

391

In particular, formulae (4.4a) determine completely a q-de formation of any maximally split real form (or normal real form), when all roots are real, = 0, .~

,~=d

0, and

(4.3) is reduced to (4.5)

i.e., this is the restriction to OR of the standard decomposition ~= ,W’C~ ~i ,and hence Uq( ~) is just the restriction of U4( ~) toUR with qeER. Thus we also inherit the property that U4(7’~~d0)and Uq(~.4~do)are Hopf subalgebras of U4( ~), since U4( ~%~,W’c) are Hopf subalgebras of U4( ~‘j. Note that a here is an antilinear involution and co-algebra homomorphism such that a( Y) = Y, V Ye U4 ( ~C). For the classical complex Lie algebras these forms are U4(sl(n, OR)), U4(so(n, n)), Uq(5O(n+ 1, n)), Uq(sp(n, OR)), which are dual to the matrix quantum groups SLq(n, OR), SOq(n, n), SO4(n, n+ 1), Spq(n, OR), introduced in ref. [FRT] from a different point of view than ours. Further note that the set of the imaginary roots A? may be identified with the root system of.,. Thus the elements in (4.4b) give the Hopf algebra Uq(A~) by the formulae: ‘~

sinh(i~ha/2) sin( a! )

[C~,C~’]=

[Ra,Cfl±C~,

(4.6a)

2elhn, q,~=q(a.a)/

~(l/~)(XaXa),

C~(j/~J~)(Xa+Xa),

Ra~jHa,

(4.6b)

ö(C~) = C,~®e’”~+e’1~°”~®C~ , aeA~ ~4S~

(4.6c)

Since .,#~= ~ ~ .f1~’is a compact reductive Lie algebra we have to choose how to do the deformation in such cases. Our choice is to preserve the reductive structure, i.e., writing in more detail ~ I

where ...#~I’is simple and algebra

~

U 4(~..1f0)

k

is one dimensional; then we shall have the Hopf

0j

1)® Uq(Jf’6

0

Uq(~”),

k

where we also have to specify that if fZ’~is spanned by K, then Uq(21’o”) is spanned by K, q ±K/4~ In particular, formulae (4.6) (with haSP) determine completely the unique qdeformation of any compact simple Lie algebra (when all roots of A are imaginary). Here one may take a as an antilinear involution and co-algebra homomorphism such that a(X~) = —X~, VaeA, a(H) = —H, VHe,,W’. Note that in

392

V.K. Dobrev / Representations, duality, realforms

this case the q-deformation inherited from Uq( ~) is often used in the physics literature without the basis change (4.6b). Returning to the general situation, so far we have chosen consistently the generators of .iY~~ d0~ff0~.V~[cf. (4.2)] as linear combinations of the generators of 0 ~40

~

aEz1

Now it remains to choose consistently the generators of 5~”~ and ..JY~ as linear combinations of the generators of the rest of ~, i.e., of ~aEAO± ~a and ~aEA ~ respectively. If aeA?, 2=aI,~, 0, then dimR ~> 1. Let 42= {aeAIaI~0=A}.If ae4?, then we have XaYa+Za, where YaC.~3~,Z~,E,)(C. Now we can see that ~=rls{~a= Ya+ jZa, VaeA2}. The actual choice of basis in ~ is a matter of convenience and is related to the choice of a and q, and to the general property that U4(,~18) and U4( .~) are Hopf subalgebras of U4( ~). 4.3. q-DEFORMATIONS WITH ARBITRARY CARTAN SUBALGEBRAS

For the purposes of q-deformations we need also to discuss Cartan subalgebras .W’ which are not conjugate to Cartan subalgebras which represent different conjugacy classes may be chosen as W’= .~d, where ~ is compact, d is noncompact, dim d
.~.

d=0.]

All notions introduced until now are easily generalized for ~W’=/t~$d nonconjugate to ~. We note the differences and notationwise we drop all 0 subscripts and superscripts. One difference is that the algebra .~#is the centralizer of d in ~ (mod d) and thus is in general a non-compact reductive Lie algebra which has the compact ~ as Cartan subalgebra (besides, in general, other noncompact Cartan subalgebras); in particular, if ~ has a compact Cartan subalgebra then for the choice d = 0 one has J/ = ~. For the purposes of the q-deformation we shall use this compact Cartan subalgebra, i.e., we set ~ /~,. Further, the classification of the roots ofA with respect to ,W’ goes as before. The difference is that if aeA, then ~. may be also non-compact. Thus for aeA, the root a is called singular, aeA~,if 2~.is non-compact, and a is called4k,compact, ae4~,, if ~ while for aeA~we is compact. Thus Aj=ASUAk. Formulae (4.4b) hold for have ~~=rls{iHa,i(Xa—X~a),Xa+X.,s}, aeA~, a

a

(4.7a)

[Ra,S7’]RS7’,

sin(hj2) q,,,

q(a.a)/2s,,,eIhc~

(4.7b)

VK. Dobrev /Representations, duality, realforms

S~(l!\/~)(Xa+Xa),

393

S(~!~J~)(Xa_Xa),

(4.7c)

~a~11a, ö(S~)=S~~

aeA~nAg.

(4.7d)

Further as before the set of the imaginary roots in A may be identified with the root system 0fJ/sc Thus formulae (4.6) and (4.7) give also the deformation U4(J/~).Since the centre of J/ is compact (it is Cartan subalgebra /t”” m)inis the given as after (4.6). Thus which is compact) then the deformation U4(~ the Hopfalgebra U 4(J/) is given. Otherwise, the considerations for the factors .A” and ~ go as before. 4.4. q-DEFORMATIONS FOR ARBITRARY PARABOLIC SUBALGEBRAS AND REDUCTIVE LIE (SUPER-)ALGEBRAS

Until now our data are the non-conjugate Cartan subalgebras W’= ~ the related Bruhat decompositions:

and (4.8)

~

In this decomposition a special role in the q-deformations is played by the subalgebra i~=J/~d~X (or equivalently by its Cartan involution conjugate ~ =J/~dEB~~~). It is called a minimal parabolic subalgebra. A standard parabolic subalgebra is any subalgebra gi’ of ~ such that P1’. The number of standard parabolic subalgebras, including ,P4~and ~, is 2~,r=dim d. They are all of the form P1’=J/’~d’~.iV”,J/’~J/, d’~d, ~‘V”c.íV;J/’ is the centralizer of d’ in ~ (mod d’); ,A~”(~‘=0,K’) is comprised of the negative (positive) root spaces of the restricted root system A’~of (~,d’). One also has the analogue of ~

(4.2), (4.8):

(4.9) Note that J/’ is a non-compact reductive Lie algebra which has a non-compact Cartan subalgebra /t”~ ~ will be where ,W~,isfornon-compact d~ .~®d’. This m of JI’ chosen the purposesand of the q-deformation. Cartan Thus subalgebra we need to ,W” extend our scheme to non-compact reductive Lie algebras. Let ~= ~~Z’=i”EB~ be a real reductive Lie algebra, where ~ is the semisimple part of ~, .9% is the centre of ~ i”, ~ are the + 1, 1 eigenspaces ofthe Cartan involution G; ~“=d’~f% 1, is the analogue of d’, ~= ~%n~. The root system of the pair (~,o~”) coincides with A ‘R and the subalgebras 7’~”’and X’ are inherited from The decomposition (4.6) then is —

~.

(4.10) where ~ !,“~ ~r”~~PI~ i”. As in the compact reductive case we choose a deformation which preserves the splitting of i.e., U4( ~) = ~‘,

VK. Dobrev / Representations, duality, realforms

394

Uq ( ~) 0 U4 ( ~),

and even further into simple Lie subalgebras and one-dimensional central subalgebras. Let us stress again the general property of the deformation U4( ~) obtained by the above procedure, which is that U4(JI0), U4(~), Uq(P1o) are Hopf subalgebras of Uq( ~). All notions above are easily generalized to the real forms of the basic classical Lie superalgebras [K], and thus our approach is immediately generalized to the deformation of such superalgebras [Do6]. The author would like to thank Professor Abdus Salam for the hospitality and financial support at the ICTP. This work was partially supported by the Bulgarian National Foundation for Science, Grant ‘~b-11.

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