Bosonic realization of the quantum superalgebra ospq(1, 2n)

Volume 242, number 3,4

PHYSICS LETTERSB

B O S O N I C REALIZATION OF T H E Q U A N T U M SUPERALGEBRA

14 June 1990

OSpq(l, 2n)

Roberto FLOREANINI Istituto Nazionale di Fisica Nucleare. Sezione di Trieste, and Dipartimento di Fisica Teorica, Universit&di Trieste, Strada Costiera I1, 1-34014 Trieste, Italy

Vyacheslav P. SPIRIDONOV California Institute of Technology, Pasadena, CA 91125, USA

and Luc VINET 2 Department of Physics, University of California, 405 Hilgard Avenue, Los Angeles, CA 90024, USA

Received 27 March 1990

A realization of the quantum superalgebraOSpq( 1, 2n) is given in terms of the annihilation and creation operators of n ordinary harmonic oscillators.

Quantum Lie algebras originally appeared in studies of the quantum inverse scattering problem [ 1 ]. Owing to their relation with the quantum YangBaxter equations, they are emerging as a universal structure in many physical situations and they are now the object of intense investigation. Mathematically, these quantum algebras (also known as quantized universal enveloping algebras) have been identified as quasi-triangular H o p f algebras [2,3 ]. The quantum deformations of all simple Lie algebras have been obtained and progress has been made in the construction of their representations [2,4]. One way to characterize these algebras is to give their generators together with the defining relations based on the Cartan matrices of the corresponding simple Lie algebras. Recently, several authors [ 5 ] have discussed the quantum generalization of the Jordan-Schwinger mapping and, using the quantum analogue of the harmonic oscillator, have Permanent address: Institute for NuclearResearchof the USSR Academy of Sciences, 6 0 th October Anniversary prospect 7a, SU-117 312 Moscow, USSR. 2 On sabbaticalleave from: Laboratoirede Physique Nucl6aire, Universit6 de Montreal, Montreal, Canada H3C 3J7.

provided explicit realizations of the quantum algebra SUq(n) of su(n). The quantum algebras corresponding to the other classical Lie algebras have also been constructed with the help of q-deformed Clifford and Weyl algebras [ 6 ]. Moreover, it has been shown [ 7 ] that the q-boson operators can actually be expressed as functions of their ordinary (classical) counterparts. This in turn allows for the construction of quantum Lie algebras in terms of classical bosonic creation and annihilation operators. These models of quantum algebras offer useful and simple techniques for performing various algebraic manipulations; they are also very practical in discussing representations. It is of course also of interest to describe the quantum deformations of Lie superalgebras. Their general definition has now been formulated and their H o p f algebra structure established [ 8 ]. The simplest examples, that is ospq( 1, 2), SIq( l, 1 ) and Slq(2, 1 ), have been examined in detail [8,9 ]. In addition, a q-oscillator representation of sly(m, n) has been given by expressing the generators as bilinears in the creation and annihilation operators of q-bosons as well as of q-fermions [ 8 ]. Such a representation for the other

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class of quantum superalgebras, namely ospq(m, 2n ), so far has remained elusive [ 8 ]. In this note we shall show that the generators of OSPo(1, 2n) can be realized in terms ofbosonic oscillator operators only. The (rank n) Lie superalgebras of classical type can be presented as follows [ 10-13,8], given their symmetric Cartan matrix A = (ao) and a subset x of the set I = { 1, ..., n). They are generated by 3n elements 0~, f, and hi, ieI, which are subject to the defining relations

Ig, [h,,f~] = -ao~,

[hi, o;]=aoO;,

(1)

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degei=degf= 1 , iez, with q = e ~ the deformation parameter. These relations should also be supplemented with the analogue of the Serre relations. A coproduct, an antipode and a co-unit can be defined [ 8 ] to set up the Hopf algebra structure [ 14 ] of the quantum superalgebras. We shall be concerned in this paper with the quantum deformations ofosp ( 1, 2n) (or B (0, n) in Kac's notation). The symmetric Cartan matrix A associated to this superalgebra is [ 13 ]

A=

--1

-.. • ..

with

2

-

'

(4)

-1

deg hi = 0 , deg ~i=deg~ = 0 ,

iCx,

deg~=deg~=l,

i~x.

and in this ease x= {n}. The complete defining system for osp¢( 1, 2n) is obtained by adding to the equations (3) the relations

The symbol [ , ] stands for the graded Lie product:

[x, y] = x y - (

)degxdegyyx .

-

(ad6)'-a°Oj=0,

(adf,)'-a'~=0,

i¢j,

(2)

with 8 o the matrix which is obtained from the nonsymmetric Cartan matrix a~K) ofref. [ 11 ] by substituting - 1 for the strictly positive elements in the rows where a , = 0 . (Multiplication by a diagonal matrix transforms a ~K) into the symmetric Cartan matrix a~, see ref. [ 13 ]. ) All the remaining algebra elements together with their commutation relations can be obtained from these data. Notice that ordinary Lie algebras correspond to x = I~. The corresponding quantum Lie superalgebras are analogously generated by 3n elements ei, f and hi, ieI, which now satisfy [ 8 ]

[ei,~] "~8ij sinh 7111i

sinh-----~'

[hi, ej] =aijej,

[h.h:]=O,

[hi,fj] = - a i j £ ,

e~ e j - 2e~eje~ cosh q+ eje 2 =0 , for I i - j l = 1, i ~ n ,

These generators should also verify the Serre relations [8,12]

(3)

e 3e n - i

--

(e~en_l en + ene~_, e~)

× (2 cosh 0 - 1 ) +e,_~e~ = 0 ,

bi b t, - q b ~b~=q -~v' , [N,, b;] = -dijb,,

(6b)

[b, bj] = [ b,, b}] = [b~, b}] = 0 , [N,, Nj] = 0 .

(6c)

In the limit q-~ 1 for r/-,0), the previous equations reduce to the standard commutation relations of bosonic operators; in particular (6a) becomes

deg hi = 0 ,

and one has

iCx,

[Ni, bf] =dijb,t,

(6a)

and for i # j

[/~,/~]] =rio,

degei=degf=O,

(5)

and [e~, ej] =0, for [ i - j l > 1, and similar ones obtained by replacing the e's b y f s . Let us now introduce the annihilation, creation and number operators, bi, b,~ and Ni, for n q-boson oscillators; they satisfy the following deformed commutation relations [ 5-8 ]:

and

384

...

~,=bT~,.

(7)

(8)

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PHYSICS LETTERS B

The q-oscillator operators can be expressed in terms of their classical analogues [7] b'= N] N , + I

/~''

b'~=q---~--, • /~'t'

N, = N , ,

(9)

where

14 June 1990

will be irreducible and infinite-dimensional. In the rank-2 case, for example, the six generators of ospq ( 1, 4) are expressed as follows in terms of the annihilation and creation operators of two deformed oscillators: h1=NI-N2,

el=b~b2,

f(~ri) - sinh r/~ sinh r/

h2 = N 2 q-½ ,

fl=b~bl, 1

(10) ez= ~

½r/

bE, f2=

1

4 2 cosh It/

b2,

(13)

(Notice that for b~and b,t to be hermitian conjugates, q has to be real or a pure phase, i.e. ~/has to be real or purely imaginary. ) It is easy to check that this identification provides a realization of the deformed commutation relations (6) as it implies in particular that

and we have deghl=deghz=dege~=degf~=O, deg e2 = degJ~ = 1. This ensures that the relations

b~lb~=f(?¢~),

[ez,f~]=0,

b~b~=f(N~ + 1 ) .

(11 )

It is known that bilinears in the annihilation and creation operators of n ordinary harmonic oscillators realize the symplectic algebra sp (2n). Adjoining the linears as odd basis elements, one gets an algebraic set which is isomorphic to osp( 1, 2n). It proves possible to generalize this construction to obtain analogously a representation of OSpq( 1, 2n) in terms of qdeformed boson operators. Consider the following operators: hk = Nk -- Nk +l , ek=b~bk+l,

fk=b~-+lbk,

(12a)

for k = 1, ..., n - 1, and

1

{e2,fE}=e2f2+f2e2-

[ht, e t ] = 2 e ~ ,

[ht, e2] = - e z ,

[h2, e l ] = - e t ,

[h2, ez]=e2,

[h,,f, ] = - 2 A ,

[h,,A] =A,

[&,ft I =ft,

sinh r/h2 sinhq '

[h-,,AI=-A,

[h~, h2] = 0

(14)

are obeyed. Another oscillator realization of OSpq( 1, 2n) can be obtained by replacing (12a) with hn_2k = N n _ 2 k + Nn_Zk+l + 1 ,

k = l ..... ~ ( n - 1 ) / 2 ~ ,

h,=U,+½, e, = x/2 cosh ½~

[ e l , f ~ ] = sinht/h-------L, [ e l , A ] = 0 , sinh q

hn_2k+l ~. -- (Nn_Zk+l "~Nn_2k+2 "~ 1 ) ,

b~, f~=

1

x/2 cosh ½~/

b,.

(lZb)

The h's are hermitian, while t h e f ' s are the hermitian conjugate of the e's. In the limit r/--,0, it is straightforward to check that these operators generate osp ( 1, 2n). Using the classical realization of the q-boson operators ( 9 ) - ( 1 1 ) , one verifies that the operators given in (12) actually satisfy the defining relations ( 3 ) and (5) for the generators of the quantum superalgebra ospo( I, 2n). Furthermore, one obtains in this fashion a representation ofospq( 1, 2n) on the Hilbert space formed by the state vectors of n ordinary harmonic oscillators. For a generic q this representation

k = l .... , ~ n / 2 ~ , • t t e,_z~=-lbn_zkb,_zk+l,

k = l .... , ~ ( n - l ) / 2 ~ ,

en-2k+l =ib,_2k+lbn_2k+2, f,_2~=-ib,_2~b,_2~+~, f~_2k+,=ib*,_2~+,b~_2k+2,

k= 1..... ~n/2~ ,

k = l , ..., ~ ( n - 1 ) / 2 ]

,

k=l,...,~n/2~,

(15)

where ~x~ is the integer part of x, while leaving the expressions (12b) for hn, e, and f~ unchanged• One can check that again the relations (3) and (5) are satisfied• However, fk is now the anti-hermitian conjugate of ek. The rank-1 quantum superalgebra ospq(1, 2) was 385

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presented in ref. [9 ] in a different basis. The three generators were d e n o t e d by H and v+, and taken to satisfy the relations

[H,v+]=+½v+,

{v+, v _ } =

sinh qH sinh 2r/ "

(16)

These can be realized by renormalizing o u r generators h, e and f a s follows:

H=th, i (

cosh ½q ~1/2 coshqcoshlqcosh½rlNJ e,

i [

cosh ~/ ~1/2 -co s h r/cosh "r/cosh ~ " r / ( N + 1 ) j/ f " ~

v+=~

(17)

We thank W. Heidenreich for a useful c o m m e n t and Y. Saint-Aubin for sharing his knowledge on quant u m groups. R.F. is grateful to the U C L A Departm e n t o f Physics for its hospitality a n d support while this work was carried out. V.P.S. thanks F. Boehm for the hospitality and support at Caltech. L.V. is supported in part by the U C L A D e p a r t m e n t o f Physics, the National Sciences and Engineering Research Council ( N S E R C ) o f C a n a d a and the F o n d s F C A R o f the Quebec Ministry o f Education.

Note added. Since the submission o f this paper, we have o b t a i n e d [ 15 ] realizations o f the q u a n t u m superalgebras corresponding to the A ( m, n), B ( m, n), C ( n + 1 ) and D (m, n) series in terms o f the creation and annihilation operators o f q-deformed Bose and F e r m i oscillators. There also recently a p p e a r e d a letter [16] in which q-oscillator representations o f OSpq(1, 2) and ospq(2n, 2) are discussed. References [ 1]L.D. Faddeev, in: Recent advances in field theory and statistical mechanics, Les Houches XXXIX, eds. J.B. Zuber and R. Stora (Elsevier, Amsterdam, 1984); P. Kulish and E. Sklyanin, in: Lecture Notes in Physics, Vol. 151 (Springer, Berlin, 1982) p. 61,

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[ 2 ] V.G. Drinfel'd, in: Proc. Intern. Congressof Mathematicians (Berkeley, 1986), vol. 1 (American Mathematical Society, Providence, RI, 1986) p. 798; M. Jimbo, Lett. Math. Phys. 10 (1985) 63; 11 (1986) 247; Commun. Math. Phys. 102 (1987) 537; S.L. Woronowicz, Commun. Math. Phys. 111 (1987) 613. [3]Yu.I. Manin, Quantum groups and non-commutative geometry (Centre de Recherches Mathematiques, Montrral, 1988), [4] M. Rosso, Commun. Math. Phys. 124 (1989) 307; P. Roche and D. Arnaudon, Lett. Math. Phys. 17 (1989) 295. [5] L.C. Biedenharn, J. Phys. A 22 (1989) L873; A.J. Mac Farlane, J. Phys. A 22 (1989) 4581; C.-P. Sen and H.-C. Fu, J. Phys. A 22 (1989) L983; Y.J. Ng, Comment on the q-analogues of the harmonic oscillator, University of North Carolina preprint IFP-365UNC ( 1989 ). [5] T. Hayashi, Commun. Math. Phys. 127 (1990) 129. [7]A.P. Polychronakos, A classical realization of quantum algebras, University of Floridapreprint, HEP-89-23 (1989). [8] M. Chaichan and P. Kulish, Phys. Lett. B 234 (1990) 72. [9] P. Kulish and N.Yu. Reshetikhin, Lett. Math. Phys. 18 (1989) 143; P. Kulish, in: Problems of modern quantum field theory, eds. A.A. Belavin, A.U. Klimyk and A.B. Zamolodichikov (Springer, Berlin, 1989); H. Saleur, Quantum osp( 1, 2) and solutions of the graded Yang-Baxter equation, Saclay preprint PhT/89-136 ( 1989); C. Derchaud, A q-analogue of the Lie superalgebra osp (2, 1) and its metaplectic representation, Freiburg University preprint THEP 89/12 ( 1989 ). [10] V. Kac, Commun. Math. Phys. 53 (1977) 31; Adv. Math. 26 (1977) 8. [11] V. Kac, in: Lecture Notes in Mathematics, Vol. 676 (Springer, Berlin, 1978) p. 597. [ 12] D.A. Leites, M.V. Saveliev and V.V. Serganova, in: Group theoretical methods in physics, eds. M.A. Markov, V.I. Man'ko and V.V. Dodonov, Vol. 1 (VNU Science Press, Utrecht, 1986). [ 13 ] L. Frappat, A. Sciarrino and P. Sorba, Commun. Math. Phys. 121 (1989) 457. [ 14] E. Abe, Hopfalgebras (Cambridge U.P,, Cambridge, 1980). [ 15] R. Floreanini, V.P. Spiridonov and L. Vinet, q-oscillator realizations of the quantum superalgebras slq(m, n) and ospq(rn, 2n), UCLA-preprint UCLA/90/TEP/21 (1990). [ 16] M. Chaichian, P. Kulish and J. Lukierski, Phys. Lett. B 237 (1990) 401.