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Quantum lie superalgebras and q-oscillators
Volume 234, number 1,2
PHYSICS LETTERS B
4 January 1990
Q U A N T U M LIE SUPERALGEBRAS A N D q - O S C I L L A T O R S M. C H A I C H I A N Department of High Energy Physics, Universityof Helsinki, Siltavuorenpenger20 C, SF-O0170Helsinki, Finland and P. K U L I S H Steklov Mathematical Institute of the USSR Academy of Sciences, Leningrad Branch, SU-191 011 Leningrad, USSR Received 2l September 1989
Quantum deformation of the simple Lie superalgebras is formulated applying both the method of the Cartan matrix and the Rmatrix approach. Using the quantum analogue of Bose and Fermi oscillators, the realization of quantum slq(nlm) generators is given. The q-oscillators are obtained from the quantum algebra itself by the contraction method. Multimode representations in terms of q-oscillators require nontrivial couplings between the different modes. Possible applications are outlined.
1. Introduction O v e r the last couple o f years a great deal o f attention has been p a i d to q u a n t u m groups both in the physical a n d the m a t h e m a t i c a l literature. Originally, this structure a p p e a r e d in the course o f d e v e l o p m e n t o f the quant u m inverse p r o b l e m m e t h o d [ 1 ] for the q u a n t u m slq(2) case [ 2 ] in studying the properties of the Y a n g - B a x t e r equation, As it turned out, this structure which can be considered as a " d e f o r m a t i o n " o f the Lie algebra with the p a r a m e t e r t / o r q = e x p ( t / ) , while reproducing the Lie algebra in the limit q ~ 0 , has a representation theory analogous and even richer than the one o f the Lie algebra itself, The precise formulation o f this structure as a quasitriangular H o p f algebra and its generalization to all simple Lie algebras has been given [ 3,4] and has been d e v e l o p e d intensively ,1 As yet a direct physical application or i n t e r p r e t a t i o n o f the q u a n t u m group structure is absent. However, for a few models o f interest these q u a n t u m algebras emerge as the underlying symmetry. For instance, the asymmetric Heisenberg spin chain (the X X Z m o d e l ) has the q u a n t u m slq ( 2 ) s y m m e t r y [ 7 ]. In this case the " q u a n t u m " p a r a m e t e r r/is connected to the coupling constant. Further, there exists an i n t i m a t e connection o f this theory [ 6 ] with the n o n c o m m u t a t i v e geometry p r o p o s e d in ref. [ 8 ] and applied, in particular, to the description o f string field and gauge theories [ 9 ]. The natural p r o b l e m o f constructing the gauge theory based on the quant u m groups has also been discussed in the literature [ 10 ]. Q u a n t u m groups have o b t a i n e d the most active applications these days in the rational conformal field theories [ 1 1 ], where the richer structure o f their C l e b s c h G o r d a n coefficients are utilized, and in the theory o f link invariants [ 12 ]. The aim o f this letter is to give the basic definitions and properties o f q u a n t u m Lie superalgebras and their realizations, The simplest q u a n t u m superalgebras o f rank one, OSpq( 1 12) and slq( 111 ) have been considered *~ The customary concept of group (implying the exponentiation property, etc.) is lacking in the theory. This concept is formulated in terms of the space of functions on a group, and the characteristics of the space are drastically changed. In this article, a dual object to the space of functions on a quantum formal group [ 5,6 ], the so-called quantum Lie algebra, will be used. Throughout this letter we use the term "'quantum" not literally. 72
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[ 13,14 ]. We give the general definition of the quantum simple Lie superalgebra using the corresponding Cartan matrix, and also independently using the R-matrix. The correspondence between these two definitions is established. As a showcase we choose the quantum superalgebra Slq(21 | ). Using the quantum analogue of Bose and Fermi oscillators [15-17 ], we give the realization of the slq(nl m ) generators. In addition, we show how these q-oscillators are obtained from the contraction of the corresponding quantum superalgebras. Finally, the nontriviality of the multimode realization in terms of q-oscillators is established.
2. Quantum superalgebra slq(n Im) As a simple case of a superalgebra with rank greater than one, consider the quantum version of sl(21 1 ). At present, there exist several approaches for the definition of quantum Lie algebras: ( 1 ) the method of generators and the defining relations based on the knowledge of Cartan matrices [ 3]; (2) a method based on the solution to the Yan-Baxter equation [ 5,6 ]; (3) a construction of the H o p f algebra quantum double [ 3 ]. In this section we utilize method ( 1 ). In contrast to Lie algebras, there exist a few Cartan matrices for the basic Lie superalgebras [ 18 ]. In particular, for sl (2 11 ) there are two of them: e/~=(ai:)=(01-~)'
' : / 2 = ( -01
10)"
(1)
A Lie superalgebra of rank r can be described by the generators H;, ei, f , corresponding to the simple roots (c~1, a2, ..., a , ) and by the Cartan matrix .~(a0). These generators satisfy the relations
[tt;,Hj]=O,
[e;,fl=c~i:H;,
[H;,e:l=aije~,
[g/,.~l=-a,j.~,
(2)
where [ , ] denotes the graded Lie product (if written in term of ( a n t i ) c o m m u t a t o r , then [e, j ] = e f ( - 1 )'(~)P~!fe, with p ( e ) , p 0 c) = 0, i, the parity of the elements e, J). In addition to (2), the generators should satisfy the Serre relations (ad
ei)l-a°ej
=0,
(3)
and similar relations for f , where the matrix ciij is obtained from a o by replacing the nonvanishing positive elements in the row with a i , = 0 by - 1. A part of the indices ielo c I = ( 1, 2, ..., r) are considered as odd and the rest as even. The roots corresponding to these indices are called odd and even, respectively. Also there is the connection a~ s) = (a,, % ) between the symmetrized Cartan matrix and the roots. The quantum Lie superalgebra corresponding to (2) has the same set of generators, denoted, e.g., by Hi, X,, Y, with slightly changed c o m m u t a t i o n relations including a deformation parameter:
[H;,H/]=O ,
[X~, Y~]=c~osinh(~lH,)/sinh~l,
[H;,A~]=aoXj,
[H;,~]=-a0Y
j.
(4)
I n our case, for the first Caftan matrix .q/l in ( 1 ) we have p ( X~ ) =p ( Y~ ) = 1, p (X2) = p ( 1:2 ) = 0. For convenience we will also use the elements k:
k ~ = e x p ( q H ; ) = q I~';, i = 1 , 2 .
(5)
(One can consider q as a complex or a real number. We shall not go into those details in this note. ) We write down explicitly the c o m m u t a t i o n relations for the quantum slq(2 11 ) corresponding to the first Cartan matrix ,.~. The sl (2 [ 1 ) superalgebra is obtained by taking the "classical" limit q = e"-~ 1:
k;ky=k:k,,
k;Xj=q~':/2X:ki,
k, tS.=q-~;H2yjk,,
[X;,Yjl=a,j(k2,-kj-2)/d,
(6)
where d = q - q - ~. let us mention that for the Lie superalgebras and for the quantum case, one has to use a matrix 2 = diag(dl ..... dr) which symmetrizes the Cartan matrix: ~.~/= ,~Tg. In particular, for .:/2 one has d~ = - d 2 = 1. So the notation q i = e x p ( d : / ) will be used. 73
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It is useful to introduce additional generators )(3, Y3, k3 (in complete analogy with the Lie superalgebra) using the quantum analogue of adjoint operation [ 19] adq (which will be explained later in the context of the H o p f algebra ),
X3=(adqX,)Xz~[XI,Xz]q=X~X2-qXzX~ ,
Y3=[Y2, Y,]q= Y 2 Y , - q - l Y i Y2,
k3=k, k2.
(7)
Taking into account the relations (6), one obtains
kiA3=q~/2X3ki,
kiY3=q;-~/2y3ki,
[X,,Y3]-X,Y~+Y3X~=k~Y2,
X~X3=-qX3X~,
Y~Y3=-qY3YI,X2X3=qX3X2,
[X2,Y3I=-X2Y3-Y3X2=q-~k~Y,,
YaY3=qY3Y2,
[X3, Y3]=(k]-k~Z)/d.
(8)
The commutation relations corresponding to the Cartan matrix ,~/2 of ( 1 ) can be written in the same way. They both describe the same q u a n t u m superalgebra slq(211 ) and can be obtained from each other by an appropriate transformation. The analogy between the representation theory of q u a n t u m (super)algebras with the one of the Lie (super)algebras is connected in the first place with the possibility to define the action of the quantum algebra generators in the tensor product of two representations. In a more abstract mathematical language, this amounts to setting the structure of the H o p f algebra [ 3,20 ] for the universal enveloping algebra J/! of a Lie algebra. The latter includes beside the associative product/1 : Y/Z®i'X/--,#z, also the operations of coproduct A : .~,'--,.y#® ~2/, antipode (or coinverse) S: ~l/--,~//and counit e : J2/-~ ~' maps, as given on the generators by
A(Hi)=H~®I+I®H~, S(H~)=-H~,
d(k,)=k~®k,,
S(k,)=k; -~,
e(H~)=e(X,)=e(Y,)=O,
A(X~)=X~®k~+ki-~®X,,
S(Z)=-q""/2X~,
e(l)=l
d(Y~)=Y~®kg+kT~®Y,,
S(Yi)=-q-~'~/:Y~,
.
(9)
These maps satisfy a certain set of axioms [ 20]. In particular, the coproduct A should be coassociative while its nonsymmetric nature (not a usual situation for physicists) is called noncocommutativity. Further, one of the axioms (id means the identical m a p ) , #o (S® id ) oA(X) = #o ( i d ® S ) od(X) = 1 . e ( X ) ,
( 10 )
explains a more complicated stucture of antipode compared with the Lie (super)algebras, in which simply
S(X) = - X . Using the above maps, one can construct the quantum adjoint operator, adq(x), x¢ 'q/, which gives the representation of # as an associative algebra:
adq(a.b)c=adq(a) (adq(b)c) , ado(' ) = (/IL®#R) ( i d ® S ) d ( " ) ,
( 11 )
where /~L and #R are the multiplication operation on the left and the right, respectively. For instance, if A(a) = ~,~a~®a~e "#® '~#,then a d q ( a ) c = E~a~cS(a~). Using (9), we obtain ad~(X, )X2 = (X~ X! -qX2X~ )k? ~ , and in (7) we have dropped the factor k~. O f course, a change of basis (e.g. X~--,X~k~ ) amounts to a change of the form of adq on the generators. By replacing in the classical Serre relations the operation of ad by adq, we arrive at the complete system of relations which define the quantum Hopfsuperalgebra.
3. R-matrix formalism The method o f the R-matrix in the theory of quantum groups [ 5,6 ] can be considered as a formalism which,
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in particular, allows to express a whole set o f properties in a compact form and to treat the dual objects (cf. footnote 1 ) on the same footing. The generators of the quantum H o p f algebra q[q, corresponding to a simple Lie algebra are arranged in the form of (upper and lower) triangular matrices L ~+-) and the relations between them has the form of the Yang-Baxter equation with an R-matrix corresponding to a given simple Lie algebra,
R (L
TM )®L
(12)
~2) ) = ( I ® L ~,2) ) (L m ) ® I ) R ,
where (e~, e2) = ( + , + ), ( + , - ), ( - , - )- The coproduct 3, counit e and antipode S maps are defined by the relations
A(L(±))=L(±)@L
~+-), e ( L ( ± ) ) = l ,
S(L(+-))L(+-)=I,
(13,14)
enl
where in ( 13 ) one has the matrix multiplication o f L's and the tensor product of their entries (A : %--, % ® % ) and I is the unit matrix. In the case of the Lie superalgebra sl (n I m) ~2 the corresponding R-matrix is as follows: n+m
R = Z Ei,.QEj)+ Z i#j
q P ( i ) E u ® E a + d Z Ea®EJ,,
i= 1
(15)
iCj
where the defining representation o f the dimension n + m is used, p(j) is the parity o f the jth basic vector, (Eo)kt=6~k @ For the showcase of the slq(2] 1 ) superalgebra and the Cartan matrix .~, we have p(1 ) = l, p ( 2 ) = p ( 3 ) = 0 and [ we omit all the vanishing elements in R (q) ] "q - I
d
1 l
d I
q
R(q) =
, R(q)=R-'(1/q), 1
R2=d/~+l,
k=.~R,
d 1 1
where ,~ is the permutation matrix: ,~ ( v®w ) = ( - 1 )"~')P~) ( w O v ) ~3,
L~+~=
I-2
112 , L ~-)=
0
~
X1
12
\x3
x2
,
(16)
13
with p (X~ .3 ) = P (Y~.3) = 1, p ()(2) = p ( Yz ) = 0. To extract from ( 12 ) the relations between the generators l~, ~, Xi and ~, one has to take into account the graded nature o f the tensor product [21 ]. In an appropriate basis we have additional signs only in the L (~')®I product:
~z The case ofn=m requires, as usual, a more careful study. ~3 The property of/~ is essential in the knot and link invariant theory [ 12]. 75
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Ii II
Ii - XI
12
L(-)®I=
X~
(17)
12 X1
-- X3
12 )(2
13
X3
X2 X~
13 X2
6
As a result, we obtain the following set of relations:
[l,, 61=o,
i ~ l i "-1
[iS¢ ~-qCi"Xyli ,
l, Y j = q - ° J Y f l i
[Xt, Y t ] + = d ( 1 2 l ? 1 - l ~ 1 l l )
,
,
[)(2, Y 2 ] = d ( l ; l
l2-13lyt)
,
X2,3 = y21, 3 = O ,
)(3 = - ( X l )(2 - X2 X1 )1~ l / d ,
[)(3, Y3]+ = d ( 1 3 1 f 1 - 1 ; 1 l ~ )
Y3 -= - ( YI 112 - Y2 YI )12/d ,
X t Y2 = q Y2 X I , X 2 YI -~ q Y t )(2 ,
,
(18)
where
CO=
-- 1
,
d=q-q
-~ .
1
From ( 18 ) one sees that the operator/i- i 1213 commutes with all the others and therefore it belongs to the centre and can be identified, for simplicity, with the unit operator. Thus the relations ( 12 ) - ( 1 4 ) define the quantum superalgebra in the R-matrix formalism. Let us establish the correspondence between the generators appearing in the R-matrix (for distinction denoted now as) J~i, Yi, l~ and the ones obtained from the method of the Cartan matrix, X,, Y~, k~. For that one should write down explicitly the product for the generators P~,, )~j. For instance, one has from (13), (16), d(Xl ) = X I ®11 + ]2®2~1 ,
(19)
and to reproduce (9) one should put X 1 =21(1112) -1/2 ,
£J(X 1) =z~(/Y 1)z~( (l 112) -1/2) -- (A~I® l I ...}_/'2 @1~ 1 ) (ll l a ® l l / 2 ) -1/2 ~-~"XI ® k l + k l "1 ®XI -
(20a)
Thus kl = (1,1~ 1 )1/2 Other relations (up to a normalization factor) are Xz=X2(1213) -1/2 ,
Yl=Yl(ll]2)
1/2 ,
Y2=Yz(1213)l/Z,k2=(lzl;l)
1/2
(20b)
To obtain the normalized commutation relations [Xi, Yj], one has to multiply J~, Yj by an additional - 1 / d factor. 76
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Let us point out that the set of R-matrices corresponding to other series of basic Lie superalgebras, osp (nl2m), can be extracted from ref. [21 ].
4. Q-oscillator representation of Slq(fi I m) The q-analogue of the quantum harmonic oscillator has been recently formulated and used for the S l q ( 2 ) algebra [ 13-15 ] and for other quantum Lie algebras of the type Ax, B~v, CN and D~v [ 16 ]. In this section we construct the q-oscillator representations for the slq(nlm) formulated in previous sections. The annihilation and creation operators a, a + for the q-oscillator satisfy the (q-commutation) relations ~4
a a + - q a + a = q -N,
[ N , a + ] = a +,
[N,a]=-a.
(21)
Thus the normalized basic vectors I n) in the q-Fock space have the form
]n)=([n]!)-'/Z(a+)~tO),
al0)=0,
NIn)=nln),
(22)
where [ n ] ! = [1] [2]...[n], [n] = ( q ~ - q - ' ) / ( q - q - ~ ) =
[n] Iq~q-~.
(23)
For the q-fermions b, b + ( b e = b + 2 = 0 ) , one has
bb++qb+b=q M,
[ M , b + ] = b +,
[M,b]=-b.
(24)
Let us consider, explicitly, the slu(211 ) case (the generalization is straightforward). The generators (6) can be represented in terms of three independent q-oscillators b ( , aS, a f :
Xl=a2b + ,
Y~=bla~,
X2=a3a + ,
Ye=a2a~,
k~=0)310)2,
k2=0)210)3,
(25)
where the notations
0)1 =q-M~/2,
0)2 _~qN2/2 ,
0.)3 =qN3/2
(26)
a r e u s e d ~,5
We would like to mention that in the general case of S l q ( n [ m ) , for the q-oscillator realization one must have as many fermionic q-oscillators as there are zero diagonal entries in the Caftan matrix since the generators corresponding to the zero length roots are the odd ones. In the same way as is the case for the usual Lie algebras of rank greater than one, the q-oscillator representation gives only a part of irreducible representations among all the existing ones of the quantum algebra. In order to enlarge this class, one can choose another set of q-oscillators (a,--*b) or, more cardinally, take several independent sets of q-oscillators. In the latter case, the habitual expression Xz = Z)o=~ a2jb ~ does not give the necessa~2¢ relations and requires modifications. We shall obtain these modifications on the example of two sets of qoscillators. Multimodes of q-oscillators. Consider a system of independent bosonic q-oscillators a~, a/~ , c~, c 3 , i= 1, 2, which satisfy the q-commutation relations (21). Let us construct the two generators of s l q ( 2 ) as consisting of a superposition of these modes,
Xl=Aa~a + +Cc~c +,
Y~=Aa2 a+ +Cc2c +,
(27)
~n We can rewrite (21 ) using the operators d = q-U/Za, d + = a + q -~v/2 in the commutator form [d, d + ] = q- 2N. ,5 We note that the R-matrix method operators l, of (20), are related in the q-oscillator representation to ~oi or to the n u m b e r operators Mi, Nj, N2 as Ii =oJ/-: =qM., 12=toF2 =q-N:, 13=O9~-2 =q-m3.
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where A, C, A, C are some weight functions (in general of operators) to be determined later. For the commutation relations of X~, Y~ we obtain [X,, YI]=
d {AA[ (0)~1032)2- (0)1-1032)-2]-1--C(~[ (~'1-1~'2)2- (}'11 ~2)-2]} ,
(28)
where the ?; are the same as the ~o; which enter (21), (26) but corresponding to the bosonic modes c;, c + . On the RHS of (28) we have dropped the two terms proportional to A C and CA which as will turn out a posteriori, vanish themselves each separately [because of (29) or ( 31 ) ]. By inspecting (28), we observe that if we choose A, A, C and C as A~z[m ( 7 i - 1 ) ) 2 ) 2 ~ k~2 ,
CC~-((D[-lo)2)-2=k~
2 ,
(29)
then (28) gives
[X,, Y, ] = ( K ~ - K F 2 ) / d ,
(30)
with K~ = o~i- l ¢o2Y7 ~Y2= k~ k'~. A solution of (29) is A=A=TFt72,
C=C=~I~
-1.
(31)
One sees from (30) that the structure of the q-oscillator realization is retained, provided the coefficients A, A, C and C are functions of number operators according to (29) or (31 ) and thus are coupled to other modes. The above result for the multimode can be generalized to any number of modes N, bosonic or fermionic, according to N
Xl = ~ Aia,ia~-, t=l
N
Y, = Z Afl2; a+ ,
(32)
i=l
with
[X,, Y,]= (K 2 - K T z ) / d , where N
A,= 1-I (~,m0~-2) 1-I ({o-' ImO,)2m) m
m>i
,
K,= [I (o~'(o2;). i~I
The relations (32 ) can easily be obtained by induction using ( 2 7 ) - ( 3 1 ). An interpretation of (32), however, can be given using the language of the R-matrix: If one has two independent sets of the quantum Hopf algebra o?/(slq(2) ) generators X~~) , Y/~) , l~~) , a = 1, 2; j = 1, 2, satisfying (12), one can obtain the o# using the coproduct (13):
A(L(+-))=LI~))LI~ ),
e.g.L/g))= \ ) ~ , , )
0l ~ ) ) .
(33)
In terms of the entries, one gets ~, = ~ , ) / { 2 ) +12{,))~2),
(34)
and so on. Taking into account the relations (20) between the generators, one obtains (27), (31) for the qoscillator realization. For the case of superalgebras and fermionic oscillators, the multimode formalism can be constructed along similar lines by invoking, in addition, the relations (24) and (26). Finally, let us mention that, as in the classical case, the q-oscillator can be obtained from the algebra by the 78
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contraction procedure: The spin s representation Vs of the slq(2) is defined by ( 2 s + 1 ) basic vectors Is, m ) , re=s, s - I, ..., -s, and by the action of the generators H. X± ,
HIs, m ) = m l s , m ) ,
X+ls, m)=([s-T-m][s+_m+l])W21s, m+_l).
(35)
We now take q~E, q > I and the limit s ~ for which qS_,~ and [s] - (qS-q-S)/d-~qS/d. Renormalizing the generators according to X+_--,X±/( [2s] )~/2 and H ~ N = s - H , for the set o f vectors Is, m ) with m satisfying l i m s ~ re~s= 1, we obtain lira X+/( [2s] )1/2=~- (q)
mq-U/2
a - (q)
,
lira
X_/ ( [2s] )~/2=d+ ( q) = a + ( q ) q - - N / 2
,
S~09
[N,a+-]=+a +,
a - a + _ q a + a - = q -N,
NIn)q=rl]n)q,
a+-(q)ln)qm([n+½+_½])~/2]n+_l)q,
(36)
provided the q-states are identified as [O)q~{S,S)
,
Ifl)q=~lS, S--t'l )
,
(37)
thus reproducing (21 ), (22). The q-fermionic oscillator (24) is nothing but the set o f generators of the Slq( 1 [ 1 ) with an additional external automorphism M (ref. [ 14 ], cf. ref. [ 23 ] ). If we apply the contraction procedure to another quantum superalgebra of rank one, namely OSpq( I [ 2 ), we obtain a different q-fermionic oscillator.
5. Concluding remarks The concept o f q u a n t u m group in physics is new and as it turns out its overlap with different fields is growing due to the rich structure o f it. Examples are the peculiarities of the representation theory for special values of q= exp(2ni/N) and its connection to a specific class ofconformal field theory. Although a direct physical meaning of the parameter involved in the q u a n t u m groups is missing, it is not inconceivable that such a structure could emerge as a manifestation o f a dynamical symmetry. Much work has to be done before one can extract all the consequences o f this structure. Due to the active interest in supersymmetry, there appears a natural demand for the quantum deformation of Lie superalgebras. This note is a step towards such a task. Although, here we have constructed the q-oscillator representations o f the sl o (n [ m ), there still remains another class, OSpq ( n 12m), a treatment of which seems not to be straightforward. This appears already for the ospq( 1 [2), for which there is no embedding of Slq(2 ). The question whether the group contraction may help is of interest. The nilpotency of some generators of the Slq(nl m) simplifies the explicit expression for the universal R-matrices. Such super R-matrices satisfy conditions of the link invariant theory, hence they can be applied in this field. Furthermore, quantum superalgebras can arise also in the description of the integrable super-spin models as the algebra of the symmetry. In many SUSY theories the infinite-dimensional (super-)Virasoro algebra is important. There is no q u a n t u m deformation of this algebra up to now [24 ], despite the attempt to include the central extension into the R-matrix formalism [25 ]. The multimode approach for the q-oscillators developed above and the operators defined in footnote 3 seem to present a tool for attacking such problems. These questions are under study.
Acknowledgement We are grateful to L. Faddeev, A. Kirillov, J. Lukierski, A. Macfarlane, N. Reshetikhin, V. Rittenberg and L. 79
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f o r u s e f u l d i s c u s s i o n s . P . K . w o u l d like t o t h a n k t h e R e s e a r c h I n s t i t u t e f o r T h e o r e t i c a l P h y s c i s , U n i -
versity of Helsinki, for the hospitality.
References [ 11 L.D. Faddeev, in: Les Houches Lectures 1982 (North-Holland, Amsterdam, 1984); P.P. Kulish and E.K. Sklyanin, Lecture Notes in Physics, Vol. 151 (Springer, Berlin, 1982) p. 61. [ 2 ] P.P. Kulish and N.Yu. Reshetikhin, Zap. Nauch. Semin. LOM1, 101 ( 1981 ) 101 [in Russian ]; J. Soviet Math. 23 ( 1983 ) 2435; E.K. Sklyanin, Funk. Anal. Appl. 16 ( 1982 ) 27; 17 (1983) 34. [ 3] V.G. Drinfeld, Proc. Intern. Congress of Mathematicians (Berkeley, 1986), Vol. 1, p. 798. [4 ] M. Jimbo, Lett. Math. Phys. 11 ( 1986 ) 247. [ 5 ]L.D. Faddeev, N. Yu. Resbetikhin and L.A. Takhtajan, Quantization of Lie groups and Lie algebras, preprint LOMI E-I 4-87 ( 1987 ) [Algebraic analysis, Vol. l (Academic Press, New York, 1988 ) p. 129 ]. [6] L.D. Faddeev, N. Yu. Reshetikhin and L.A. Takhtajan, Algebra i Analys 1 (1989) 178 [in Russian]. [71 V. Pasquier and H. Saleur, Saclay preprint SPhT/88-187 (1988). [8] A. Connes, Publ. Math. IHES 62 (1985) 257. [9] E. Witten, Nucl. Phys. B 268 (1986) 253; Princeton preprint IASSNS-HEP-89/32 ( 1989 ). [ 10] J.M. Maillet and F. Nijhoff, CERN preprint CERN TH.5449/89 (1989). [ 11 ] L. Alvarez-Gaum6, C. Gomez and G. Sierra, Phys. Lett. B 220 (1989) 142; CERN preprint CERN TH.5369/89 ( 1989 ); G. Moore and N. Yu. Reshetikhin, Princeton preprint IASSNS-HEP-89/18 ( 1989 ). [ 12 ] N.Yu. Reshetikhin, preprints LOMI E-4-87, E-17-87 ( 1988 ); Y. Akutsu and M. Wadati, J. Phys. Soc. Jpn. 56 ( 1987 ) 839; Phys. Rep. 180 (1989) 247. [ 13 ] P.P. Kulish, Kyoto preprint RIMS-615 ( 1988 ); P.P. Kulish and N.Yu. Reshetikhin, Lett. Math. Phys. 18 (1989) 143. [ 14] P.P. Kulish, Proc. Alushta School on Quantum field theory (Springer, Berlin, 1989). [ 15 ] A.J. Macfarlane, J. Phys. A 22 ( 1989 ) 4581. [ 16] L.C. Biedenharn, J. Phys. A 22 (1989) L873. [ 17 ] T. Hayashi, Nagoya University preprint ( 1989 ). [ 18] V. KaY, Lecture Notes in Mathematics, Vol. 676 (Springer, Berlin, 1978) p. 597; L. Frappat, A. Sciarrino and P. Sorba, Commun. Math. Phys. 121 ( 1989 ) 457. [ 19] M. Rosso, Commun. Math. Phys. 124 (1989) 307; N. Burroughs, Cambridge preprint D A M T P / R - 8 9 / 4 (1989). [20] E. Abe, Hopf algebras (Cambridge U.P., Cambridge, 1980). [21 [ D. Leites, Soy. Math. Usp. 35 (1980) 3. [22 ] V. Bazhanov and A. Shadrikov, Teor. Mat. Fiz. 73 ( 1987 ) 402. [23] M. de Crombrugghe and V. Rittenberg, Ann. Phys. (NY) 151 (1983) 99. [ 24 ] D. Bernard and A. Le Clair, princeton preprint PUPT-1123 ( 1989 ). [ 251 N. Reshetikhin and M. Semenov-Tian-Shansky, Berkeley preprint MSRI03808-89 ( 1989 ).
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Q U A N T U M LIE SUPERALGEBRAS A N D q - O S C I L L A T O R S M. C H A I C H I A N Department of High Energy Physics, Universityof Helsinki, Siltavuorenpenger20 C, SF-O0170Helsinki, Finland and P. K U L I S H Steklov Mathematical Institute of the USSR Academy of Sciences, Leningrad Branch, SU-191 011 Leningrad, USSR Received 2l September 1989
Quantum deformation of the simple Lie superalgebras is formulated applying both the method of the Cartan matrix and the Rmatrix approach. Using the quantum analogue of Bose and Fermi oscillators, the realization of quantum slq(nlm) generators is given. The q-oscillators are obtained from the quantum algebra itself by the contraction method. Multimode representations in terms of q-oscillators require nontrivial couplings between the different modes. Possible applications are outlined.
1. Introduction O v e r the last couple o f years a great deal o f attention has been p a i d to q u a n t u m groups both in the physical a n d the m a t h e m a t i c a l literature. Originally, this structure a p p e a r e d in the course o f d e v e l o p m e n t o f the quant u m inverse p r o b l e m m e t h o d [ 1 ] for the q u a n t u m slq(2) case [ 2 ] in studying the properties of the Y a n g - B a x t e r equation, As it turned out, this structure which can be considered as a " d e f o r m a t i o n " o f the Lie algebra with the p a r a m e t e r t / o r q = e x p ( t / ) , while reproducing the Lie algebra in the limit q ~ 0 , has a representation theory analogous and even richer than the one o f the Lie algebra itself, The precise formulation o f this structure as a quasitriangular H o p f algebra and its generalization to all simple Lie algebras has been given [ 3,4] and has been d e v e l o p e d intensively ,1 As yet a direct physical application or i n t e r p r e t a t i o n o f the q u a n t u m group structure is absent. However, for a few models o f interest these q u a n t u m algebras emerge as the underlying symmetry. For instance, the asymmetric Heisenberg spin chain (the X X Z m o d e l ) has the q u a n t u m slq ( 2 ) s y m m e t r y [ 7 ]. In this case the " q u a n t u m " p a r a m e t e r r/is connected to the coupling constant. Further, there exists an i n t i m a t e connection o f this theory [ 6 ] with the n o n c o m m u t a t i v e geometry p r o p o s e d in ref. [ 8 ] and applied, in particular, to the description o f string field and gauge theories [ 9 ]. The natural p r o b l e m o f constructing the gauge theory based on the quant u m groups has also been discussed in the literature [ 10 ]. Q u a n t u m groups have o b t a i n e d the most active applications these days in the rational conformal field theories [ 1 1 ], where the richer structure o f their C l e b s c h G o r d a n coefficients are utilized, and in the theory o f link invariants [ 12 ]. The aim o f this letter is to give the basic definitions and properties o f q u a n t u m Lie superalgebras and their realizations, The simplest q u a n t u m superalgebras o f rank one, OSpq( 1 12) and slq( 111 ) have been considered *~ The customary concept of group (implying the exponentiation property, etc.) is lacking in the theory. This concept is formulated in terms of the space of functions on a group, and the characteristics of the space are drastically changed. In this article, a dual object to the space of functions on a quantum formal group [ 5,6 ], the so-called quantum Lie algebra, will be used. Throughout this letter we use the term "'quantum" not literally. 72
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[ 13,14 ]. We give the general definition of the quantum simple Lie superalgebra using the corresponding Cartan matrix, and also independently using the R-matrix. The correspondence between these two definitions is established. As a showcase we choose the quantum superalgebra Slq(21 | ). Using the quantum analogue of Bose and Fermi oscillators [15-17 ], we give the realization of the slq(nl m ) generators. In addition, we show how these q-oscillators are obtained from the contraction of the corresponding quantum superalgebras. Finally, the nontriviality of the multimode realization in terms of q-oscillators is established.
2. Quantum superalgebra slq(n Im) As a simple case of a superalgebra with rank greater than one, consider the quantum version of sl(21 1 ). At present, there exist several approaches for the definition of quantum Lie algebras: ( 1 ) the method of generators and the defining relations based on the knowledge of Cartan matrices [ 3]; (2) a method based on the solution to the Yan-Baxter equation [ 5,6 ]; (3) a construction of the H o p f algebra quantum double [ 3 ]. In this section we utilize method ( 1 ). In contrast to Lie algebras, there exist a few Cartan matrices for the basic Lie superalgebras [ 18 ]. In particular, for sl (2 11 ) there are two of them: e/~=(ai:)=(01-~)'
' : / 2 = ( -01
10)"
(1)
A Lie superalgebra of rank r can be described by the generators H;, ei, f , corresponding to the simple roots (c~1, a2, ..., a , ) and by the Cartan matrix .~(a0). These generators satisfy the relations
[tt;,Hj]=O,
[e;,fl=c~i:H;,
[H;,e:l=aije~,
[g/,.~l=-a,j.~,
(2)
where [ , ] denotes the graded Lie product (if written in term of ( a n t i ) c o m m u t a t o r , then [e, j ] = e f ( - 1 )'(~)P~!fe, with p ( e ) , p 0 c) = 0, i, the parity of the elements e, J). In addition to (2), the generators should satisfy the Serre relations (ad
ei)l-a°ej
=0,
(3)
and similar relations for f , where the matrix ciij is obtained from a o by replacing the nonvanishing positive elements in the row with a i , = 0 by - 1. A part of the indices ielo c I = ( 1, 2, ..., r) are considered as odd and the rest as even. The roots corresponding to these indices are called odd and even, respectively. Also there is the connection a~ s) = (a,, % ) between the symmetrized Cartan matrix and the roots. The quantum Lie superalgebra corresponding to (2) has the same set of generators, denoted, e.g., by Hi, X,, Y, with slightly changed c o m m u t a t i o n relations including a deformation parameter:
[H;,H/]=O ,
[X~, Y~]=c~osinh(~lH,)/sinh~l,
[H;,A~]=aoXj,
[H;,~]=-a0Y
j.
(4)
I n our case, for the first Caftan matrix .q/l in ( 1 ) we have p ( X~ ) =p ( Y~ ) = 1, p (X2) = p ( 1:2 ) = 0. For convenience we will also use the elements k:
k ~ = e x p ( q H ; ) = q I~';, i = 1 , 2 .
(5)
(One can consider q as a complex or a real number. We shall not go into those details in this note. ) We write down explicitly the c o m m u t a t i o n relations for the quantum slq(2 11 ) corresponding to the first Cartan matrix ,.~. The sl (2 [ 1 ) superalgebra is obtained by taking the "classical" limit q = e"-~ 1:
k;ky=k:k,,
k;Xj=q~':/2X:ki,
k, tS.=q-~;H2yjk,,
[X;,Yjl=a,j(k2,-kj-2)/d,
(6)
where d = q - q - ~. let us mention that for the Lie superalgebras and for the quantum case, one has to use a matrix 2 = diag(dl ..... dr) which symmetrizes the Cartan matrix: ~.~/= ,~Tg. In particular, for .:/2 one has d~ = - d 2 = 1. So the notation q i = e x p ( d : / ) will be used. 73
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It is useful to introduce additional generators )(3, Y3, k3 (in complete analogy with the Lie superalgebra) using the quantum analogue of adjoint operation [ 19] adq (which will be explained later in the context of the H o p f algebra ),
X3=(adqX,)Xz~[XI,Xz]q=X~X2-qXzX~ ,
Y3=[Y2, Y,]q= Y 2 Y , - q - l Y i Y2,
k3=k, k2.
(7)
Taking into account the relations (6), one obtains
kiA3=q~/2X3ki,
kiY3=q;-~/2y3ki,
[X,,Y3]-X,Y~+Y3X~=k~Y2,
X~X3=-qX3X~,
Y~Y3=-qY3YI,X2X3=qX3X2,
[X2,Y3I=-X2Y3-Y3X2=q-~k~Y,,
YaY3=qY3Y2,
[X3, Y3]=(k]-k~Z)/d.
(8)
The commutation relations corresponding to the Cartan matrix ,~/2 of ( 1 ) can be written in the same way. They both describe the same q u a n t u m superalgebra slq(211 ) and can be obtained from each other by an appropriate transformation. The analogy between the representation theory of q u a n t u m (super)algebras with the one of the Lie (super)algebras is connected in the first place with the possibility to define the action of the quantum algebra generators in the tensor product of two representations. In a more abstract mathematical language, this amounts to setting the structure of the H o p f algebra [ 3,20 ] for the universal enveloping algebra J/! of a Lie algebra. The latter includes beside the associative product/1 : Y/Z®i'X/--,#z, also the operations of coproduct A : .~,'--,.y#® ~2/, antipode (or coinverse) S: ~l/--,~//and counit e : J2/-~ ~' maps, as given on the generators by
A(Hi)=H~®I+I®H~, S(H~)=-H~,
d(k,)=k~®k,,
S(k,)=k; -~,
e(H~)=e(X,)=e(Y,)=O,
A(X~)=X~®k~+ki-~®X,,
S(Z)=-q""/2X~,
e(l)=l
d(Y~)=Y~®kg+kT~®Y,,
S(Yi)=-q-~'~/:Y~,
.
(9)
These maps satisfy a certain set of axioms [ 20]. In particular, the coproduct A should be coassociative while its nonsymmetric nature (not a usual situation for physicists) is called noncocommutativity. Further, one of the axioms (id means the identical m a p ) , #o (S® id ) oA(X) = #o ( i d ® S ) od(X) = 1 . e ( X ) ,
( 10 )
explains a more complicated stucture of antipode compared with the Lie (super)algebras, in which simply
S(X) = - X . Using the above maps, one can construct the quantum adjoint operator, adq(x), x¢ 'q/, which gives the representation of # as an associative algebra:
adq(a.b)c=adq(a) (adq(b)c) , ado(' ) = (/IL®#R) ( i d ® S ) d ( " ) ,
( 11 )
where /~L and #R are the multiplication operation on the left and the right, respectively. For instance, if A(a) = ~,~a~®a~e "#® '~#,then a d q ( a ) c = E~a~cS(a~). Using (9), we obtain ad~(X, )X2 = (X~ X! -qX2X~ )k? ~ , and in (7) we have dropped the factor k~. O f course, a change of basis (e.g. X~--,X~k~ ) amounts to a change of the form of adq on the generators. By replacing in the classical Serre relations the operation of ad by adq, we arrive at the complete system of relations which define the quantum Hopfsuperalgebra.
3. R-matrix formalism The method o f the R-matrix in the theory of quantum groups [ 5,6 ] can be considered as a formalism which,
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in particular, allows to express a whole set o f properties in a compact form and to treat the dual objects (cf. footnote 1 ) on the same footing. The generators of the quantum H o p f algebra q[q, corresponding to a simple Lie algebra are arranged in the form of (upper and lower) triangular matrices L ~+-) and the relations between them has the form of the Yang-Baxter equation with an R-matrix corresponding to a given simple Lie algebra,
R (L
TM )®L
(12)
~2) ) = ( I ® L ~,2) ) (L m ) ® I ) R ,
where (e~, e2) = ( + , + ), ( + , - ), ( - , - )- The coproduct 3, counit e and antipode S maps are defined by the relations
A(L(±))=L(±)@L
~+-), e ( L ( ± ) ) = l ,
S(L(+-))L(+-)=I,
(13,14)
enl
where in ( 13 ) one has the matrix multiplication o f L's and the tensor product of their entries (A : %--, % ® % ) and I is the unit matrix. In the case of the Lie superalgebra sl (n I m) ~2 the corresponding R-matrix is as follows: n+m
R = Z Ei,.QEj)+ Z i#j
q P ( i ) E u ® E a + d Z Ea®EJ,,
i= 1
(15)
iCj
where the defining representation o f the dimension n + m is used, p(j) is the parity o f the jth basic vector, (Eo)kt=6~k @ For the showcase of the slq(2] 1 ) superalgebra and the Cartan matrix .~, we have p(1 ) = l, p ( 2 ) = p ( 3 ) = 0 and [ we omit all the vanishing elements in R (q) ] "q - I
d
1 l
d I
q
R(q) =
, R(q)=R-'(1/q), 1
R2=d/~+l,
k=.~R,
d 1 1
where ,~ is the permutation matrix: ,~ ( v®w ) = ( - 1 )"~')P~) ( w O v ) ~3,
L~+~=
I-2
112 , L ~-)=
0
~
X1
12
\x3
x2
,
(16)
13
with p (X~ .3 ) = P (Y~.3) = 1, p ()(2) = p ( Yz ) = 0. To extract from ( 12 ) the relations between the generators l~, ~, Xi and ~, one has to take into account the graded nature o f the tensor product [21 ]. In an appropriate basis we have additional signs only in the L (~')®I product:
~z The case ofn=m requires, as usual, a more careful study. ~3 The property of/~ is essential in the knot and link invariant theory [ 12]. 75
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Ii II
Ii - XI
12
L(-)®I=
X~
(17)
12 X1
-- X3
12 )(2
13
X3
X2 X~
13 X2
6
As a result, we obtain the following set of relations:
[l,, 61=o,
i ~ l i "-1
[iS¢ ~-qCi"Xyli ,
l, Y j = q - ° J Y f l i
[Xt, Y t ] + = d ( 1 2 l ? 1 - l ~ 1 l l )
,
,
[)(2, Y 2 ] = d ( l ; l
l2-13lyt)
,
X2,3 = y21, 3 = O ,
)(3 = - ( X l )(2 - X2 X1 )1~ l / d ,
[)(3, Y3]+ = d ( 1 3 1 f 1 - 1 ; 1 l ~ )
Y3 -= - ( YI 112 - Y2 YI )12/d ,
X t Y2 = q Y2 X I , X 2 YI -~ q Y t )(2 ,
,
(18)
where
CO=
-- 1
,
d=q-q
-~ .
1
From ( 18 ) one sees that the operator/i- i 1213 commutes with all the others and therefore it belongs to the centre and can be identified, for simplicity, with the unit operator. Thus the relations ( 12 ) - ( 1 4 ) define the quantum superalgebra in the R-matrix formalism. Let us establish the correspondence between the generators appearing in the R-matrix (for distinction denoted now as) J~i, Yi, l~ and the ones obtained from the method of the Cartan matrix, X,, Y~, k~. For that one should write down explicitly the product for the generators P~,, )~j. For instance, one has from (13), (16), d(Xl ) = X I ®11 + ]2®2~1 ,
(19)
and to reproduce (9) one should put X 1 =21(1112) -1/2 ,
£J(X 1) =z~(/Y 1)z~( (l 112) -1/2) -- (A~I® l I ...}_/'2 @1~ 1 ) (ll l a ® l l / 2 ) -1/2 ~-~"XI ® k l + k l "1 ®XI -
(20a)
Thus kl = (1,1~ 1 )1/2 Other relations (up to a normalization factor) are Xz=X2(1213) -1/2 ,
Yl=Yl(ll]2)
1/2 ,
Y2=Yz(1213)l/Z,k2=(lzl;l)
1/2
(20b)
To obtain the normalized commutation relations [Xi, Yj], one has to multiply J~, Yj by an additional - 1 / d factor. 76
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Let us point out that the set of R-matrices corresponding to other series of basic Lie superalgebras, osp (nl2m), can be extracted from ref. [21 ].
4. Q-oscillator representation of Slq(fi I m) The q-analogue of the quantum harmonic oscillator has been recently formulated and used for the S l q ( 2 ) algebra [ 13-15 ] and for other quantum Lie algebras of the type Ax, B~v, CN and D~v [ 16 ]. In this section we construct the q-oscillator representations for the slq(nlm) formulated in previous sections. The annihilation and creation operators a, a + for the q-oscillator satisfy the (q-commutation) relations ~4
a a + - q a + a = q -N,
[ N , a + ] = a +,
[N,a]=-a.
(21)
Thus the normalized basic vectors I n) in the q-Fock space have the form
]n)=([n]!)-'/Z(a+)~tO),
al0)=0,
NIn)=nln),
(22)
where [ n ] ! = [1] [2]...[n], [n] = ( q ~ - q - ' ) / ( q - q - ~ ) =
[n] Iq~q-~.
(23)
For the q-fermions b, b + ( b e = b + 2 = 0 ) , one has
bb++qb+b=q M,
[ M , b + ] = b +,
[M,b]=-b.
(24)
Let us consider, explicitly, the slu(211 ) case (the generalization is straightforward). The generators (6) can be represented in terms of three independent q-oscillators b ( , aS, a f :
Xl=a2b + ,
Y~=bla~,
X2=a3a + ,
Ye=a2a~,
k~=0)310)2,
k2=0)210)3,
(25)
where the notations
0)1 =q-M~/2,
0)2 _~qN2/2 ,
0.)3 =qN3/2
(26)
a r e u s e d ~,5
We would like to mention that in the general case of S l q ( n [ m ) , for the q-oscillator realization one must have as many fermionic q-oscillators as there are zero diagonal entries in the Caftan matrix since the generators corresponding to the zero length roots are the odd ones. In the same way as is the case for the usual Lie algebras of rank greater than one, the q-oscillator representation gives only a part of irreducible representations among all the existing ones of the quantum algebra. In order to enlarge this class, one can choose another set of q-oscillators (a,--*b) or, more cardinally, take several independent sets of q-oscillators. In the latter case, the habitual expression Xz = Z)o=~ a2jb ~ does not give the necessa~2¢ relations and requires modifications. We shall obtain these modifications on the example of two sets of qoscillators. Multimodes of q-oscillators. Consider a system of independent bosonic q-oscillators a~, a/~ , c~, c 3 , i= 1, 2, which satisfy the q-commutation relations (21). Let us construct the two generators of s l q ( 2 ) as consisting of a superposition of these modes,
Xl=Aa~a + +Cc~c +,
Y~=Aa2 a+ +Cc2c +,
(27)
~n We can rewrite (21 ) using the operators d = q-U/Za, d + = a + q -~v/2 in the commutator form [d, d + ] = q- 2N. ,5 We note that the R-matrix method operators l, of (20), are related in the q-oscillator representation to ~oi or to the n u m b e r operators Mi, Nj, N2 as Ii =oJ/-: =qM., 12=toF2 =q-N:, 13=O9~-2 =q-m3.
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where A, C, A, C are some weight functions (in general of operators) to be determined later. For the commutation relations of X~, Y~ we obtain [X,, YI]=
d {AA[ (0)~1032)2- (0)1-1032)-2]-1--C(~[ (~'1-1~'2)2- (}'11 ~2)-2]} ,
(28)
where the ?; are the same as the ~o; which enter (21), (26) but corresponding to the bosonic modes c;, c + . On the RHS of (28) we have dropped the two terms proportional to A C and CA which as will turn out a posteriori, vanish themselves each separately [because of (29) or ( 31 ) ]. By inspecting (28), we observe that if we choose A, A, C and C as A~z[m ( 7 i - 1 ) ) 2 ) 2 ~ k~2 ,
CC~-((D[-lo)2)-2=k~
2 ,
(29)
then (28) gives
[X,, Y, ] = ( K ~ - K F 2 ) / d ,
(30)
with K~ = o~i- l ¢o2Y7 ~Y2= k~ k'~. A solution of (29) is A=A=TFt72,
C=C=~I~
-1.
(31)
One sees from (30) that the structure of the q-oscillator realization is retained, provided the coefficients A, A, C and C are functions of number operators according to (29) or (31 ) and thus are coupled to other modes. The above result for the multimode can be generalized to any number of modes N, bosonic or fermionic, according to N
Xl = ~ Aia,ia~-, t=l
N
Y, = Z Afl2; a+ ,
(32)
i=l
with
[X,, Y,]= (K 2 - K T z ) / d , where N
A,= 1-I (~,m0~-2) 1-I ({o-' ImO,)2m) m
m>i
,
K,= [I (o~'(o2;). i~I
The relations (32 ) can easily be obtained by induction using ( 2 7 ) - ( 3 1 ). An interpretation of (32), however, can be given using the language of the R-matrix: If one has two independent sets of the quantum Hopf algebra o?/(slq(2) ) generators X~~) , Y/~) , l~~) , a = 1, 2; j = 1, 2, satisfying (12), one can obtain the o# using the coproduct (13):
A(L(+-))=LI~))LI~ ),
e.g.L/g))= \ ) ~ , , )
0l ~ ) ) .
(33)
In terms of the entries, one gets ~, = ~ , ) / { 2 ) +12{,))~2),
(34)
and so on. Taking into account the relations (20) between the generators, one obtains (27), (31) for the qoscillator realization. For the case of superalgebras and fermionic oscillators, the multimode formalism can be constructed along similar lines by invoking, in addition, the relations (24) and (26). Finally, let us mention that, as in the classical case, the q-oscillator can be obtained from the algebra by the 78
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contraction procedure: The spin s representation Vs of the slq(2) is defined by ( 2 s + 1 ) basic vectors Is, m ) , re=s, s - I, ..., -s, and by the action of the generators H. X± ,
HIs, m ) = m l s , m ) ,
X+ls, m)=([s-T-m][s+_m+l])W21s, m+_l).
(35)
We now take q~E, q > I and the limit s ~ for which qS_,~ and [s] - (qS-q-S)/d-~qS/d. Renormalizing the generators according to X+_--,X±/( [2s] )~/2 and H ~ N = s - H , for the set o f vectors Is, m ) with m satisfying l i m s ~ re~s= 1, we obtain lira X+/( [2s] )1/2=~- (q)
mq-U/2
a - (q)
,
lira
X_/ ( [2s] )~/2=d+ ( q) = a + ( q ) q - - N / 2
,
S~09
[N,a+-]=+a +,
a - a + _ q a + a - = q -N,
NIn)q=rl]n)q,
a+-(q)ln)qm([n+½+_½])~/2]n+_l)q,
(36)
provided the q-states are identified as [O)q~{S,S)
,
Ifl)q=~lS, S--t'l )
,
(37)
thus reproducing (21 ), (22). The q-fermionic oscillator (24) is nothing but the set o f generators of the Slq( 1 [ 1 ) with an additional external automorphism M (ref. [ 14 ], cf. ref. [ 23 ] ). If we apply the contraction procedure to another quantum superalgebra of rank one, namely OSpq( I [ 2 ), we obtain a different q-fermionic oscillator.
5. Concluding remarks The concept o f q u a n t u m group in physics is new and as it turns out its overlap with different fields is growing due to the rich structure o f it. Examples are the peculiarities of the representation theory for special values of q= exp(2ni/N) and its connection to a specific class ofconformal field theory. Although a direct physical meaning of the parameter involved in the q u a n t u m groups is missing, it is not inconceivable that such a structure could emerge as a manifestation o f a dynamical symmetry. Much work has to be done before one can extract all the consequences o f this structure. Due to the active interest in supersymmetry, there appears a natural demand for the quantum deformation of Lie superalgebras. This note is a step towards such a task. Although, here we have constructed the q-oscillator representations o f the sl o (n [ m ), there still remains another class, OSpq ( n 12m), a treatment of which seems not to be straightforward. This appears already for the ospq( 1 [2), for which there is no embedding of Slq(2 ). The question whether the group contraction may help is of interest. The nilpotency of some generators of the Slq(nl m) simplifies the explicit expression for the universal R-matrices. Such super R-matrices satisfy conditions of the link invariant theory, hence they can be applied in this field. Furthermore, quantum superalgebras can arise also in the description of the integrable super-spin models as the algebra of the symmetry. In many SUSY theories the infinite-dimensional (super-)Virasoro algebra is important. There is no q u a n t u m deformation of this algebra up to now [24 ], despite the attempt to include the central extension into the R-matrix formalism [25 ]. The multimode approach for the q-oscillators developed above and the operators defined in footnote 3 seem to present a tool for attacking such problems. These questions are under study.
Acknowledgement We are grateful to L. Faddeev, A. Kirillov, J. Lukierski, A. Macfarlane, N. Reshetikhin, V. Rittenberg and L. 79
Volume 234, number 1,2 Takhtajan
PHYSICS LETTERS B
4 January 1990
f o r u s e f u l d i s c u s s i o n s . P . K . w o u l d like t o t h a n k t h e R e s e a r c h I n s t i t u t e f o r T h e o r e t i c a l P h y s c i s , U n i -
versity of Helsinki, for the hospitality.
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