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The quantum Heisenberg group H(l), Enrico Celeghini Dipartimento di Fisica dell ‘Universith and I. N.F. N. Sezione di Firenze, Largo E. Fermi, 2-I 50125 Firenze, Italy Riccardo Giachetti Dipartimento di Matematica, 50125 Firenze, Italy Emanuele Universitb di Bologna and I.N.F.N. Sezione di Firenze, Largo E. Fermi, 2-I Sorace and Marco Tarlini I. N.F. N. Sezione di Firenze, Largo E. Fermi, 2-I 50125 Firenze, Italy (Received 18 July 1990; accepted for publication 21 November 1990) The structure of the quantum Heisenberg group is studied in the two different frameworks of the Lie algebra deformations and of the quantum matrix pseudogroups. The R-matrix connecting the two approaches, together with its classical limit r, are explicitly calculated by using the contraction technique and the problems connected with the limiting procedure discussed. Some unusual properties of the quantum enveloping Heisenberg algebra are shown. I. INTRODUCTION In a previous paper,’ we have shown how one can determine new quantum groups by using the technique of contraction. Indeed the contraction procedure defined in Ref. 1 as an extension of the usual one allows us to obtain quantum groups (Hopf algebras) from semisimple quantum groups making the following diagram commutative: SQG q-l ‘lo CQG 1 I SL -t q- 1, CL t--O where SQG and CQG, SL and CL, respectively, mean simple and contracted quantum group, simple and contracted Lie algebras. We note that in general, because of the nonlinearity of the algebraic structure, the limit procedures produce q-deformations of Lie algebras that may be not Hopf algebras.2V3 Contracting the well-known SU(2),, we havedefinedin Ref. 1 the quantum Heisenberg group H ( 1) 4, whose three generators A, A + , and H satisfy the relations [H,A ] =o, [H,A+] =o, [A,A ‘I = sh(Iy) , (1.1) with coproduct: wH/~~A +ABeWH/4 hA=e hAt=e-“‘H/4eAt+Atee”H/4, hH=1@H+H@1, (1.2) the antipod and counit are y(A) = -A, y(A+) E(A) =E(A+) =E(H) = -A+, =O, y(H) = -H, e(1) = 1. The quantum content of this structure is clearly given by the noncocommutative coproduct ( 1.2) while the algebraic relations ( 1.1) can be obviously redefined, because His central, to yield the classical Heisenberg algebra. 1155 J. Math. Phys. 32 (5), May 1991 The Hopf algebra H( 1) 4 just defined is clearly different from the algebra of the q-deformed creation and annihilation operators used in the Jordan-Schwinger map of SU (2) 4;4 as it has been shown in Ref. 5 the right quantum structure for these q-deformed operators is B( O( 1) 9. This fact is related to the following properties of the classical a, a’; they close an algebra [a,at ] = 1, quantized as our H( 1) 4, but also with H = 2at a + 1 a super algebra {a,~+ ) = H, [ H,a] = - 2a, [ H,at ] = 2a’, quantized as B( 011 )9 in Ref. 5, that gives rise to the Jordan-Schwinger approach to SU(2),. The purpose of this work is to gain a deeper insight into the structure of H ( 1) ~. We thus derive the universal R-matrix for H( 1 )4 by a contraction on the R-matrix of the quantum group SU(2),.“*’ We find that the leading R-matrix term emerging from the procedure turns out to be singular. This singularity can be discussed and removed and we shall see that the finite contribution to R gives rise to an operator Nanalogous to the occupation number operator of the usual Lie algebra of the harmonic oscillator and indeed satisfies the same commutation laws with A and A +. However, we see that, as in the classical harmonic oscillator enveloping algebra, N is not a primitive element, and it is reminiscent of its quantum structure at the coproduct level: this fact provides a concrete and instructive example of the very stringent requirements imposed by the Hopf algebra structure. Moreover, by using the explicit form of AN, it is easy to see that the quasitriangular property of Hopf algebras6 fails to be satisfied. The universal and some parameter-dependent solutions of the QYBE and CYBE (quantum and classical Yang-Baxter equations) are provided. The R-matrix so far determined will be used to make an explicit connection with the formalism of matrix quantum pseudogroups due to Woronowicz’ and to define the Hopf algebra of the representative elements. In this framework, we also determine the infinitesimal generators of the matrix pseudogroup and we calculate both commutation relations and coproducts. As a matter of fact, the former turn out to be coincident with the commutators defining the Heisenberg Lie algebra h ( 1) : It is again the coproducts that give a clear indication of the underlying quantum structure. 0022-2488191 I051 155-04$03.00 @ 1991 American Institute of Physics 1155 Downloaded 20 Aug 2009 to 136.152.180.163. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp ILTHER-MATRIX FOR H(l), AND ITSCLASSICAL LIMIT The universal R-matrix of a quantum group Q is defined by its property of being a sort of intertwining operator on the coproduct A, namely, RA(u)R -’ = ah(u), u&P, tions of the coproduct, counit, and antipode of the element N, that will be of use later on. Proposition(2.7): In U, [h ( 1) ] we have A.(N) ewuI 2 (2.1) wherea(amb) = b@a. For SU( 2)y the R-matrix has the well-known expression7 R = /JeJJ c (l -e-‘)k X (erkJ”’b-J+lk8e - rkJ.s/2(J- At=~1’2JJ_, +e(w’4F’HeH o Fob P (w’/2e - &/4~ ,“I, (2.2) fi=2EJ,, w=&-‘2, (&/2&0H/4AIk . f)k (2.4) nevertheless the E-+O limit of the first exponential is singular, and its meaning must be properly clarified. A first observation can be immediately made from the defining equation (2.1) of the R-matrix. Indeed, we see that the multiplication of R by any central factor leaves (2.1) invariant. When taking the limit of (2.4), we shall thus neglect any possibly emerging central factor, even a singular one: This is just the case of the leading term of order E - ’ in the exponent of exp [ (w/4&) He H] . A second observation arises then naturally from the previous point. Since the leading singular term of the above exponential will be neglected because it is central, the finite contribution to the limit of (2.4) is coming from the next order of the expansion in E. Setting J3 = H/( 2~) - N, this contribution results in exp[ - (w/2) finite x (Hs N + N@ H) ] as can be deduced by looking at the limiting procedure yielding, using the relations (2.3), the representations of H( 1 )4:’ J- [E-$,I’) = ([%‘],[&-‘&‘-I’+ J+ /E-&J’) = ([Y+ 1]s)“2/&->,V- l].,[E-‘p-y]s)1’21E--Ip,~+ I), I), (2.5) J3 IE - ‘p,Y) = (E - ‘p/2 - Y) I& - ‘p,V), where s = e’“, p = 2&j, Y = j - m, and Nis the diagonal operator with eigenvalues Y. The explicit expression of N in terms of A, A *, H is given by the series N=;A+A -I [ 1 sh? =wAfA 2 ,-Ck+l/ZWH, tA ciD e - kwH (k+3/4)wHAtQpe-(k+I/4)WHA -Fe- J. Math. Phys., Vol. 32, No. 5, May 1991 e(N)=O, (k +3/4,&A o e - (k + b’4)WH~ t) (2.6) computa- y(N)= --N-l. Prooj Indeed, using the definitions, some lengthy but otherwise straightforward computations show that the Hopf algebra identities m(idsy)A(N) = e(N), (e@id)A(N) =N,and(A@id)A(N) = (ideA)A(N) (Ref,9),aresatisfied. Remark: We observe that U4 [h ( 1) ] is an example of nonquasitriangular Hopf algebra,’ since the defining relations of the latter, = R,,R,, (Aeid)R = R,,R,, and (ideA)R are not satisfied, as can be easily checked using ( 2.7) and (2.10). We also observe that A, A +, H, N satisfy the commutation relations [A,A +] = (2/w) sh(wH/2), [NJ ] = -A, [%A+] =A+, (2.8) with Hcentral, which constitute a quantum deformation ofa four-dimensional Lie algebra, that could be also obtained as contraction of an SU(21, with central extension. In this point of view N is a primitive generator, with A(N)=N691fIsN,~(N)=O,andy(N)= -N,and the Hopf algebra becomes quasitriangular. Proposition (2.9): (i) The nonsingular part of the contraction of (2.2) leads to the element REU, [h( 1 f ] Q Uq [h( 1) 1 given by R _ e - (to/2)(HsN+ NeH) W1/2ewH/4A) kzo$ 8 (wl/2e- W11/4A - (w/Z)CHeN+ =e k ( f)k NGM)~B~B+ (2.10) 3 t = w1/2 , 3 = WI” exp[wH/4] and A where xexp[ - wH/4lA +. (ii) The equation (2.1) is satisfied in U, [h ( 1) I. (iii) The R-matrix (2.10) solves the QYBE: (2.11) &R13&3 = R,&,R,z. Proof: The point (i) has already been discussed. (ii) and (iii) are again proved by direct calculations. As far as the QYBE is concerned, we may observe that R 12 =e- R 13 =e k=O Equation (2.6) also permits straightforward 1156 (k + I/‘lfwH~ and (2.3) and then letting ~-0. Therefore, by a straightforward substitution and neglecting higher orders in E, we find that R e - -l-e- with ez = q. We shall determine the R-matrix of H( 1 )9 by contracting (2.2) with the rules that produce H( 1 I9 from SU(2), (see Ref. I), i.e., by defining A = &“ZJ +, + e-zkck-It/4 [k I! k>O (e-‘k+ItluN~e-(k+I/2’~HAtA k=O Rz3 = e- t~/2’(NeH~l’~- WZ)WC+N~~~~BIB+~I (w/Z)(NheleHie- ~ZmfsleN~~BellepB+ f , ~~/2~(I~N~,H~~-truff~~leH~N~~leBoS’ The problem is thus reduced to proving the equality Celeghini etal, 1156 Downloaded 20 Aug 2009 to 136.152.180.163. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp eBaB’ol eBe = exp[wH/Z)eB exp[ - eleBeB+eBe The same formula (2.20) then gives a trigonometric of the CYBE for H(l),, where now eleBeB+ wH/21eBteBeB+el , s=;(aea+-a+ea), which indeed holds true, due to the commutation law [B,B+ ] = 2 sh(wH/2). From the universal R-matrix, by standard techniques,” we can readily deduce a new R-matrix depending on a parameter x. In fact, defining the operator TX by its action on the generators T,A=x.4, TxAt=A+/.x, t=;(aea++a+@a--hhn-nnh). As in the SU ( 2)4 case, t /u is the rational solution of (2.19). III. THE MATRIX QUANTUM GROUP H(l), T,H=H, We shall now present the quantum group H( 1)4 as a matrix quantum group b la Woronowicz.’ We therefore consider a 3 x 3 matrix of the form we can define R(x) = (T@l)R =e-(w’2)(HoN+NeH)exBeBt (2.12) 1 a P 1 S, (3.1) 0 0 1 ( 1 where the matrix elements a, 8, and S generate a C *-algebra d whose relations are to be determined. This will be done by first giving a 3 x 3 representation of the generators A, A +, H, N of U, [h ( 1) 1, with commutation relations (2.8). As a consequence, the R-matrix (2.10) will be represented on the tensor product and the defining equation” and again by a direct calculation we have: Proposition (2.13): The matrix R(x) defined in (2.12) satisfies the QYBE R,2 (x)R,, WV&3 (Y) = A,, (yM13 W)R,, T=O (xl. (2.14) Corollary (2.15): The classical r-matrix corresponding to (2.10) is r=asa+-t(h@n+n@h), (2.16) where a, at, h, n are defined to be the classical limit ofA, A +, H, N, and satisfy the commutation rules [a,a+J = h, Moreover [n,a] = -a, RT,T, = T,T,R, (3.2) with T, = TB 1 and T2 = 1 o T, will provide the required relations. Proposition (3.3): A representation of (2.8) is given by [ n,a+] = a+. (2.16) solves the CYBE (2.1 7) + [r122r23] $- [r13yr23] = O. [r 129r13] Similarly, the classical limit of (2.12), with x = e”, is r(u) = e”asa+ -$(hen+neh) and solves the parameter-dependent p(A)=! 9 3, p(A+)=[ ; ;) p(H)=[ ; i), p(N) =[ 8 3. (2.1 8) CYBE [r,2 (u),r,3 (u + VI] + [r,2 (u),r23 to)] + [r,3 (u f u),r23 (VI] = 0. Correspondingly, (2.19) 13 ProoJ Recalling that the classical r-matrix is obtained p(R) = by a first-order expansion in w of the quantum R-matrix, i.e., R = 181 + wr,itisimmediatetoverifythat (2.16)~(2.18) are the classical counterparts of (2.10)-(2.12) and (2.14), respectively. Remark: The contraction technique can again be used to produce trigonometric and rational solutions of (2.19). Indeed it is well known that for SU( 2), such a solution is of the form r(u) =s-t+2t/(l -e’), 0 ( 0 (2.10) is represented by the 9 X 9 matrix w/M +I - (d2)pW) 13 - (w/2)&H) 0 0 13 , 1 1, being the 3 X 3 identity matrix. Prooj Immediate. In the following we shall suppress the explicit indication of the representationp and we shall simply denote the representatives of the generators by the generators themselves. Proposition (3.4): The relations between the generators of LZ?are the following: (2.20) a@ - pa = (w/2)a, aS - Sa = 0, ps - c5p= - (w/2)& Pro08 Indeed the explicit form of (3.2) reads where s=J, t=J, solution eJ_ -JBJ,, @J, +J2@J2 +J38J3. I T 0 0 1157 aTf wA+T T- (w/2)HT 0 PT-t wA+6T- (w/2)NT 6T- (w/2)HST = T I( J. Math. Phys., Vol. 32, No. 5, May 1991 T 0 0 wTA++ Ta- (w/2)TaH T- (w/2)TH 0 - (w/2)TN+ TS T Tfi Celeghini etal. 1157 Downloaded 20 Aug 2009 to 136.152.180.163. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp We shall now use the tensor product of representative elements of type (3.1) in order to define a Hopf algebra structure on d. Since lel 0 TeT= 1e/9-t-pe1+CYe 1@8fG@l 181 Isa+aal lel 0 i (3.5) 3 and -a a&--/3 1 -6 ) , 1 0 we see that a, ,B, and y with the relations specified in (3.4) have coproduct, counit, and antipode given by T-lEio A(a) = 1 @a + a@ 1, y(a) = -a, c(a) = 0, A(B) = leB+Bsl r(P) = -B+aS, E(P) =O, y(S) = - 4 E(S) = 0. +aeS, A(S) = 1~6+Sel, (3.6) (3.7) I The verification of the Hopf algebra identities’ is straightforward. Let us recall that in the Woronowicz approach,’ three infinitesimal generators A,, A, , A,, are defined for SU ( 2) 4, and they satisfy the relations q,‘2A,A, qA,A, - ( l/q,‘2)AoA2 - (l/q)A,A, = A,, = (1 -kq)A,, (3.8) = (1 -I- q)A,. @,A, - (l/qM,A, We shall now find the analogous generators and relations for H(l), by using again the contraction technique. However, in order to do that, we need an explicit expression ofA,, A,, A, in termsofJ+ , J- , J3,12 l/4 A, =q A, = _ - 5,,/2 q’/4- J +, A, =q,‘2--‘[2J&, Jd2~- , (3.9) where, as usual, [ 2J3 ] = sh ( zJA ) /sh (z/2 ) and z = log q. Proposition (3.10): The infinitesimal generators of the quantum matrix pseudogroup H( 1 )9 obtained by contracting the generators A,, A, , A, of SU (2) I are, respectively, B. =e- wH’4A, B, = (l/w)(l--e-“H), & = -e-wH’4At, (3.11) and they satisfy the relations BIB0 - BOB, = 0, B,& -BOB2 B, B2 - B,B, = 0, =B,. (3.12) Prooj It is a straightforward evaluation of the contraction limit. Remark: From (3.12) it appears that the intinitesimal generators B,, B, , B, satisfy the very same commutation relations as their classical analogs. Of course the quantum structure has not disappeared and can be recognized in the 1158 J. Math. Phys.. Vol. 32, No. 5, May 1991 coproduct, where the deformation parameter w enters. In fact, from ( 3.11)) we have AB, = (1--~~3,)“~s3~ +B,sl, +B, ol-wB, eB,, A4 =lsB, AB, = (1-wB,),‘2@BB2 t&c31 and we see that, while the classical generators (w = 0) are primitive elements in the corresponding Hopf algebra, the quantum generators are not such. ACKNOWLEDGMENTS One of us (R.G. ) would like to acknowledge Professor E. Taft for a critical reading of the manuscript. ’E. Celeghini, R. Giachetti, E. Sorace,and M. Tarlini, J. Math. Phys. 31, 2548 (1990). ‘P. P. Kulish, “Kontraktzia kvantovikh algebr i q-ostzilliatori,” LOMI preprint ( 1990). ‘M. Chaichian and D. Ellinas, J. Phys. A: Math. Gen. 23, L291 ( 1990); Y. J. Ng, J. Phys. A: Math. Gen. 23, 1023 (1990). 4L. C. Biedenharn, J. Phys. A: Math. Gen. 22, L873 ( 1989); A. J. Macfarlane, J. Phys. A: Math. Fen. 22,458l ( 1989). ‘E. Celeghini, T. D. Palev, and M. Tarlini, “The quantum superalgebra B, (011) and q-deformed creation and annihilation operators,” Kyoto Univ. preprint YITP/K-865 ( 1990), Mod. Phys. Lett. B (in press). ‘V. G. Drinfeld, Proceedings of the KM, Berkeley, CA, edited by A. M. Gleason (American Mathematical Society, Providence, RI, 1986), p, 798. ‘A. N. Kirillov and N. Yu. Reshetikhin, LOMI preprint E-9-88 ( 1988). *S. Woronowicz, Publ. RIMS (Kyoto Univ.) 23, 117 ( 1987); Commun. Math. Phys. 111,613 (1987). 122, 125 (1989). 9E. Abe, “Hopf algebras,” Cambridge Tracts in Math. No. 74, (Cambridge UP., Cambridge, 1980). “M. Jimbo, Int. J. Mod. Phys. A 4,3759 ( 1989). ” L. Faddeev, N. Yu. Reshetikhin, and L. Takhtajan, in Braid Group, Knot Theory and Statistical Mechanics, edited by C. N. Yang and M. L. Ge (World Scientific, Singapore, 1989). “T. L. Curtright and C. K. Zachos, Phys. Lett. B 243,237 ( 1990). Celeghini et al. 1158 Downloaded 20 Aug 2009 to 136.152.180.163. Redistribution subject to AIP license or copyright; see http://jmp.aip.org/jmp/copyright.jsp