ANYONIC REALIZATION OF THE. QUANTUM AFFINE LIE ALGEBRA U q ( d A N?1 ) Laboratoire de Physique Theorique ENSLAPP 2

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1 Groupe d'annecy Laboratoire d'annecy-le-vieux de Physique des Particules Groupe de Lyon cole Normale Superieure de Lyon ANYONIC RALIZATION OF TH QUANTUM AFFIN LI ALGBRA U q ( d A N? ) L. Frappat a, A. Sciarrino b, S. Sciuto c, P. Sorba a a Laboratoire de Physique Theorique NSLAPP 2 Chemin de Bellevue BP, F-7494 Annecy-le-Vieux Cedex, France b Dipartimento di Scienze Fisiche, 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 Abstract We give a realization of quantum ane Lie algebra U q ( b AN?) in terms of anyons dened on a two-dimensional lattice, the deformation parameter q being related to the statistical parameter of the anyons by q = e i. In the limit of the deformation parameter going to one we recover the Feingold-Frenkel [] fermionic construction of undeformed ane Lie algebra. NSLAPP-AL-552/95 DSF-T-4/95 DFTT-6/95 October 95 supported in part by MURST. 2 URA 436 du CNRS, associee a l'cole Normale Superieure de Lyon et a l'universite de Savoie.

2 Introduction The connection between anyons and quantum groups was originally discovered in [2] for the case of Sl q (2) and later extended in [3] to Sl q (n) and in [4] to the other classical deformed algebras. For a review see [5]. Anyons are particles with any statistics [6, 7] that interpolate between bosons and fermions of which they can be considered, in some sense, as a deformation (for reviews see for instance [8, 9]). Anyons exist only in two dimensions because in this case the con- guration space of collections of identical particles has some special topological properties allowing arbitrary statistics, which do not exist in three or more dimensions. Anyons are deeply connected to the braid group of which they are abelian representations, just like bosons and fermions are abelian representations of the permutation group. In fact, when one exchanges two identical anyons, it is not enough to compare their nal conguration with the initial one, but instead it is necessary to specify also the way in which the two particles are exchanged, i.e. the way in which they braid around each other. In the present paper we will use creation and destruction anyonic operators, equipped with a prescription about their braiding; such a prescription also induces an ordering among the anyons themselves [2]. However we remark that anyons can consistently be dened also on one dimensional chains; for simplicity, in this paper mainly we will use anyons dened on an innite onedimensional chain; at the end we will extend our results to anyons dened on a twodimensional lattice. Here we extend the realization of classical deformed algebras to the case of the deformed ane algebra U q ( A b N? ). As the fermionic construction of undeformed ane algebras, the anyonic construction on a linear chain gives a value of the central charge equal to one. However at the end, see Sect. 3, we briey discuss how to get central charges equal to an integer positive number, considering anyons in a two-dimensional lattice. In the case of deformed ane algebras two kinds of explicit realization are known and only for the case of ane Sl q (n): a construction in terms of an (innite) combination of the Cartan-Chevalley generators of Sl q () build up with bilinears of deformed bosonic or fermionic oscillators ([]) or a vertex operator construction ([]). The paper is organized as follows. In Sect. 2, after recalling the fermionic realization of ba N?, we present an anyonic realization by the use of anyons dened on a one-dimensional lattice Z. In Sect. 3 we explain how more general representations of the quantum ane algebras can be built by means of anyons dened on a two-dimensional lattice; then we make some nal remarks. The Appendix is devoted to a discussion of the ordering of anyons on a chain and on a two-dimensional lattice.

3 2 Anyonic contruction of U q ( c A N? ) Consider the ane Lie algebra A b N? with generators denoted in the Cartan-Weyl basis by h m i (Cartan generators) and e m a (root generators) where i = ; : : : ; N? and m 2 Z. The root system of A N? is given by = f"? " ; 6= Ng, the " 's spanning the dual of the Cartan subalgebra of gl N. The generators satisfy the following commutation relations: h m i ; h n j h m i ; en a e m a ; e n b = h m i ; = = m m;?n ij (2.a) = a i e m+n a 8 >< >: "(a; b)e m+n a+b if a + b is a root a i h m+n i + m m;?n if b =?a otherwise e m a ; (2.b) (2.c) = (2.d) where "(a; b) = is the usual 2-cocycle and is the central charge of the algebra. One can also use the Serre-Chevalley presentation, in which the algebra is described in terms of simple generators and relations (the Serre relations), the only data being the entries of the Cartan matrix (a ) of the algebra. Let us denote the generators in the Serre-Chevalley basis by h and e where = ; ; : : : ; N?. The commutation relations are: h ; h = (2.2a) together with the Serre relations h ; e e + ; e? = a e (2.2b) = h (2.2c)?a X `= (?)`? a `! e?a?` e e ` = (2.3) The correspondence between the Serre-Chevalley and the Cartan-Weyl bases is the following (i = ; : : : ; N? ): h i = h i e + i = e " i?" i+ e? i = e " i+?" i h =? P (2.4) N? j= h j e + = e " N?" e? = e? "?" N 2

4 Let us recall now the fermionic realization of AN? b in terms of creation and annihilation operators. Consider an innite number of fermionic oscillators c (r); c y (r) with = ; : : : ; N and r 2 Z+ =2 = Z, which satisfy the anticommutation relations c (r); c (s) = c y (r); c y (s) = and c y (r); c (s) = rs (2.5) the number operator being dened as usual by n (r) = c y (r)c (r). These oscillators are equipped with a normal ordered product such that ( c : c y y (r)c (s) := (r)c (s) if s >?c (s)c y(r) if s < (2.6) and therefore: : n (r) := ( n (r) if r > n (r)? if r < (2.7) Then a fermionic oscillator realization of the generators of b AN? is given by h m i = X r2z : c y i(r)c i (r + m) :? : c y i+(r)c i+ (r + m) : (i = ; : : : ; N? ); (2.8a) e m "?" = X r2z c y (r)c (r + m); 6= ; ; N (2.8b) Now, taking e m "?" and e?m "?" with let us say, > and m, one can compute e m "?" ; e?m "?" 2 = 4 X X 3 c y (r)c (r + m); c y (s)c (s? m) 5 X r2z s2z = c y (s? m)c (s? m)? c y (s)c (s) s2z = X =? X s62[;m] (: n (s? m) :? : n (s) :) + h k k= s2[;m] (: n (s? m) :? : n (s) : +) A + m (2.9) which proves that the central charge has value = in the fermionic representation. Note that the value of the central charge is intimately related to the denition of the normal ordered product q. (2.7); dierent denitions, like: : n (r) := n (r); 8r 2 Z (2.a) or : n (r) := n (r)? ; 8r 2 Z ; (2.b) 3

5 would lead to =. It follows that a fermionic oscillator realization of the simple generators of AN? b Serre-Chevalley basis is given by ( = ; ; : : : ; N? ) X h = h (r) r2z X e = e r2z (r) in the (2.a) (2.b) where (i = ; : : : ; N? ) h i (r) = n i (r)? n i+ (r) =: n i (r) :? : n i+ (r) : h (r) = n N (r)? n (r + ) =: n N (r) :? : n (r + ) : + r;?=2 (2.2a) (2.2b) e + i (r) = c y i(r)c i+ (r); e + = c y N(r)c (r + ) (2.2c) e? i (r) = c y i+(r)c i (r); e? (r) = c y (r + )c N (r) (2.2d) Inserting q. (2.2b) into q. (2.a) and taking into account that the sum over r can be splitted into a sum of two convergent series only after normal ordering, one again checks that that is =. X X N? X h = + : n N (r) :? : n (r) : =? r2z r2z j= h j ; (2.3) In order to obtain an anyonic realization of U q ( b A N? ), we will replace the fermionic oscillators by suitable anyons in the expressions of the simple generators of U q ( b A N? ) in the Serre-Chevalley basis. We introduce therefore the following anyons dened on a onedimensional lattice (r 2 Z ): a (r) = K (r)c (r) N; (2.4) and similar expressions for the conjugated operator a y (r), where the disorder factor K (r) is expressed as K (r) = Y t6=r e?i(r;t):n(t): (2.5) 2 [; 2) being the statistical parameter and (r; t) a suitably dened angle (see Refs. [2, 5]). The anyonic ordering being the natural order of the integers, in q. (2.5) the anyonic angle (see Appendix and Ref. [5]) can be chosen as (r; t) = +=2 (r; t) =?=2 if r > t if r < t (2.6) 4

6 By means of q. (2.6) the disorder factor K (r) can be rewritten, using the sign function "(t) = jtj=t if t 6= and "() =, P K (r) = q? 2 t2z "(t?r):n(t): (2.7) with q = exp(i). By a direct calculation, one can prove that the a-type operators satisfy the following braiding relations for r > s: and a (r)a (s) + q? a (s)a (r) = a y (r)a y (s) + q? a y (s)a y (r) = a y (r)a (s) + q a (s)a y (r) = a (r)a y (s) + q a y (s)a (r) = (2.8) a (r)a y (r) + a y (r)a (r) = a (r) 2 = a y (r) 2 = (2.9) which shows that the operators a (r); a y (s) are indeed anyonic oscillators with statistical parameter. Notice that the last equations (2.9) do not contain any q-factor, the operators a (r); a y (s) at the same point thus satisfying standard anticommutation relations. Following Ref. [2], we also introduce twiddled anyonic oscillators: ~a (r) = ~ K (r)c (r) N; (2.2) where: ~K (r) = Y t6=r e?i ~ (r;t):n (t): ; (2.2) with the same statistical parameter and opposite braiding (and ordering) prescription, corresponding to the choice ~(r; t) =?=2 if r > t (2.22) ~(r; t) = +=2 if r < t and therefore to replace q with q? in qs. (2.7), (2.8). Let us add that anyons with dierent indices ; obey the standard anticommutation relations, for all values of r; s 2 Z and for all values of the anyonic parameter. Finally, the braiding relations between a-type and ~a-type anyons are given by ~a (r); a (s) ~a y(r); a (s) = ~a y (r); a y (s) = for all r; s 2 Z (2.23) = ~a (r); a y (s) = for all r 6= s 2 Z (2.24) 5

7 and ~a (r); a y (r) ~a y (r); a (r) = q P t2z "(t?r):n (t): = q?p t2z "(t?r):n(t): (2.25) It is useful to remark that the following identities hold: a y (r)a (r) = ~a y (r)~a (r) = n (r) (2.26) and that normal ordering among anyons is dened exactly as in q. (2.7). verywhere in this paper the generators + will be expressed in terms of the oscillators a and the generators? in terms of the ~a as, in the deformed case, the? 's are not simply equal to ( + ) y, but also the exchange q $ q? must be performed. We can now build an anyonic realization of the quantum ane Lie algebra U q ( b A N? ) "anyonizing" qs. (2.a-2.b), i.e. replacing the fermionic oscillators c with the a 's in q. (2.2c), with the ~a 's in q. (2.2d), and, indierently (see q. (2.26)), with the a 's or the ~a 's in the Cartan generators h i (see qs. (2.2a) and (2.2b)). Theorem An anyonic realization of the simple generators of the quantum ane Lie algebra U q ( b A N? ) with central charge = is given by ( = ; ; : : : ; N? ) where ( i N? ) X X H = H (r) r2z = r2z (r) (2.27) H i (r) = h i (r) =: n i (r) :? : n i+ (r) : + i (r) = a y i(r)a i+ (r)? i (r) = ~a y i+(r)~a i (r) (2.28a) H (r) = h (r) =: n N (r) :? : n (r + ) : + r;?=2 + (r) = q? 2 "(r+=2) a y N(r)a (r + )? (r) = q? 2 "(r+=2) ~a y (r + )~a N (r) (2.28b) Proof We must check that the realization qs. (2.28a-2.28b) indeed satisfy the quantum ane Lie algebra U q ( b A N? ) in the Serre-Chevalley basis, i.e. for ; = ; ; : : : ; N?, H ; H H ; + ;? = (2.29a) = a (2.29b) = [H ] q (2.29c) 6

8 with the quantum Serre relations?a X `= (?)` "? a ` # q where the notations are the standard ones, i.e. [x] q = qx? q?x q? q? ; " m n # q =?a?` ` = (2.3) [m] q! [n] q![m? n] q! ; [m] q! = [] q : : : [m] q (2.3) a being the Cartan matrix of b AN? given by (N 3) and, for N = 2, by a = B 2??? 2?...? ?? 2??? 2 a = Moreover in q. (2.29c) and q. (2.3) one has: 2?2?2 2! C C A (2.32) (2.33) q = q (a;a)=2 (2.34) (a ; a ) being the length of the th simple root a, and therefore q = q for all = ; ; : : : ; N?. The proof of this theorem will be based on the observation that the qs. (2.29a-2.29c) and (2.3), which dene a generic deformed ane algebra U q ( b A), reduce to U q (A) if the dot number is removed, and to another nite dimensional algebra U q (A ) if the dot number is kept and one or more other suitable dots are removed. In general, the relations dening U q ( b A) coincide with the overlap of those dening U q (A) and U q (A ); therefore it will be enough to check that the equations dening U q (A) and U q (A ) are satised. Thus our strategy will be the following: step ) { prove that the generators written in the anyonic representation (see q. (2.27)) satisfy the relations (2.29a-2.29c) and (2.3) for ; 6=. This proof is performed using the technique outlined in Sect. 4 of [5], that is going through the following steps: 7

9 { step a) { individuate a set of generators fh i ; e i ; i 6= g of a low dimensional representation of U q (A); { step b) { show that, once that the anyonic oscillators a, ~a are expressed in terms of the fermionic ones, the set fh i ; i ; i 6= g is related to the set fh i ; e i ; i 6= g by iterated coproduct in U q (A) and thus is a representation of U q (A). step 2) { Repeat the same procedure for U q (A ), through the following steps: { step 2a) { individuate a set of generators fh ; e ; 2 Ig of a low dimensional representation of U q (A ); I is a suitable subset of the extended Dynkin graph of U q ( b A), with 2 I; { step 2b) { show that, once that the anyonic oscillators a, ~a are expressed in terms of the fermionic ones, the set fh ; ; 2 Ig is related to the set fh ; e ; 2 Ig by iterated coproduct in U q (A ) and thus is a representation of U q (A ). In order to prove the present theorem, the step ) is easily done once one realizes that, inserting qs. (2.4) and (2.2), the expressions (2.28a-2.28b) simplify and become P (r) = e (r) q 2 t2z "(t?r)h(t) (2.35) where the generators h (r) = H (r)j q= = H (r) and e (r) = (r)j q=, coincide with those dened in qs. (2.2a-2.2d), corresponding to the undeformed ane algebra A b N?. In fact, for any xed r 2 Z+=2, the set fh i (r); e i (r); i = ; : : : ; N?g is a representation of A N? of spin and /2, and thus also of U q (A N? ) [4]; thanks to qs. (2.27), (2.35), H ; are the correct coproduct in U q (A N? ). Therefore qs. (2.29a-2.29c), (2.3) surely holds for ; 6=. For the step 2), we cut the extended Dynkin diagram by deleting a dot 6= ; for any xed r 2 Z, the set fh (r); e (r); h i (r); e i (r); h j (r + ); e j (r + )g, where + i N? and j?, is a spin, /2 representation of A N?, and thus a representation of U q (A N? ) (step 2a)); again, thanks to qs. (2.27), (2.35), fh ; ; 6= g is the correct coproduct in U q (A N? ). Therefore, qs. (2.29a-2.29c) and the quantum Serre relations (2.3) hold for any value of the indices ;, only excluding the couples = ; = or = ; =. However, one notes that these equations obviously hold for j? j 2 where? is taken modulo N. Therefore, for N 4, we take, for instance, = 2 and for N = 3 it is enough to repeat the argument twice, once for = and once for = 2. To complete the proof, one has still to consider the case of U q ( A b ), whose Cartan matrix is reported in q. (2.33). In this case the previous argument fails to prove q. (2.29c) and the quantum Serre relations (2.3), which take now the following form 3? 3? ? 2? 3 = 3 = ; (2.36a) (2.36b) 8

10 where = q 2 + q?2 +. Such equations can however be explicitly checked by using the braiding properties of the anyonic oscillators a and ~a. The central charge is equal to exactly for the same reasons of the undeformed case. As the deformation of any ane Lie algebra does not change its central charge, we will not repeat this argument in the following. It is worthwhile to remark that the presence of a power of q in front of the anyonic oscillators in the expressions (2.28b) of (r) (necessary in order to reproduce q. (2.35)) reects the non vanishing of the central charge and is related to the denition q. (2.7) of normal ordering; no power of q in front of the anyonic oscillators in q. (2.28b) would be needed if normal ordering were dened according to q. (2.a) or to q. (2.b), corresponding to vanishing central charge. Let us remark that we could prove Theorem by using the embedding U q ( b A N? ) U q (A ) according to a theorem due to Hayashi [], that we report here in our notations: Theorem 2 Let k i ; f i equations: (i 2 Z) be the simple generators of U q (A ), satisfying the following k i ; k j = (2.37a) k i ; f j f + ; f? i j?a X ij `= = a ij f j (2.37b) = ij [k i ] q (2.37c) (?)` "? aij ` with the innite dimensional Cartan matrix a ij = # q f?aij?` i f j ` f i = ? 2?? 2? C A (2.37d) (2.38) Then H = X`2Z k +N ` and = X`2Z f +N ` q 2 P m2z "(m?`)k +Nm (2.39) where = ; ; : : : ; N?, give a representation of the simple generators of U q ( b A N? ). By means of q. (2.35) one easily checks that the generators H and dened in q. (2.27) coincide with those dened in q. (2.39), once that the identication h (`? =2) = k +N ` e (`? =2) = f +N ` (2.4) 9

11 where ` 2 Zand = ; ; : : : ; N?, is made. 3 More general representations and two dimensional anyons In the previous section, we have built a representation of the deformed ane Lie algebras U q ( b A N? ) by means of anyons dened on an innite linear chain; as the corresponding fermionic representation, it has central charge =. Representations with vanishing central charge could be built in the same way by using alternative normal ordering prescriptions q. (2.a-2.b). Representations with = and = can be combined together by associating a representation to any horizontal line of a two-dimensional square lattice, innite in one direction, let us say the horizontal one. So by M copies of one-dimensional representations with central charge equal to one can get representations with the value of the central charge equal to M. Note that by combining one representation with central charge equal to M with a nite number of (one-dimensional) representations with vanishing value of the central charge one obtains an inequivalent representation with the same value of the central charge. We do not discuss here the problem of the irreducibility of these representations. The extensions to two dimensional lattice innite in both directions can also be done, but it requires some care in the denition in order to avoid convergence problems. We will show now that the use of anyons dened on a two-dimensional lattice naturally gives the coproduct of representations with the correct powers of the deformation parameter q. ach site of the two-dimensional square lattice is labelled by a vector x = (x ; x 2 ); the rst component x 2 Z is the coordinate of a site on the line x 2 2 Z. As it is shown in the Appendix in q. (4.8), in a suitable gauge, the angle (x; y) which enters into the denition of two-dimensional anyons can be chosen in such a way that (x; y) = ( +=2 if x2 > y 2?=2 if x 2 < y 2 (3.) while when x and y lie on the same horizontal line, that is x 2 = y 2, the denitions of Sect. 2 hold. Two-dimensional anyons still satisfy the braiding and anticommutations relations expressed in the general form in qs. ( ). Replacing in Sect. 2 one-dimensional anyons with two-dimensional ones, and making the corresponding replacements in the sums over the sites, we easily get that H = X x H (x) = X x (x) (3.2) where the sum over the vector of the lattice has to be read as the sum on the (innite) line

12 x and the sum over the (nite) line x 2, that is H = X x 2 H (line x 2 ) P y 2 "(y 2?x 2 )H (line y2 ) X = 2 q (line x 2) (3.3) x 2 where q = q (a;a)=2 and H (line x 2 ), (line x 2 ) are the simple generators of the U q ( b A) dened in the previous section. These equations show that the coproduct rules are fullled and the set fh ; g gives a representation of the algebra U q( b A) with central charge equal to the sum of the central charges associated to each line of the two-dimensional lattice. It is important to remark that, for sake of simplicity, we have proved all the theorems of Sect. 2 by exploiting the representation of anyons in terms of fermions, the disorder operators giving the correct coproducts; however, this is not necessary and purely "anyonic" proofs of our Theorems can be given: it is easy to get convinced that the braiding and anticommutations relations of qs. ( ) are the only relations necessary to prove that the simple generators built by means of anyons satisfy the commutations relations and the Serre equations dening the deformed ane algebras. We emphasize that anyons, dened by the braiding and anticommutations relations of qs. ( ), have nothing to do with q? oscillators, that were even used to build representations of deformed algebras. In the previous sections we have discussed the case of jqj =. The case of q real can also be discussed and we refer to [2] for the denition of anyons for generic q. In conclusions we have presented a method to get representations (in general reducible) of the ane untwisted quantum algebra U q ( A b N? ) with positive integer value of the central charge. The role of the denition of the ordering of the anyons and of the normal ordering is essential in the above construction. One can naturally ask if this construction can also be generalized to the case of the other ane untwisted quantum algebras (B; C; D series) and to the twisted ane algebras, or if a construction of a deformed Virasoro algebra can be obtained by means of anyons, or if anyons can also be used to realize ane deformed Lie superalgebras. On the same line, one could think that anyons are natural objects to use in statistical mechanical models in which quantum groups arise. We hope to address these issues in some future work. 4 Appendix A crucial role in the construction of anyons in terms of fermions coupled to a Chern-Simons eld, which endow them with a magnetic ux, is given by the angle (x; y) under which the point x is seen from the point y. As the angle (x; y) appears in the exponent of the disorder operator K(x) (2.5) multiplied by i, being not integer, it must be thoroughly dened without any ambiguity (see [2] and references quoted therein). On a two-dimensional lattice, we x a base point B and associate to any point x a path from B to x (see gure ).

13 x???? t y t y t gure The angle (x; y) is dened as (x; y) = By d x + (4.) where is a constant and By d x is the angle under which the oriented path Bx is seen from a point y that will be eventually sent to y inside one of the four cells which have y as vertex. The point y is shifted from the point y by an arbitrarily small vector in order to dene unambiguously the angle By d x even if the path Bx passes through the site y. The simplest choice is to x B as the innity point of the positive x-axis and to associate to each point x the horizontal straight line coming from B (see gure 2). t B t x B gure 2 Choosing y = y + "(u x + u y ) with "! + and u x ; u y unit vectors of the axis x and y, one easily gets for two points on the same line (x 2 = y 2 ) and therefore, xing = =2, By d x = if x > y (4.2) By d x =? if x < y (x; y) = ( +=2 if x > y?=2 if x < y (4.3) reproducing q. (2.6). For two generic points x; y on the two-dimensional lattice, the angle By d x can get any value in the interval [?; ). However, in the braiding relations, and therefore in all calculations of this paper, the only quantities coming into play are the dierences (x; y)?(y; x). It is easy to check that, for any couple x; y, the denition of gure 2 gives (x; y)? (y; x) = 8 >< >: +? 2 x 2 > y 2 x > y if x 2 = y 2 x 2 < y 2 x < y if x 2 = y 2 (4.4)

14 The denition of the angle (x; y) induces an ordering among the points of the lattice [2]. We say x y if (x; y)? (y; x) = + x y if (x; y)? (y; x) =? (4.5) This ordering coincides with the natural one for points on a horizontal line: x y () x > y for x 2 = y 2 (4.6) As it is possible [5] to deform continuously the functions (x; y) with the only restriction of keeping xed the quantities (x; y)? (y; x); we will choose (x; y) = 2 () x y (4.7) and therefore (x; y) = ( +=2 if x2 > y 2?=2 if x 2 < y 2 (4.8) 3

15 References [] A.J. Feingold, I.B. Frenkel, Adv. Math. 56 (985) 7. [2] A. Lerda and S. Sciuto, Nucl. Phys. B4 (993) 63. [3] R. Caracciolo and M.A. R.-Monteiro, Phys. Lett. 38B (993) 58. [4] M. Frau, M.A. R.-Monteiro and S. Sciuto, J. Phys. A27 (994) 8. [5] M. Frau, A. Lerda and S. Sciuto, Anyons and deformed Lie algebras, DFTT 33/94-DFT- US 2/94, to appear in Proc. of the Int. School of Physics ". Fermi", Course CXXVII Quantum Groups and Their Application in Physics. [6] J.M. Leinaas and J. Myrheim, Nuovo Cim. 37B (977). [7] F. Wilczek, Phys. Rev. Lett. 48 (982) 4 and Phys. Rev. Lett. 49 (982) 957. [8] F. Wilczek, in Fractional Statistics and Anyon Superconductivity edited by F. Wilczek (World Scientic Publishing Co., Singapore 99). [9] A. Lerda, Anyons: Quantum Mechanics of Particles with Fractional Statistics (Springer- Verlag, Berlin, Germany 992). [] T. Hayashi, Commun. Math. Phys. 27 (99) 29. [] I.B. Frenkel, N. Jing, Proc. Natl. Acad. Sci. USA 85 (988)