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International Centre for Theoretical Physics 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 <j-deformed Lorentz algebra is Hopf subalgebra of of Cartan subalgebras will correspond to the unique conjugacy class of Cartan subalgebras of <JC both Weyl algebras. and will be "freezed" under she q - deformation (cf. (lb) below). For each such choice we shall get a different q - deformation. Our approach is easily generalized for the real forms of the basic classical Lie superalgebras and of the corresponding afrine Kac-Moody (super-) algebras. 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^) restricted roots: 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 <? = ?f ffi H ® ff? , and hence c iH, resp., H\ as an element of the basis of K . Uq(G) is just the restriction of [/,(Q°j to R. For the classical complex Lie algebras these forms are sl(,n, JR), so(n,n), so{n + 1,n), sp(n, R), which are dual to the matrix quantum groups It is important for our procedure to choose consistently the basis of the rest of Q and c SL,(n, JR), SO (n,n), SO (n,n+ 1), Sp,(n, R), introduced in [1] from another point of view Q . For this we use the classification of the roots from A with respect to H. The set A, = {a £ q q than ours. A | a | Hm = 0 } is called the set of real roots, A, = {a € A | a \ A - 0} - the set of imaginary roots, Ac s A\(Ar U A,) - the set of complex roots [10].
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