IC/92/13
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
q-DEFORMATIONS OF NONCOMPACT LIE (SUPER-) ALGEBRAS: THE EXAMPLES OF q-DEFORMED LORENTZ, WEYL, POINCARE' AND (SUPER-) CONFORMAL ALGEBRAS
V.K. Dobrcv INTERNATIONAL ATOMIC ENERGY AGENCY
UNITED NATIONS EDUCATIONAL. SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE-TRIESTE
T
IC/92/13
1. Introduction International Atomic Eneigy Agency The Lorentz, Poincare and conformal algebras, and also other non-compact Lie algebras and and groups play a very important role in physics. Thus the problem the q - deformation of these and United Nations Educational Scientific and Cultural Organization other noncompact algebras is of utmost importance. Actually, the deformation of compact simple INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS Lie algebras is used in the physics literature without much explanation assuming implementation of the Weyl unitary trick. In [1] considering the real forms of the matrix quantum groups [1],[2],[3] were introduced the compact matrix quantum groups St/,(n), (for n = 2 first in [3]), 50,(n), 1 Sp,(n) and the maximally split real noncompact forms S£,(n, jR), SOq(n,n), S O,(n,n + 1), Sp,{ n,R). From our point of view it is not accidental that these cases were obtained first q - DEFORMATIONS OF NONCOMPACT LIE (SUPER-) ALGEBRAS: since the root systems of these real forms coincide (up to multiple of»' in the compact case) with THE EXAMPLES OF q - DEFORMED LORENTZ, WEYL, the root systems of their complexifications (cf. the description of our approach below). Besides POINCARE' AND (SUPER-) CONFORMAL ALGEBRAS* the above C/,(au( 1,1)) was considered in f41, Uq(su{n, 1)) were introduced in [5], A quantum Lorentz group was introduced and studied in [6] and a seven - dimensional quantum Lorentz algebra V.K. Dobrev" was introduced in [7]. The q-deformation of Heisenberg, Galilei and Eucledean algebras in two dimensions were studied in [8]. International Centre for Theoretical Physics, Trieste, Italy. Thus there is still lacking an universal approach to the q - deformation of real simple algebras. Such an approach was proposed in [9] and is reviewed and explained here. It is well ABSTRACT known that the real forms Q of a complex simple Lie algebra C?care in 1-to-l correspondence with We review and explain a canonical procedure for the q - deformation of the real forms 5 of the Cartan automorphisms 9 of Q°. This allows to study the structure of the real forms and to find complex Lie (super-) algebras associated with (generalized) Cartan matrices. Our procedure gives their explicit embeddings as real subalgebras of £cinvariant under 9. This is one basic ingredient different q - deformations for the non-conjugate Cartan subalgebras of Q. We give several in detail of our approach which is enough for the compact case. The other basic ingredient is related to the the q - deformed Lorentz and conforms! (super-) algebras. The q - deformed conformal algebra fact that a real noncompact simple Lie algebra has in general (a finite number of) non-congugate contains as a subalgebra a q - deformed PoincartS algebra and as Hopf subalgebras two conjugate Cartan subalgebras [10]. This is very important since we have to choose which conjugacy class 11-generator ^-deformed Weyl algebras. The The organization of the paper is as follows. In Section 2 we recall the q - deformation MIRAMARE-TRIESTE of complex simple Lie algebras. In Section 3 we present our approach. In Sections 4 and 5 we present the q - deformation of the Lorentz algebra so{ 3,1) and of the conformal algebra su( 2,2) January 1992 (the exposition is much more detailed than [9] and also some misprints there are corrected). In Section 6 we discuss the q - deformed Weyl and Poincare algebras. In Section 7 we recall the q - deformation of complex Lie superalgebras and present the q - deformation of the conformal superalgebra su(2,2/N). To appear in the Proceedings of the Quantum Grotips Workshop of the II Wigner Symposium, 2. Synopsis on the q • deformation of complex simple Lie algebras (Goslar, Germany, July 1991). At ICTP until June 30, 1992; Permanent address: Bulgarian Academy of Sciences, Institute of Let 5c be a complex simple Lie algebra; then the q - deformation t/,(Cc) of the universal Nuclear Research and Nuclear Energy, 72 Boul. Tntkia, 1784 Sofia, Bulgaria. enveloping algebras U{QC) is defined [11),[12] as the associative algebra over (Dwith Chevalley generators X* , H, , j = I,...,£ = rank Qc and with relations: e(Hj) = = 0 t(H}) = -Hi Xf , (5c) [H, , Hj] = 0, Itf, , Xf] = ±o,;Xf , e action where p e Kc corresponds to p = j £Q£i. a, p = j *£,ae&' H« • Th of 5, e, 7 t, , Xj ] - &ij l/2 ^ _1/2 = £,;[//,],, , ft = ^ on the Cartan-Weyl generators Hp, Xp is obtained easily from (5) since Hp (see above) and Xg (cf. [12],[14] and, e.g., formulae (24) below) are given algebraically in terms of the Chevalley generators. (Of course, (5b) holds for all Cartan-Weyl generators.) 3. q - deformation of real semisimple Lie algebras where (a,t) = (2(a,-,at)/(a;,a;)) is the Cartan matrix of Qc , the scatar product of the roots (•,-) is normalized so that (a, a) 6 2JW, 3.1. Synopsis on real semisimple Lie algebras Let Q be a real semisimple Lie algebra, 0 be the Cartan involution in Q, and Q = K. © V (2a) be the Cartan decomposition of Q, so that SX = X, X € £, 6X = -X, X 6 V ; K is the maximal compact subalgebra of Q. Let A be the maximal subspace of V which is an abelian subalgebra of (26) S ; T =dim A is the real (or split) rank of Q, 0 < r < £ = rank 5, Let AH be the root system of the pair (Q, A), also called {.A -) restricted root system: s, «•<• -!) = ,;'• (2c) The elements i/;- span the Cartan subalgebra Wc of (Jc , white the elements Xf generate + + the subalgebras Qf = © 5ca , where A = A U A ~ is the root system of Qc, A , A " are the (6) flea* sets of positive, negative, roots, respectively. Thus one has the standard decomposition The elements of &R = A£ U A^ are called {A -) restricted roots ; if i G AJJ , & are called (.4.-) restricted root spaces , dim^ Si > 1. Now we can introduce the subalgebras corresponding to the (3) positive (Ajj) and negative (A^) restrictedroots : We recall that Hj correspond to the simple roots a, of Qc, and if 0^ = £;. UjCtj,^ = 20/(f},P), n then to fi corresponds Hp = J^j j H; • The elements of Qc which span ^cg, (dim Qcp = 1), are 0 denoted by Xg . These Cartan-Weyl generators [121,114] are normalized so that ft = (7) eAn where Jf1, A^2, resp., is the direct sum of with dim^ 5x = 1, dim# Sx > 1, resp., and [XP , X-fi = . (4) analogously for A^a = 0^a. Then we have the (Bruhat) decompositions which we shall use for our ^-deformation : The algebra £/,(&) is a Hopf algebra [15] with co-multiplication 6, co-unit E (homo- morphisms) and antipode q (untihomomorphism) defined on the generators of Ut(Qc) as follows (8) [1UF12]: 5 = where Mis the centralizer of A in£,i.e.,M = {X e£\ [X,Y] =0, W e A). IngeneralM is a compact reductive Lie algebra, and we shall write M = Ms ® Zm . whereMj = [M,M] is 81 Hj) = Hj , HXf) (5o) the semisimple part of M, and Zm is the centre of M.. 3 Further let Hm be the Cartan subalgebra of M, i.e., Hm = 7i'n ® 2m , where rl'n is the maximally split real form (or normal real form), when all roots are real, M = 0, and Wo = A, and Caitan subalgebra of M,. Then Ho = Hm © .A is a Cartan subalgebra of $, the most noncompact (3) is reduced to one. Let Hcbe the Cartan subalgebra of the complexification CJcof Q. Of course £ = rank gD= c dimc H = dimfl H = dimfl H'm+ dimn 2m + r. Q = (10) Next it is natural to choose the basis in K^so that the elements of A take real values on c c iHm © A, namely, if H, resp., H\ is an element of the basis of Hm , resp., A, then we shall take i.e., this is the restriction to R of the standard decomposition dimjj £„ = 3 iff the a e A,. U Aj 110]. If a g A, then Ca is noncompact. For a € Aj the root a is called, singular, n e Aa, if Ca is noncompact, and a is called compact, a € At, if Ca is compact. sivh(H h l2) a a (lla) Thus Aj = A, U At . Let H = Ho, then As = 0 and the algebras £„ are given by : a-X-a) . Ha = -iHo , (lit) La = r,ls,{Ha , _a} , a 6 (9a) (lie) Ca = r.l.s.{iHa, XQ-X (96) where As is the simple root system, and for the non-simple roots as explained after (5). Note that where r.l.s. stands for real linear span. Note that X £ "PEif a £ Ar , X £ JC^if a 6 Aj . If a a formulae (11) (with ha € M) determine completely the unique q - deformation of any compact H ^f Ho we have to consider also the singular roots. Explicitly, for a £ Aa we have: semisimple Lie algebra [11] (when all roots of A are imaginary). Note that in this case the q - deformation inherited from U^G13) is often used in the physics literature without the basis change (11). Ca = r.ls.{iHa,i(.Xa-X-a), Xa (9c) Note that there is a 1-to-l correspondence between the real roots a € A and the restricted All notions above are easily generalized for the real forms of the basic classical Lie su- roots X g An with dim/; (?), = 1 and naturally this correspondence is realized by the restriction peralgebras [13]. : A = a j A . Further note that the set of the imaginary roots in A may be identified with the root system of Mf. 3.2. q - deformation of real semisimple Lie algebras Thus so far we have chosen consistently the generators of flx © A © M © N1 (cf. (8)) as linear combinations of the generators of Ho ffi ©aei,ui( Qa • Now it remains to choose Let Q be a real semisimple Lie algebra. We shall use the standard deformation from consistently the generators of fl2 © M2 as linear combinations of the generators of the rest of Qc, section 2. to the simple components of the complexification Qcof Q. i.e., ffioea, 1. Let Ax = {a e A | a \ A = \}. c The first step in our procedure is the choice of Cartan subalgebra H of Q. First we shall If a € Ac , then we have Xa = Ya + Za , where Ya eV ,Za€ KF. NOW we can see that & = give the g-deformation using the most non-compact Cartan subalgebra Ho. We consider the basis r.l.s. {Xa = Ya + iZa , Va 6 Aj, }. The actual choice of basis in Q\ is a matter of convenience, elements in (9) also as basis elements of the q - deformation US(Q) with commutation relations cf. the examples below. and Hopf algebra structure inherited from [7,(5^. A general property of the deformation Uq{Q) obtained by the above procedure is that For the real roots, a G A^, the generators in (9a) obey (4), if a £ A^ n As also (5), and U,(M), U,(Vo), U,(f>o) are Hopf subalgebras of Ut(g), where VQ = M © A ffi N, Po = otherwise as explained after (5). Thus formulae (9a) determine completely a q - deformation of any 5 We recall that a real noncompact simple Lie algebra has in general (a finite number of) non-congugate Cartan subalgebras. The conjugacy classes may be represented by Cartan subalge- bras H' = H'k © A', where Hm C H'k CHk ,Hk being the Cartan subalgebra of K, A' C A. The Cartan subalgebras with maximal dimension of A' are conjugate to K ; also those with minimal where ,M = M, © Zm, Ms = Ma ,Zm = Zm 4. q - deformed Lorentz algebra U,(so(3,l)) sinh(Haha/2) <12o) 4.1. so(p,r) = (1/N/2)(JCO .) , S~ = (i/y/2)(Xa-X-a) , Ha = -iHa , (126) Let 5 = soip^), withp > r > 2 or p > r > 1 with generators : MAB = -MBA, A,B = l,...,p+ T,7)AB = diag( + •••+), (p times minus, r times plus) which obey : ±"M< "M*± (I2c) Thus our scheme provides a different q - deformation for each conjugacy class of Cartan subalgebras. , Men] - — VBDMAC + VADMBC) • (15) 3.3. q - deformations with other parabolic subalgebras and q - deformation of Besides the "physical" generators MAB we shall also use the "mathematical" generators YAB = reductive Lie algebras and superalgebras -iMAB • One has: JC = so(p) © $o(r) if r > 2 and C = so(p) if r = 1. The generators of JC are MAB with 1 < A < B < p and p+1 With the notation of Section 2. we recall chat Vo = M®A@N(-M®A@J\f) is M * Mp - r), if p - r > 2 and M = 0 if p - r = 0,1, dim # = dim N = r(p - 1). the minimal parabolic subalgebra of Q. A standard parabolic subalgebra is any subalgebra V' of Furthermore the dimensions of the roots in the root system A of so(p+ T,(D), and in A^ depending Q such that Vo C V. The number of standard parabolic subalgebras, including T>Q and Q, is 2 \ on the parity K of p + r are given by: They are all of the form V = M'® A' <&N', M' D M,A' C A,N' C N ; M' is the centralizer of A' in Q(mod A') ; N' (resp. A7' = QN') is comprised from the negative (resp. positive) root (16a) spaces of the restricted root system of (9, A'). One also has the analogue of (8): (p-r-n)(P-r-2 + (166) r(p-r-K) (16 c) Q = ti'® A'S> M'® AT , (13) 0 for p + r even We would like to have a deformation of Q which is compatible with this decomposition. 1 for p + r odd In the scheme desribed in subsection 3.2. we have used the fact that the deformation of M, is inherited f;om the deformation of M^- However, in general M' is a noncompact reductive Lie Note that the algebra so(2n+ 1,1) has only one conjugacy class of Cartan subalgebras. Thus algebra. in these cases our q - deformation is unique in our procedure. The algebra so(2n, 1) has two conjugacy classes of Cartan subalgebras and in these cases there are two q - deformations. Thus we need to extend our scheme to reductive Lie algebras. Let Q - Q® 2 = be i real reductive Lie algebra, where (? is the semisimple part of Q, Z is the centre of Q; JC, V are 4.2. U,(so(3,l)) the +1,~1 eigenspaces of the Canan involution fl;>i = A<&ZP, is the analogue of A,Zp = ZlTP. The root system of the pair {Q,A) coincides with An a.id the subalgebras # andN are inherited With A,B = 1,2,3,0,{ +), choose D = M30 for the generator of A and H = from (y. The decomposition (g) then is: Mn for the generator of M. All roots of the complexification Qc = so(4,(U) = so(3,© 3 8 ao(3, ± [J/,A' ] = (184) (23 a) (18 c) -D, (236) = ±25, 5. q - deformed conformal alytbra U,(su(2,2)) the rotation subalgebra being given by (18a). Using 30(4,(0 af so(3,(£) © so(3, (196) the simple roots are: (a,, a2) = (02,03) = -1. The roots ±ai, ±113 are real, while the other roots are complex. The Cartan-Weyl basis for the non-simple roots is given by (cf. [12],[17]): Theweuse(7,(so(4,(£)) = © £/,(ao(3,W)) given by: = \Ha], [Ha,X±\ = f , a = 1,2, (20) X% = f), (}k) = (12),(23), (24o) and with Hopf algebra structure given by (5) replacing X{ with Xo. This q - Lorentz algebra obtained in [9] as an application of our procedure was actually first proposed in [ 16] as the quantum (246) group of Liouville theory in the strong coupling regime. All other commutation relations follow from these definitions. Besides those in (4) we have ( Thus we obtain the following Uq( so{ 3, 1)) relations with q = e* € M: \ = 2[H]cos(Dh/2) (21a) /2 fft/2 = -2[H]cos(Dk/2) (216) [X: , X;b] = -?"- X- 16 , [X[ , X;b] = Xi_,,- , 1 < a < 6 < 3 , (25a) (21c) 2 ff IX- , Xi] = Xlnq-^ , [X,- , Xil = -9 '/*XBV, , 1 < a < 6 < 3 , (256) X-iX*l, = ff +J? + Hl [X;2 , X,-3] = -« ' 'X3- , [ f2 , Xf3] = X3 9- (25e) = M±®e7ih/icos(bh/4) - N± [X^ , X,-3] = Xffl-*-* , [ £ , X,*3] = -a*'* (25/) (22 a) [Xf, , Xo] = iXfX* , [X,*2 10 1 r T ] 1L — 1/2 / <*y£\ ), K = (31) s* — Q — Q • \ *"/ Ko =-i 2 Note that for q —» 1 the RHS of eqs. (25g) vanishes. Now we can derive the relations in C7V{ a«( 2,2)) : 5.2. l),(su(2,2)) 1) According to our general scheme the deformed Lorentz subalgebra is a Hopf subalge- bra. This is seen also directly since formulae (2 8) are just the inverse of (19), only Xf ,Hj should Let Q - su(2 ,2) = JO(4 ,2). It has three non-conjugate classes of Cartan subalgebras be replaced by X* , Hi . The two commuting subalgebras here are generated by X* , Hi and represented, say, by H" . a = 0,1,2 with a non-compact generators. We shall work with the Xf ,Hi and they commute because (en ,03) = 0 . Thus the deformation of the Lorentz inc-st nor.compact Cartan subalgebra H - Ho = H1. Using the notation from subsection 4.1. subalgebra is described by formulae (21),(22),(23). with A, 3 = 1,2,3,5,6,0,( ++), choose K30 and Y^ as generators of A and Yn 2) Formulae (27a) are not deformed since D eH . for the generator of M. Since su( 2,2) is the conformal algebra of 4 - dimensional Minkowski space - time we would like to deform it consistently with the subalgebra structure relevant for the 3) The deformation of the translation subalgebra is given by: physical applications. These subalgebras are : the Lorentz subalgebra M' * so(3,1) generated by YpV, fi, v - 1,2,3,0, the subalgebra JV' of translations generated by P^ = Y^ + Y^ , ±1/2 the subalgebra JV' of special conformal transformations generated by K^ = Y^s - Y^t , the P.(PO±PJ) = 9 (Po±P3)Pa, a =1,2, (32a) dilatations subalgebra A' generated by D = F56 - The commutation relations besides those for the [Pi,P2l=0, [Po-P3,Po + P3] f (32*) Lorentz subalgebra are: 4) For the subalgebra of special conformal transformations we have: ID , = 0 , [D , FM] = [D , KA = -Ku (27 a) u2 Ka(K0 ±Ki) =q* (KQ ±K3)Ka, 0= 1,2, (33o) [ i\.\ , J\2 ] — u, [ A|] — 1V3 , IY0 ~r {Y3 ] — A( IV] ~r 1^2 ) \33D) [PM. ^"] = 2^ + 2i]M1,D, (27c) 5.1) Formulae (27b) are not deformed for the Cartan generators (28a) : From formulae (27a) we see that the Lorentz subalgebra M' is a maximal subalgebra of Q com- muting with A' , and thai M', resp., JV' , is a 4 - dimensional root vector space of the restricted root system A^ = {±X ; A(D) = 1} of the pair (Q,A') , corresponding to the restricted root X, [H,Pi ±tPj] = ±(P, ±iP2) , [ff,P3±Po] = 0 , (34o) resp., -\ . In short, the algebra Vmal = M! ®A' ®N' (or equivalently Vmax = M'©j4'ffiJV') [D,P3±Po] = ±i(Pi±Pb) , ID,Pi ±iP2] = 0, (346) is the so called maximal parabolic subalgebra of 5. where JV', resp.,JV' , is the root vector space [H,Ki±iK1] = ±(Ki±iK2), [H,Ki±K0] = 0, (35a) of the restricted root system A^ = {±X ; \{D) = 1} of (5,.4'h corresponding to X, resp., —X, lD,Ki±Ka] = ±i(Jf3±ifo), ID.ATi ±iK ] = 0, (356) (cf. subsection 3.3.). 2 For the Lorentz algebra generators we have the following expressions: 5.2) For the commutation relations between the Lorentz generators and the translation generators instead of those in (27b) we have: H = (28o) ± + 1/2 + M = X^ + Xf, N* = HXf-X?) , (286) M (Ps + Po) - + D = (29) - Y"(P| -.P2)M = -y^F, -il 1/2 Po = ^) , Pi = HX^+Xl) , Pi HX^-X^) , (30) + (P0-P3) - 11 12 \M~ , Pi + P ] = - (37 a) Q (42 0) {ib H) 2 [M~ , Po - P3] = (Pi - iP7)q - ' (37 b) (424) IM- , P, + iP>] = (P 3 - Po) , (37c) (f IN* , Ki - iK2] = i (38 a) + (P3 -Po) - tP2) , (38fc) (43 a) (436) (38 c) (43 c) - ^-(P, -iP2)N+ = y(Pi-( i N~(Ki+iK2) - (38d) •i(K3 K3) (43d) , Po + Pi) = -iqW (39 a) The commutation relations between M* and Kv may be obtained from those between M* and P^ , ft - Po] = i(Pi - (396) ± T + 5 by the following changes: M >-* M V JV K-. -N~, H K-+ -H, D I-» D, P,, I-» (-1) *' /fw , P, + iP ] = i«(ifl+ -HP6 (39 c) 2 gi/2 K+ g-i/i. These follow from the automorphism of Uq(G°) : Xf <—> Xf , H\ <—• -H3 , 1/2 1 1 Pi - iPi] = 0 (39d) H-» -Xf , Hi n. -Hi , q q"' ' , (then -Xf3). The ± commutation relations between JV* and P^, K^ may be obtained from those between M and PM 5.3) For the commutation relations between the Lorentz generators and the generators ± ± by the changes M *•—> iAf* , Po *—> P3 , Pi <—• iPz and from those between M and Kp of special conformal transformations instead of those in (27b) we have: ± by the changes M <—* —iAT* , tfo <—• ^3 , K1 -—• -iK2. 6) For the commutation relations between translations and special conformal transfor- mations we have: , Ka + K3] = g<" + iKi) , (4Oa) , X3 - Ka\ = (if, If, - iK2] = ^" - K3) - (Ko (40c) [P ±Po , Ki±K ] = ±\q±l-H-D)/2 (44a) K\ + 1K2] = 0 . 3 0 [fi±Po , K-i^Ko] = 4[?iD-D] (446) [Pj±iP2 , Ki ?iK2] = (44c) l/1 - q {K0 (41a) [Pi ±iP2 , Ki±iK2] = 0 , (44d) - iK2) , (416) [Pj + Po ,Jfi+iff2] = 2 ( (44e) y(ir,- (41c) [P3-P0 ,/f, (D ifi [P3 -Po ,/fi -%K2] = -2 , " Ki-Ka (41d) [Pi -tPj ,/fa + Jf0] = 2 (M- + (44t) 13 14 "T" [Pi + iP2 ,K3 + Ko] = -2 (44;) [Pi - iP2 , Ki - Ko] = 2 ,< - .W") , (44 It) -1/4^1/4/---1 _ 9 [Pi + (Pi ,/C3 - /Co] = -2 (44() i* The comultiplication for the Lorentz subalgebra is given by (22), for the translations, special conformal transformations and dilataions we have: (45 a) * = P3 ± Po , (466) + T = P3 + Po, T~ = Ki + Ko, f * = P\ - iP2 , iK2 = ~D (46 c) _ A f T Q(-HTib)l* g, q(D-H)/4 {M + r = /f3 - KB , r~ = P3 - Po ; Consistently with the general scheme formulae (45),(46) tell us that the deformed subal- gebras of translations and special conformal transformations are not Hopf subalgebras of (?, rather the algebras [/,CPmilI), U,CPmax) are Hopf subalgebras of 17,(5). 7* = P, ± iPi , Kx ? iK2 6. q - deformed l'oincare and Weyl algebras (456) The Poincare1 algebra is not a semi simple (or reductive) Lie algebra and our procedure is + not directly applicable. One may try to use the fact that it is a subalgebra of the conformal algebra. = P, + iP2 , T- = /f, - tJT2 , f' = ft + Po , Indeed, there is a g - deformed Poincare algebra with generators M±, N*, H, D = i£>, P^ , and + - = If3 + Ko , 7"' = P3 - Po , 7"'- = Ki - Ko , with relations given by (21), (32), (34). (36) - (39), (45) (restricted to PM), (46a). This q - deformed =0 ; Poincare1 is a (commutation) subalgebra of the deformed conformal algebra (cf. (21), (32), (34), (36) - (39)), however, from formulae (45) follows that it is not a Hopf subalgebra of Uq( au(2,2)). 6(D) = D®\ + 1 (45 c) On the other hand the deformation r/,CPm0I) of the 11 - generator Weyl subalgebra = Poincare & dilatations = "?„„ - is a Hopf subalgebra of Uq( &u( 2,2)). Another Weyl algebra conjugate to this The antipode for the Lorentz subaigebra is given by (23), for the translations, special is (7,CPmal) with generators M*. N*, H,D = it), K^, D, and with commutations relations given conformal transformations and dilataions we have: by (21), (33), (35), (40) - (43), (45) (restricted to #„), (46b) [9], Consistently with our procedure one may choose other relations instead of (28) • (30), as is done in [18], and obtain a Weyl algebra _ i\ which is seemingly different from UqCPmllI) • l±1/2 = -? (Po±p3) (Pi Other deformed Poincare algebras may be obtained by contraction of t/,(so(4,1)) and U, (so( 3,2)). Let us denote by [/*, , k = 1,2, U*2 , k = 0,1,2, respectively, the deformations of U{so( 4,1)), U(so( 3,2)), respectively, with Cartan subalgebras with k compact generators. As (46O) explained in [9], one cannot obtain in this way a Poincare algebra with a deformed Lorentz algebra (9- as a subalgebra, since one has to use contractions which involve Cartan generators. This may be a -(Pi -»• noncompact generator which is possible for [/4', and U$2 . a = 0,1, or a compact generator which is possible for [/J, , a = 1,2, and U^2 • (The last case was studied in [19].) The resulting deformed 15 16 Poincare" algebras will have a noncompact Hopf subalgebra in the case U%i and in one of the U^ The Hopf algebra structure is given by formulae (5), however, with p = pa - pj , p, cases and a compact Hopf subalgebra in the other four cases. * <*• <\o) • AfT) • resP--is lhe set of even- od Let Qc = Qc= sl( M/N; ©,£= M + W-l.We choose a Cartan matrix with elements: 7. q - deformed con formal superalgebras U4(su(2,2/N)) ajj = a'jj = 2(1 - SJU) , o/>±i = a^±[ = -1 except for o;;+i = 1, all other elements are zero; dj = \, j < M,dj = -\, j > M. Consistently the products between the simple roots 1 7.1. q - deformed complex superalgebras are: (ajtaj) = 2,0,-2 for; < M, ; = 0, ; > M, resp., (a;,a;+i) = -1,1 for; < ± M, j > M, respectively, all other products are zero. The root system is given by: A = {±a;* = Let Qc be a complex Lie superalgebra with a symmetrizable Cartan matrix A = (a,*) ±(ay + O(j+i + - •+ak) \ I <} the q - deformation U,( Qc) of the universal enveloping algebras U{QC) is defined [14],[20] as the inductively in analogy to (24) (cf. also [14]): associative algebra over W with generators Xf , Hj ,j 6 J = {1 £} and with relations: (i) (la) with ojit replaced by a'jk and [,] being the supercommutator: d [Y,Z] = YZ - (_i) <*r (iii) [21-24], for every three simple roots, say, a, , aJ±1, such that (a;-,aj) = 0, 7.2. U,(su(2,2/N)) (otj±\,aj±]) i 0,(«j+i,a;_i) = 0,(o;-,aj+i + a;vi) = 0, also holds: The Lie superalgebra 5s = su(2,2/JV) [13] is a real noncompact form of C?c= 3i(4/JV*;(R s s s = o , (476) with Cartan decomposition and splitting into even and odd parts: Q = K + V = (?f0) + 5(5) such that gf0) s au(2,2) ffiu(l) © su( JV), Kf0) * u(2) e u(2) © su(JV), dimfl-pf0) = 8, where: f The parabolic subalgebras of Qs are determined by the parabolic subalgebras of the non- compact subalgebra su(2,2) of the even part Qf0) . As for au( 2,2) in the present paper we s TV* (48) consider only q - deformation of Q consistent with the maximal parabolic subalgebra V^ = M ® in (47a,b) one uses the deformed supercommutator: (51Q) (51fc) .Vs = 5f © 5» . Gk = G->* • K) = St * N1, (51c) (49) S + = Q\®Qi, g*k = Qi> = og;, JV( 0) = g2 s fr, When T = 0 relations (47a) for « = 1 are the same as for K = —1 and coincide with (lc). The = 1/2 , i2 = 2Xi , dim fff = 4W , dim ^ = 4 , necessity of the extra relations (iii) was communicated to the author in May 1991 independently by s s M. Scheunert [21], V.G. Kac [22], and D.A. Leites [23]. These relations were written first in [21] where the primed objects are su( 2,2) subalgebras. The Caitan subalgebra H C ?fo> ' chosen for U,{sl( M(N; 17 18 [4] M. Chaichian, D. Ellinas and P. Kulsih, Phys.Rev.Lett. 65 (1990) 980; Hs = H (52) T. Curtright and C. Zachos, Phys.Lett. 243B (1990) 237; P.P. Kulish and E.V. Damashinsky, J.Phys.A: 23 (1990) L415. where K is the Cartan subalgebra of su(2,2), KAT is the Cartan subalgebra of 3u(N). [5] A. Chakrabarti, J. Math. Phys. 32 (1991) and these Proceedings; s We express the generators of Uq(g ) in terms of those of U,(g°). Fort/,(su(2,2)) we L.L. Vaksman and L.I. Korogodsky, Funkts. Anal. Prilozh. 25 (1991) No. 1, 60-62. use formulae (28) - (31), and for U,(su(.N)) formulae (11). For the latter we note that {±iajk | [6] P. Podles and S.L. Woronowicz, Comm. Math. Phys. 130 (1990) 381-431; 5 Af+3 [7] W.B. Schmidke, J. Wess and B. Zumino, preprint MPI-PAE/PTh 15/91 (1991). (53) t-i *-5 [8] E. Celeghini, R. Giachetti, E. Sorace and M. Tarlini, J. Math. Phys. 31 (1990) 2548; J. Math. Phys. 32,1155 & 1159 (1991). Note that e4 coincides with Z4 described above. Next we have to express the 8 TV generators of 5f,j. Let us denote the generators of A^> = Sf by P*k , and of A& = 5,"" by K% . Then we (9] V.K. Dobrev, Goettingen Univ. preprint, (July 1991). have [10] N. Bourbaki, Groupes at algibres de Lie, Chapitres 4,5 et 6, (Hermann, Paris, 1968). [11] V.G. Drinfeld, Soviet.Math.Dokl. 32 (1985) 2548; Proceedings ICM (1986) 798. = ^oV4-«Jf.Vt+«. «= 1,2, * = 1 J (54o) [12] M. Jimbo, Lett. Math. Phys. 10 (1985) 63-69; Lett. Math. Phys. 11 (1986) 247-252. [13] V.G. Kac, Adv. Math. 26 (1977) 8-96. = Xa*2,kU ~ iX*Mt> a= 1.2. * = l [14] V.N. Tolstoy, Quantum Groups Workshop (Clausthal, 1989) Proceedings, Eds. H.D. The commutation and Hopf algebra relations of U,(su(2,2/N)) can be explicitly writ- Doebner and J.D. Hennig, Lect. Notes in Physics, Vol. 370 (Springer-Verlag 1990) p. ten now usingformulae (28) - (31), (11), (53), (54), (50), (1), (47), (5). These formulae are omitted 118. here for the lack of space. [15] E. Abe, HopfAgebras (Cambridge Univ. Press, 1980). Acknowledgments [16] C. G6mez and G. Sierra, Phys. Lett. 255B (1991)51-60. [17] V.K. Dobrev, Proceedings of the Quantum Groups Workshop, (Clausthal, 1989) Eds. The author would like to thank Professor Abdus Salam for hospitality and financialsup - H.-D. Doebner and J.-D. Hennig, Lecture Notes in Physics, Vol. 370 (Springer-Verlag, port at the ICTP. He would like to thank L. Castellani for noticing a misprint in [9]. This work was Berlin, 1990) p. 107; ICTP Trieste internal report IC/89/142 (June 1989) & Proceedings partially supported by the Bulgarian National Foundation for Science, Grant [2] Yu.I. Manin, Ann. Inst. Fourier 37 (1987) 191-205; Montreal University preprint, CRM- [20) M Chaichian and P.P. Kulish, Phys. Lett. 234B (1990) 72-80. 1561 (1988). [21] M. Scheunert, Bonn University preprint BONN-HE-9M0 (May 1991) and these Pro- [3] S.L. Woronowicz, Comm.Math.Phys. Ill (1987) 613; Lett.Math.Phys. 21 (1991) 35. ceedings. 19 20 [22] V.G. Kac, private communication (May 1991). [23] D.A. Leitcs and V. Serganova, private communication (May 1991). [24] S.M. Khoroshkin and V.N. Tolstoy, these Proceedings. 21