1 Introduction Superalgebras, wic are te matematical framework for describing symmetry betwen bosons and fermions ave by now found several interesting

Transcription

1 ANYONIC REALIZATION OF THE QUANTUM AFFINE LIE SUPERALGEBRAS U q ( c A(M? 1; N? 1)) L. Frappat a 1, A. Sciarrino b, S. Sciuto c, P. Sorba d a Centre de Recerces Matematiques Universite de Montreal, Canada b Dipartimento di Scienze Fisice, Universita di Napoli \Federico II" and I.N.F.N., Sezione di Napoli, Italy c Dipartimento di Fisica Teorica, Universita di Torino and I.N.F.N., Sezione di Torino, Italy d Laboratoire de Pysique Teorique ENSLAPP Annecy-le-Vieux et Lyon, France Abstract We give a realization of te quantum ane Lie superalgebras U q ( b A(M? 1; N? 1)) in terms of anyons dened on a one or two-dimensional lattice, te deformation parameter q being related to te statistical parameter of te anyons by q = e i. Te construction uses anyons contructed from usual fermionic oscillators and deformed bosonic oscillators. As a byproduct, realization deformed in any sector of te quantum superalgebras U q (A(M? 1; N? 1)) is obtained. 1 On leave of absence from Laboratoire de Pysique Teorique ENSLAPP. CRM-2383 DSF-T-35/96 DFTT-39/96 ENSLAPP-AL-603/96 q-alg/ September 1996

2 1 Introduction Superalgebras, wic are te matematical framework for describing symmetry betwen bosons and fermions ave by now found several interesting applications in pysics, even if te fundamental supersymmetry between elementary constituents of te matter as not yet been supported by experimental evidence. A furter enlarged concept of symmetry represented by quantum algebras as sown up in a large number of areas in pysics.te fusion of tese new enlarged symmetry structures is a natural step and it leads to te so-called q-superalgebras. Moreover, te connection between quantum algebras and generalized statistics as been pointed out in several contexts. Anyons are typical objects of generalized statistics wose importance in two-dimensional pysics is relevant. Tey ave been used to construct Scwinger- Jordan like realization of te deformed classical nite Lie algebras [7, 6] and of te deformed ane Lie algebras of te unitary and symplectic series [3, 4]. So it is natural to ask wic kind of oscillators are necessary to build realizations of q-superalgebras. We will sow tat te deformation of te ane unitary superalgebras (and terefore also of te nite unitary superalgebras) can be realized by means of anyons and of a new type of generalized statistics objects wic satisfy braiding relations and wic will be called bosonic anyons for reasons wic will be clear from teir denitions, see Sec. 4. Let us empasize tat te construction we propose may be interesting in te study of systems of correlated electrons. In fact te so-called t? J model [14], wic as been suggested as an appropriate starting point for te teory of te ig-temperature superconductivity, is supersymmetric for particular values of te coupling constants and of te cemical potential, te Hamiltonian commuting wit a su(1j2). Moreover in ref. [8] it as been sown tat te t? J model at te supersymmetric point can be written in terms of anyons, wic gives a new realization of supersymmetry. Altoug in tis model no deformation appears, it is conceivable tat anyons can be used to describe furter deformed generalizations of tis or of similar models, like te Hubbard model [10]. Te article is organized in te following way: in Sec. 2 we briey recall te structure of ba(m? 1; N? 1) in te Cartan-Weyl and in te Serre-Cevalley bases and ten we write its deformation in te distinguised Serre-Cevalley basis; in Sec. 3 te fermionic-bosonic oscillators realization of b A(M? 1; N? 1) is presented and nally in Sec. 4 te anyonic realization of U q ( b A(M? 1; N? 1)) is given in terms of one-dimensional anyons and bosonic anyons; in Sec. 5 te generalization of te construction to two-dimensional anyons is discussed and a few conclusions are presented. 1

3 2 Presentation of te superalgebra c A(M? 1; N? 1) We will recall in tis section te presentation of te ane Lie superalgebra b A(M? 1; N? 1), were M; N 1, bot in te Cartan{Weyl basis and in te Serre{Cevalley basis. We set R = M + N? 1 and we exclude te case R = 1 (obtained wen M = N = 1). 2.1 Cartan{Weyl presentation of c A(M? 1; N? 1) In te Cartan{Weyl basis, te generators of te ane Lie superalgebra A(M b? 1; N? 1) are denoted by m a (Cartan generators) and e m a (root generators) were a = 1; : : : ; R and m 2 Z. Te even root system of A(M b? 1; N? 1) is given by 0 = f(" i? " j ); ( k? l )g and te odd root system by 1 = f(" i? k )g were 1 i < j M and 1 k < l N, te " i and k spanning te dual of te Cartan subalgebra of gl(mjn). To eac root generator e m a one assigns a Z 2 -grading dened by deg(e m a ) = 0 if a 2 0 and deg(e m a ) = 1 if a 2 1. Te generators satisfy for M 6= N te following commutation relations: i m a ; n b = m m+n;0 K( a ; b ) (2.1a) m a ; e n a i = aa e m+n a 8 i > < e m a ; e n b = >: "(a; b) e m+n a+b m a ; i = e m a ; i = 0 if a + b is a root a a a m+n + m m+n;0 K(e a ; e?a ) if b =?a 0 oterwise (2.1b) (2.1c) (2.1d) were "(a; b) = 1 is te usual 2-cocycle, K is te (non-degenerate) Killing form on te orizontal superalgebra A(M? 1; N? 1) and is te central carge. [ ; ] denotes te super-commutator: [e m a ; e n b ] = e m a e n b? (?1) deg(em a ): deg(en b ) e n b e m a. Note tat by virtue of Eqs. (2.1a-d) te value of te central carge of ^A M?1 is opposite to tat of ^A N?1. In te case M = N, altoug te Killing form is zero, it is possible to dene a non-degenerate bilinear form K on A(N? 1; N? 1) suc tat Eqs. (2.1a-d) still old. 2.2 Serre{Cevalley presentation of c A(M? 1; N? 1) In te Serre{Cevalley basis, te algebra is described in terms of simple root and Cartan generators, te only data being te entries of te Cartan matrix (a ) of te algebra. Let us denote te generators in te Serre{Cevalley basis by and e were = 0; 1; : : : ; R. If is a subset of f0; 1; : : : ; Rg, te Z 2 -gradation of te superalgebra is dened by setting deg(e ) = 0 if =2 and deg(e ) = 1 if 2. (super)commutation relations ; i = 0 2 Te superalgebra is described by te (2.2a)

4 i ; e = a e (2.2b) i e + ; e? = e + e?? (?1) deg(e+ ) deg(e? ) e? e + = (2.2c) n o e ; e = 0 if a = 0 (2.2d) and by te following relations: Serre relations for all 6= (ad e ) 1?~a e = 0 (2.3) supplementary relations for suc tat a = 0 i (ad e?1) e ; (ad e +1) e = 0 (2.4) were te matrix ~A = (~a ) is deduced from te Cartan matrix A = (a ) of b A(M? 1; N? 1) by replacing all its positive o-diagonal entries by?1. Here ad denotes te adjoint action: (ad X) Y = XY? (?1) deg X: deg Y Y X. One as to empasize tat for superalgebras, te description given by te Serre relations (2.3) leads in general to a bigger superalgebra tan te superalgebra under consideration. It is necessary to write supplementary relations involving more tan two generators, tat for A(M? 1; N? 1) take te form (2.4), in order to quotient te bigger superalgebra and recover te original one, see [11] for more details. As one can imagine, tese supplementary conditions appear wen one deals wit isotropic fermionic simple roots (tat is a = 0). Note tat tese supplementary relations are unnecessary wen M = 1 or N = 1. In te following, we will only use te Serre{Cevalley description of te ane Lie superalgebra in te distinguised basis, suc tat te number of odd simple roots is te smallest one. In te case of b A(M? 1; N? 1), te distinguised basis is dened by taking = f0; Mg. Te corresponding Dynkin diagram is te following, wit te labels identifying te corresponding simple roots: m?@ H 0 H H H H H H?@ m m m m m 1 M? 1 M M + 1 M + N? 1 3

5 associated to te Cartan matrix ?1?1 2? ? ?1 0?1 2?1?1 0 1?1 2?1 0..? ?1? ?1 2 1 C A (2.5) Te correspondence between te distinguised Serre{Cevalley and te Cartan{Weyl bases is te following (i = 1; : : : ; M? 1 and k = 1; : : : ; N? 1): i = 0 i e + i = e 0 " i?" i+1 e? i = e 0 " i+1?" i M = 0 M e + M = e 0 " M? 1 e? M = e 0 1?" M M +k = 0 M +k e + M +k = e 0 k? k+1 e? M +k = e 0 k+1? k 0 =? + P M i=1 0 i? P N?1 k=1 0 M +k e + 0 = e 1 N?" 1 e? 0 = e?1 " 1? N (2.6) Notice tat in te Serre{Cevalley picture te central carge is uniquely dened by te following equation: 0 =? + MX i=1 i? N X?1 k=1 M +k (2.7) 2.3 Serre{Cevalley presentation of U q ( c A(M? 1; N? 1)) We consider now te universal quantum ane Lie superalgebra U q ( A(M b? 1; N? 1)). Te Serre{Cevalley description in te quantum case is very similar. Te dening relations take te form i ; = 0 (2.8a) ; e e + ; e? i = a e i = e + e?? (?1) deg(e+ ) deg(e? ) e? e + = q 4? q? q? q?1 (2.8b) (2.8c)

6 n o e ; e = 0 if a = 0 (2.8d) were q = q d and te numbers d symmetrize te Cartan matrix a of A(M? 1; N? 1): d a = d a (; 6= 0) and d 0 = 1 (in te distinguised basis, q = q for = 0; : : : ; M and q = q?1 for = M + 1; : : : ; M + N? 1). In terms of te generators E = e q?=2 te usual Serre relations are for all 6= (ad q E ) 1?~a E = 0 (2.9) wile te supplementary relations read now for suc tat a = 0 (te denition of te quantum adjoint action ad q is given below Eq. 2.13) [1, 11, 13] i (adq E?1) E ; (ad q E+1) E = 0 (2.10) or in terms of te generators e e i i i?1; e ; e q ; e +1 = 0 (2.11) q te q-commutator being dened as usual by X; Y i q = XY? qy X. Te universal quantum ane Lie superalgebra U U q ( b A(M? 1; N? 1)) is endowed wit a Hopf algebra structure, wit coproduct : U! U U, counit " : U! C and antipode S : U! U suc tat ( = 0; 1; : : : ; R) ( ) = and (e ) = e q =2 + q?=2 e (2.12a) "( ) = "(e ) = 0 and "(1) = 1 (2.12b) S( ) =? and S(e ) =?q a=2 e (2.12c) Te quantum adjoint action ad q can be explicitly written in terms of te coproduct and te antipode as (ad q X) Y = (?1) deg X (2): deg Y X (1) Y S(X (2) ) (2.13) using te Sweedler notation for te coproduct: (X) = X (1) X (2) (summation is understood). 3 Oscillator realization of te ane Lie superalgebra c A(M? 1; N? 1) Let us recall now te oscillator realization of A(M b? 1; N? 1) in terms of creation and anniilation operators. We consider an innite number of fermionic oscillators c i (r); c y i(r) wit i = 1; : : : ; M and r 2 Z + 1=2 = Z 0, wic satisfy te anticommutation relations n ci (r); c j (s) o = n c y i(r); c y j(s) o = 0 and n ci (r); c y j(s) o = ij rs (3.1) 5

7 and an innite number of bosonic oscillators d k (r); d y k(r) wit k = 1; : : : ; N and r 2 Z 0, wic satisfy te commutation relations dk (r); d l (s) i = d y k(r); d y l (s) i = 0 and dk (r); d y l (s) i = kl rs (3.2) te two sets c i (r); c y i(r) and d k (r); d y k(r) commuting eac oter: ci (r); d k (s) i = c i (r); d y k(s) i = c y i(r); d k (s) i = c y i(r); d y k(s) i = 0 (3.3) Te fermionic and bosonic number operators are dened as usual by n i (r) = c y i(r)c i (r) and n 0 k(r) = d y k(r)d k (r). Tese oscillators are equipped wit a normal ordered product suc tat and Terefore and : c y i(r)c j (s) := : d y k(r)d l (s) := : n i (r) := : n 0 k(r) := ( c y i(r)c j (s) if s > 0?c j (s)c y i(r) if s < 0 ( d y k(r)d l (s) if s > 0 d l (s)d y k(r) if s < 0 ( ni (r) if r > 0 n i (r)? 1 if r < 0 ( n 0 k (r) if r > 0 n 0 k(r) + 1 if r < 0 (3.4) (3.5) (3.6) (3.7) Ten an oscillator realization of te generators of b A(M? 1; N? 1) in te Cartan{Weyl basis wit = 1 is given by m i = X r2z 0 : c y i(r)c i (r + m) :? : c y i+1(r)c i+1 (r + m) : (i = 1; : : : ; M? 1) ; (3.8a) X m M = : c ym(r)c M (r + m) : + : d y 1(r)d 1 (r + m) : ; (3.8b) r2z X 0 m M +k = : d y k(r)d k (r + m) :: d y k+1(r)d k+1 (r + m) : (k = 1; : : : ; N? 1) ; (3.8c) r2z 0 X e m " i?" j = c y i(r)c j (r + m) r2z X 0 e m k? l = d y k(r)d l (r + m) r2z X 0 e m " i? k = c y i(r)d k (r + m) r2z X 0 e m k?" i = d y k(r)c i (r + m) r2z 0 (3.8d) (3.8e) (3.8f) (3.8g) 6

8 A fermionic oscillator realization of te simple generators of b A(M? 1; N? 1) in te distinguised Serre-Cevalley basis is given by ( = 0; 1; : : : ; R) = X r2z 0 (r) and e = X r2z 0 e (r) (3.9) were (i = 1; : : : ; M? 1 and k = 1; : : : ; N? 1) i (r) = n i (r)? n i+1 (r) = : n i (r) :? : n i+1 (r) : (3.10a) M (r) = n M (r) + n 0 1(r) = : n M (r) : + : n 0 1(r) : (3.10b) M +k (r) = n 0 k(r)? n 0 k+1(r) = : n 0 k(r) :? : n 0 k+1(r) : (3.10c) 0 (r) = n 0 N(r) + n 1 (r + 1) = : n 0 N(r) : + : n 1 (r + 1) :? r+1=2;0 (3.10d) e + i (r) = c y i(r)c i+1 (r) ; e? i (r) = c y i+1(r)c i (r) (3.10e) e + M(r) = c y M(r)d 1 (r) ; e? M(r) = d y 1(r)c M (r) (3.10f) e + M +k(r) = d y k(r)d k+1 (r) ; e? M +k(r) = d y k+1(r)d k (r) (3.10g) e + 0 (r) = d y N(r)c 1 (r + 1) ; e? 0 (r) = c y 1(r + 1)d N (r) (3.10) Inserting Eq. (3.10d) into Eq. (3.9) and taking into account tat te sum over r can be splitted into a sum of two convergent series only after normal ordering, one can ceck tat 0 =?1 + X r2z 0 : n 0 N(r) : + X r2z 0 : n 1 (r) : =?1 + M X?1 i=1 0 i + 0 M? N X?1 k=1 0 M +k ; (3.11) tat is te central carge is = 1. Note tat te value of te central carge is related to te denition of te normal ordered product. A dierent denition like (i = 1; : : : ; M and k = 1; : : : ; N) would lead to = 0. : n i (r) := n i (r) and : n 0 k(r) := n 0 k(r) for any r 2 Z 0 (3.12) 4 Anyonic realization of U q ( c A(M? 1; N? 1)) In order to obtain an anyonic realization of U q ( b A(M? 1; N? 1)), we will replace te fermionic and bosonic oscillators by suitable anyons in te expressions of te simple generators of U q ( b A(M? 1; N? 1)) in te distinguised Serre{Cevalley basis. Since we ave to deal wit fermionic and bosonic generators, we ave to introduce two dierent types of anyons. Let us rst dene fermionic anyons on a one-dimensional lattice Z 0 [7, 5]: a i (r) = K i (r)c i (r) and ~a i (r) = ~K i (r)c i (r) (4.1) 7

9 and similar expressions for te conjugated operators a y i(r) and ~a y i(r), were te disorder factors K i (r) and ~K i (r) are expressed as K i (r) = q? 1 2 ~K i (r) = q 1 2 P P t2z 0 "(t?r):n i (t): t2z 0 "(t?r):n i (t): (4.2a) (4.2b) Te function "(t) = jtj=t if t 6= 0 and "(0) = 0 is te sign function. It is easy to prove tat te a-type anyons satisfy te following braiding relations for r > s: and a i (r)a i (s) + q?1 a i (s)a i (r) = 0 a y i(r)a y i(s) + q?1 a y i(s)a y i(r) = 0 a y i(r)a i (s) + q a i (s)a y i(r) = 0 a i (r)a y i(s) + q a y i(s)a i (r) = 0 (4.3) a i (r)a y i(r) + a y i(r)a i (r) = 1 a i (r) 2 = a y i(r) 2 = 0 (4.4) Te braiding relations between te ~a-type anyons are obtained from eqs. (4.3) and (4.4) by replacing q $ q?1. Finally, te braiding relations between a-type and ~a-type anyons are given by n ~ai (r); a i (s) o = n ~a y i(r); a y i(s) o = 0 for all r; s 2 Z 0 (4.5) n ~a y i(r); a i (s) o = n ~a i (r); a y i(s) o = 0 for all r 6= s 2 Z 0 (4.6) and Moreover, te following identity olds: n o P ~ai (r); a y i(r) = q t2z 0 "(t?r):n i (t): n o P ~a y i(r); a i (r) = q? t2z 0 "(t?r):n i (t): (4.7) a y i(r)a i (r) = ~a y i(r)~a i (r) = n i (r) (4.8) te normal ordering between a-type and ~a-type anyons being dened as in eq. (3.4). Now we will dene anyonic-like operators based on q-deformed bosons. Let us recall tat q-deformed bosons can be constructed from ordinary ones by te following procedure [12]: n 0 k(r) = d y k(r)d k (r) b k (r) = d k (r) b y k(r) = vu u vu u t [n0 k(r)] q n 0 k(r) t [n0 k(r)] q n 0 k(r) = d y k(r) = d y k(r) 8 vu u t [n0 k(r) + 1] q (4.9a) n 0 k(r) + 1 d k(r) (4.9b) vu u t [n0 k(r) + 1] q n 0 k(r) + 1 (4.9c)

10 Te q-deformed bosons b k (r); b y k(r) satisfy te following q-commutation relations: b k (r)b y l (s)? q kl rs b y l (s)b k(r) = q?n0 k (r) kl rs b k (r)b y l (s)? q? kl rs b y l (s)b k (r) = q n0 k (r) kl rs b k (r)b l (s)? b l (s)b k (r) = b y k(r)b y l (s)? b y l (s)b y k(r) = 0 n 0 k (r); b l (s) i =?b k (r) kl rs n 0 k (r); b y l (s) i = b y k(r) kl rs (4.10a) (4.10b) (4.10c) (4.10d) (4.10e) from wic it follows tat b y k (r)b k(r) = [n 0 k(r)] q and b k (r)b y k (r) = [n0 k(r) + 1] q (4.11) Now, let us dene anyonic-like operators as follows: A k (r) = K 0 k(r)b k (r) and ~A k (r) = ~K 0 k(r)b k (r) (4.12) and similar expressions for te conjugated operators A y k(r) and ~A y k(r), were te disorder factors are given by Kk(r) 0 = q 1 2 ~K k(r) 0 = q? 1 2 P P t2z 0 "(t?r):n 0 k (t): t2z 0 "(t?r):n 0 k (t): (4.13a) (4.13b) It can be proved tat te operators A k (r); A y k(r) satisfy te following braiding relations for r > s: and A k (r)a k (s)? qa k (s)a k (r) = 0 A y k (r)ay k (s)? qay k (s)ay k (r) = 0 A y i(r)a k (s)? q?1 A k (s)a y i(r) = 0 A k (r)a y k(s)? q?1 A y k(s)a k (r) = 0 (4.14) A k (r)a y k(r)? qa y k(r)a k (r) = q?n0 k (r) A k (r)a y k(r)? q?1 A y k(r)a k (r) = q n0 k (r) (4.15) Terefore, te operators A k (r); A y k(r) satisfy te q-commutation relations of te q-deformed bosonic oscillator at te same point, wile tey satisfy braiding relations wen taken at dierent points. Te braiding relations between te ~A-type anyons are obtained from eq. (4.14) by replacing q $ q?1. Note tat for tese anyonic q-deformed bosons dened by te above relations, we do not ave any pysical interpretations, on te contrary of te a-type anyons, see refs. [7, 5]. Is is 9

11 wort to point out tat te above introduced bosonic anyons dier from te ones introduced in ref. [9] by te non trivial fact tat our anyons are dened on a lattice wile in ref. [9] are dened in te continuum and by te local braiding relation (4.15). Replacing in Eq. (4.12) te q-boson by a standard boson, we nd in te lattice te bosonic anyons of ref. [9]. We will come back on te dierence between te two approaces in te next section. Now we can build an anyonic realization of U q ( b A(M? 1; N? 1)) by "anyonizing" te oscillator realization eqs. (3.10a-3.10), tat is replacing te fermionic oscillators c i by te anyonic oscillators a i and ~a i and te bosonic oscillators b i by te operators A i and ~A i. More precisely, one as: Teorem 1 An anyonic realization of te simple generators of te quantum ane Lie superalgebra U q ( b A(M? 1; N? 1)) wit central carge = 1 is given by (wit = 0; 1; : : : ; R) H = X r2z 0 H (r) and E = X r2z 0 E (r) (4.16) were (i = 1; : : : ; M? 1 and k = 1; : : : ; N? 1) H i (r) = n i (r)? n i+1 (r) =: n i (r) :? : n i+1 (r) : H M (r) = n M (r) + n 0 1(r) =: n M (r) : + : n 0 1(r) : H M +k (r) = n 0 k(r)? n 0 k+1(r) =: n 0 k(r) :? : n 0 k+1(r) : H 0 (r) = n 0 N(r) + n 1 (r + 1) =: n 0 N(r) : + : n 1 (r + 1) :? r+1=2;0 (4.17a) (4.17b) (4.17c) (4.17d) E + i (r) = a y i(r)a i+1 (r) ; E? i (r) = ~a y i+1(r)~a i (r) (4.17e) E + M(r) = a y M(r)A 1 (r) ; E? M(r) = ~A y 1(r)~a M (r) (4.17f) E + M +k(r) = A y k(r)a k+1 (r) ; E? M +k(r) = ~A y k+1(r) ~A k (r) (4.17g) E + 0 (r) = A y N(r)a 1 (r + 1) ; E? 0 (r) = ~a y 1(r + 1) ~A N (r) (4.17) Proof We must ceck tat te realization eqs. (4.16) and (4.17a-4.17) indeed satisfy te quantum ane Lie superalgebra U q ( A(M b? 1; N? 1)) in te distinguised Serre{Cevalley basis (2.8a-2.8d) togeter wit te quantum Serre relations (2.9) and (2.11). Te proof follows te lines of te algebraic case [3]: te equations (2.8a-2.11) wic dene a generic deformed ane superalgebra U q ( b A) reduce to U q (A) wen te ane dot is removed and to anoter nite dimensional superalgebra U q (A 0 ) if te ane dot is kept and one or more oter suitable dots are removed. Te relations dening U q ( b A) coincide wit te union of tose dening U q (A) and U q (A 0 ): terefore, it will be enoug to ceck tat te equations dening U q (A) and U q (A 0 ) are satised. Consider te non-extended Dynkin diagram of A(M? 1; N? 1) to wic te set of generators fh ; E g (wit 6= 0) corresponds. Inserting Eqs. (4.1), (4.2a-b), (4.11), (4.12), te expressions (4.17e-g) become P t2z 0 "(t?r) : (t): E (r) = ^e (r) q 1 2 (4.18) 10

12 were te generators ^e (r) are obtained from te generators e (r) in Eqs. (3.10e-g) replacing te bosonic oscillators d k by te q-deformed bosons b k. Te generators ^e (r) coincide locally, i.e. for xed r, wit te generators of U q (A(M? 1; N? 1)) of refs. [1, 2] as te q-deformed fermions i of [2] are equivalent to te usual fermionic oscillators c i. It follows tat te generators fh ; E g of Eq. (4.16) are a representation of U q (A(M? 1; N? 1)) as tey are obtained wit te correct coproduct, see Eqs. (2.12a) and (4.18), by te representation in terms of f ; ^e g. Let us remark tat due to te equivalence betwen q-fermions and standard fermions te realization of nite q-superalgebras of ref. [2] are realizations of deformed algebras only for te subalgebra realized in terms of q-bosons wile te subalgebra realized in terms of q-fermions is left undeformed. On te contrary te ere presented anyonic realization is completely deformed in any sector. Finally let us remark tat te dierence q! q?1 in te disorder factor of te A-type anyons (in te site r) wit respect to te a-type anyons (in te same site) { see te q -factor in Eq. (4.18) { is essential for te consistency of te q-superalgebra structure. We consider ten te extended Dynkin diagram of A(M? 1; N? 1) and we delete a dot wic is not te ane dot. For example, cutting te dot number 2, we obtain te following Dynkin diagram: 1 0 M+N-1 M+1 M M-1 3 m m m m m m m?@ wic corresponds to te Lie superalgebra A(M? 1; N? 1) in a non-distinguised basis. For a xed r 2 Z 0, it is possible to sow tat te set f j (r); ^e j (r); 1 (r + 1); ^e 1 (r + 1)g (j = 0; 3; : : : ; M +N?1) is a representation of U q (A(M? 1; N? 1)) in te non-distinguised basis specied by te above Dynkin diagram. We empasize tat in tis case we ave to satisfy two more supplementary Serre relations tan in te distinguised basis. Of course for particular values of M and N one or bot relations can be absent. Note tat deleting te M-t dot, we reobtain te superalgebra U q (A(M? 1; N? 1)) in te distinguised basis again. Ten it follows tat te generators fh ; E g wit 6= 2 are a representation of U q (A(M? 1; N? 1)) as tey are obtained by te generators of a representation of te nite q-superalgebra wit te correct coproduct. Tis completes te proof. 5 General representations and conclusions In te previous section, we ave built a representation of te deformed ane Lie superalgebras U q ( b A(M? 1; N? 1)) by means of anyons dened on an innite linear cain; as te corresponding fermionic representation, it as central carge = 1. Representations wit vanising central carge could be built in te same way by using alternative normal ordering prescriptions Eq. (3.12). Representations wit = 0 and = 1 can be combined togeter to get representations wit arbitrary positive integer central carges (we do not discuss ere te problem of te irreducibility of tese representations). Associating a representation to any orizontal line 11?@

13 of a two-dimensional square lattice, innite in one direction (say te orizontal one), and taking K copies of representations in one-dimensional lattice wit central carge equal to 1, one can get representations wit te value of te central carge equal to K. Note tat by combining one representation wit central carge equal to K wit a nite number of representations (in one-dimensional lattice) wit vanising value of te central carge one obtains an inequivalent representation wit te same value of te central carge. Te extensions to two-dimensional lattice innite in bot directions can also be done, but it requires some care in te denition in order to avoid convergence problems. We ave sown in ref. [3] tat te use of a-anyons on a two-dimensional lattice naturally gives te correct coproduct wit te correct powers of te deformation of te representations of a q-algebra dened in a xed site of te lattice. For completeness we recall tat eac site of te two-dimensional lattice is labelled by a vector ~x = (x 1 ; x 2 ), te rst component x 1 2 Z 0 being te coordinate of a site on te line x 2 2 Z. Te angle (~x; ~y) wic enters in te denition of two-dimensional a-anyons troug te disorder factor, see e.g. ref. [5], may be cosen in suc a way tat K(~x) = exp 0 i ~y6=~x 1 (~x; ~y) n(~y) A (5.1) (~x; ~y) = ( +=2 if x2 > y 2?=2 if x 2 < y 2 (5.2) wile if ~x and ~y lie on te same orizontal line, tat is x 2 = y 2, te denitions of Sec. 4 old. Two-dimensional anyons still satisfy te braiding and anticommutations relations expressed in te general form in Eqs. (4.3)-(4.7). Analogous relations old for te A-anyons. Let us replace in te equations of Sec. 4 te one-dimensional anyons by two-dimensional ones and sum over te sites of te two-dimensional lattice. Tis sum as to be read as a sum over te innite line x 1 and a sum over te nite number of lines labelled by x 2. Ten te generators are given by a sum, wit te correct coproduct, of te generators of a U q ( b A(M? 1; N? 1)) representations dened in a line. Terefore tey dene a U q ( b A(M? 1; N? 1)) representation wit value of te central carge given by te sum of te values (0 or 1, see discussion in Sec. 4) of te central carges associated to eac line of te two-dimensional lattice. In te previous sections we ave discussed te case of jqj = 1. Te case of q real can also be discussed and we refer to [7] for te denition of anyons for generic q. One can naturally ask if te realization in terms of a-anyons and A-anyons ere presented can be used to realize te deformation of oter nite or ane superalgebras. It seems tat tis procedure can be extended to te oter basic nite superalgebras, i.e. te series B(0; N), B(M; N), C(N + 1), D(M; N), wile it is not clear its extension to te exceptional or strange nite superalgebras or to te ane case. Finally we want briey to comment 12

14 on te dierence between our approac and te approac of [9], even if ere we present te realization of a q-superalgebra and in [9] a realization of a q-algebra is presented. Te approac of [9] is made on te continuum and te autors do not use, as already remarked, q-bosons, as in te present paper, but standard bosons before \anyonization". However one as to stress tat teir approac is based on a Fock space realization wic guarantees te consistency of te commutation relations. It is wort noticing tat on te Fock space fundamental representation of te deformed algebra is indistinguisable from te fundamental representation of te undeformed algebra. It follows tat a sum wit te correct product of te fundamental representation gives a representation of te deformed algebra. On te contrary as we fulll te commutation relations in abstract way, we are not allowed to consider only te fundamental representation of te deformed algebra realized by bosons and in order to acieve te consistency of te representation we are lead to use q-bosons. Acknowledgements Work supported by te European Commission TMR programme ERBFMRX-CT A.S. tanks te Laboratoire de Pysique Teorique ENSLAPP for kind ospitality during te period in wic tis paper was nised. 13

15 References [1] R. Floreanini, D. Leites, L. Vinet, On te dening relations of quantum superalgebras, Lett. Mat. Pys. 23 (1991) 127. [2] R. Floreanini, V. Spiridonov, L. Vinet, q-oscillator realizations of te quantum superalgebras sl q (mjn) and osp q (mj2n), Commun. Mat. Pys. 137 (1991) 149. [3] L. Frappat, A. Sciarrino, S. Sciuto, P. Sorba, Anyonic realization of te quantum ane Lie algebra U q ( b A N ), Pys. Lett. B 369 (1996) 313. [4] L. Frappat, A. Sciarrino, S. Sciuto, P. Sorba, Anyonic realization of te quantum ane Lie algebras, Proceedings of te 5t International Colloquium on Quantum Groups and Integrable Systems, Prague (1996), and Proceedings of te 10t International Conference on Problems of Quantum Field Teory, Alusta (1996). To appear in te Czecoslovak Journal of Pysics. [5] M. Frau, A. Lerda and S. Sciuto, Proceedings of te International Scool of Pysics "E. Fermi", Course CXXVII, L. Castellani and J. Wess Eds, Amsterdam [6] M. Frau, M. Monteiro and S. Sciuto, q-deformed classical Lie algebras and teir anyonic realization, J. Pys. A: Mat. Gen. 27 (1994) 801. [7] A. Lerda and S. Sciuto, Anyons and quantum groups, Nucl. Pys. B 401 (1993) 613. [8] A. Lerda and S. Sciuto, Slave anyons in te t? J model at te supersymmetric point, Nucl. Pys. B 410 (1993) 577. [9] A. Liguori, M. Mintcev and M. Rossi, Representations of U q ( ^A N ) in te space of continuous anyons, preprint IFUP-96. [10] A. Montorsi, M. Rasetti, Quantum symmetry induced by ponons in te Hubbard model, Pys. Rev. Lett. 72 (1994) [11] M. Sceunert, Te presentation and q-deformation of special linear Lie superalgebras, J. Mat. Pys. 34 (1993) [12] X. Song, Te construction of te q-analogues of te armonic oscillator operators from ordinary oscillator operators, J. Pys. A: Mat. Gen. 23 (1990) L821. [13] H. Yamane, Quantized enveloping algebras associated to simple and ane Lie superalgebras, Proceedings of te XXt ICGTMP, Toyonaka (1994); World Scientic, Singapore (1995). [14] P.C. Zang, T.M. Rice, Pys. Rev. B 37 (1988)