ANYONIC REALIZATION OF THE. Chemin de Bellevue BP 110, F Annecy-le-Vieux Cedex, France. and I.N.F.N., Sezione di Napoli, Italy

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1 ANYONIC REALIZATION OF TE QUANTUM AFFINE LIE ALGEBRAS L. Frappat a, A. Sciarrino b, S. Sciuto c,. Sorba a a Laboratoire de hysique Theorique ENSLA Chemin de Bellevue B 0, 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 algebras U q ( b AN?) and U q ( b CN ) in terms of anyons dened on a one-dimensional chain (or 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 algebras. roceedings of the 5th International Colloquium on Quantum Groups and Integrable Systems, rague (Czech Republic), June 996, to appear in the Czechoslovak Journal of hysics, Talk presented by L. Frappat, and roceedings of the 0th International Conference on roblems of Quantum Field Theory, Alushta (Crimea, Ukraine), 3-8 May 996, Talk presented by S. Sciuto. ENSLA-AL-602/96 DSF-T-3/96 DFTT-38/96 q-alg/ July 996 URA 436 du CNRS, associee a l'ecole Normale Superieure de Lyon et a l'universite de Savoie.

2 Introduction Anyons are particles with any statistics that interpolate between fermions and bosons. In the rst quantization scheme, the notion of statistics is related to the symmetry properties of the wave function of N identical particles, the bosons (resp. fermions) corresponding to symmetric (resp. antisymmetric) wave functions under the exchange of particles. In two dimensions, there exist more possibilities: the anyons. In that case, the wave function of N identical particles picks up a phase factor under the exchange of particles. More precisely, if we consider a system of N identical hard-core particles in d dimensions, the conguration space is M N;d = [(R d ) N? ]=S N where is the set of points of (R d ) N with at least two equal coordinates and S N is the permutation group of N elements. The fundamental group (M N;d ) = S N for d > 2 but (M N;d ) = B N for d = 2 where B N is the braid group. Anyons appear as abelian representations of the braid group 2. Let us remark that anyons can consistently be dened also on a one-dimensional lattice. For simplicity, this will be used here to construct anyonic realizations of the quantum ane Lie algebras U q ( b A N? ) and U q ( b C N ). In sect. 4 the construction by means of two-dimensional anyons [2] will be briey recalled. 2 The unitary case: Uq( c A N? ) The strategy to construct an anyonic realization of the quantum ane Lie algebra U q ( b A N? ) with non-vanishing central charge will be the following: () start from the description of U q ( b A N? ) in the Serre{Chevalley basis, (2) nd a fermionic realization of ba N? in terms of creation and annihilation operators, (3) construct anyonic oscillators on a one-dimensional lattice and (4) 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. Let us denote by and E where = 0; ; : : : ; N? the simple generators of U q ( A b N? ) in the Serre{Chevalley basis. The commutation relations are: h i ; = 0 (2.a) and the quantum Serre relations read as:?a X `=0 (?)` "? a ` h i ; E = a E h i E + ; E? = [ ] q # (2.b) (2.c) q E?a?` E E ` = 0 (2.2) 2 for a review on anyons see for instance [5]; for a review on anyonic realization of deformed Lie algebras see [6].

3 where the notations are the standard ones, i.e. [x] q = qx? q?x q? q? ; " m n # q = [m] q! [n] q![m? n] q! ; [m] q! = [] q : : : [m] q (2.3) a being the Cartan matrix of b AN? and q = q for all = 0; ; : : : ; N?. A fermionic realization of U q ( A b N? ) in terms of creation and annihilation operators is obtained by taking an innite number of fermionic oscillators c (r); c y (r) with = ; : : : ; N and r 2 Z 0 Z+, which satisfy the anticommutation relations 2 n c (r); c (s) o = n c y (r); c y (s) o = 0 and n c y (r); c (s) o = rs (2.4) 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 and therefore : c y (r)c (s) := : n (r) := ( c y (r)c (s) if s > 0?c (s)c y (r) if s < 0 ( n (r) if r > 0 n (r)? if r < 0 (2.5) (2.6) Then a fermionic oscillator realization of the simple generators of b AN? in the Serre{ Chevalley basis is given by ( = 0; ; : : : ; N? ) (we use small letters for the simple generators of the undeformed Lie algebra b A N? ) h = X r2z 0 h (r) and e = X r2z 0 e (r) (2.7) where (i = ; : : : ; N? ) h i (r) = n i (r)? n i+ (r) = : n i (r) :? : n i+ (r) : h 0 (r) = n N (r)? n (r + ) = : n N (r) :? : n (r + ) : + r;?=2 (2.8a) (2.8b) e + i (r) = cy i(r)c i+ (r) ; e + 0 = cy N (r)c (r + ) (2.8c) e? i (r) = c y i+(r)c i (r) ; e? 0 (r) = c y (r + )c N (r) (2.8d) Inserting Eq. (2.8b) into Eq. (2.7) and taking into account that the sum over r can be splitted into a sum of two convergent series only after normal ordering, one can check that that is the central charge is =. X h 0 = + : n N (r) :? X : n (r) : =? N X? r2z r2z 0 0 We now introduce anyons dened on a one-dimensional lattice: j= h j ; (2.9) a (r) = K (r)c (r) and ~a (r) = ~ K (r)c (r) ( N) (2.0) 2

4 and similar expressions for the conjugated operator a y (r) and ~ay (r), where the disorder factors K (r) and K ~ (r) are expressed as K (r) = q? 2 t2z 0 "(t?r):n (t): and ~ K (r) = q 2 t2z 0 "(t?r):n (t): (2.) using the sign function "(t) = jtj=t if t 6= 0 and "(0) = 0. 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) = 0 a y (r)a (s) + q a (s)a y (r) = 0 a y (r)a y (s) + q? a y (s)a y (r) = 0 a (r)a y (s) + q a y (s)a (r) = 0 (2.2) a (r)a y (r) + ay (r)a (r) = a (r) 2 = a y (r)2 = 0 (2.3) which shows that the operators a (r); a y (s) are indeed anyonic oscillators with statistical parameter such that q = e i. The ~a-type anyons have the same statistical parameter but opposite braiding (and ordering) prescription. 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 ( = 0; ; : : : ; N? ) where ( i N? ) = X r2z 0 (r) and E = X r2z 0 E (r) (2.4) i (r) = h i (r) =: n i (r) :? : n i+ (r) : (2.5a) E i + (r) = a y i(r)a i+ (r) E i? (r) = ~a y i+(r)~a i (r) (2.5b) 0 (r) = h 0 (r) =: n N (r) :? : n (r + ) : + r;?=2 (2.5c) E + 0 (r) = q? 2 "(r+=2) a y N(r)a (r + ) E? 0 (r) = q? 2 "(r+=2) ~a y (r + )~a N (r) (2.5d) roof We must check that the realization Eqs. (2.5a-2.5d) indeed satisfy the quantum ane Lie algebra U q ( b A N? ) in the Serre{Chevalley basis. Fist of all, inserting eqs. (2.0), the expressions (2.5a-d) become E (r) = e (r)q 2 t2z 0 "(t?r):h (t): (2.6) where the generators h (r) and e (r) coincide with those dened in eqs. (2.8a-d) corresponding to the undeformed ane Lie algebra A b N?. Now, consider the non-extended Dynkin diagram of AN b?, to which corresponds the set of generators fh (r); e (r)g where 6= 0. For a xed r 2 Z 0, the set fh (r); e (r)g for 6= 0 is a representation of A N? of spin 0 and /2, and thus of U q (A N? ) [4]. Thanks 3

5 to eqs. (2.4) and (2.6), ; E are the correct coproduct in U q(a N? ). Therefore, the relations (2.a-c) hold for ; 6= 0 (step ). We consider then the extended Dynkin diagram of b A N? and we delete a dot which is not the ane dot. For a xed r 2 Z 0, the set fh (r); e (r)g for 6= is a representation of A N? of spin 0 and /2, and thus of U q (A N? ). The eqs. (2.4) and (2.6) give once again the correct coproduct in U q (A N? ) and hence ; E form a representation of U q(a N? ). Therefore, the relations (2.a-c) hold for ; 6= (step 2). Finally, in the case U q ( b A ) the previous arguments fail (in particular to prove the quantum Serre relations). Such equations can however be checked explicitly by using the braiding properties of the anyonic oscillators a and ~a. 3 The symplectic case: Uq ( c C N ) Let us consider now the U q ( b C N ) case. As in the U q ( b A N? ) case, a guideline is given by the fermionic realization of the (undeformed) ane algebra. Such a fermionic realization in terms of creation and annihilation operators is easily obtained by noticing that the folding of the Dynkin diagram of b A2N leads to the Dynkin diagram of b CN (see gure below). h 0 m m m m m m h h N? h N h N + h 2N? ba 2N # # m m m m h 0 0 h 0 h 0 h 0 2 N?2 h 0 N? h bc 0 N N Denoting the simple generators of A b 2N by h i ; e i with i = 0; : : : ; 2N?, the lifting of the automorphism associated to the symmetry of the Dynkin diagram of A2N b leads to the following expression of the simple generators of C b N : h 0 = h 0 0; h 0 i = h i + h 2N?i ; h 0 N = h N e 0 0 = e 0 ; e 0 i = e i + e 2N?i; e 0 N = e N (3.) In the undeformed case, it is immediate to see that the Serre relations of b C N are satised (actually one can write them entirely in terms of the Serre relations of b A2N ). Now, we go to the deformed case. The main idea of the construction is to use the folding b A 2N?! b C N to obtain a realization of U q ( b C N ) in terms of anyons, following with some modications the procedure used in Ref. [4] to build U q (C N ). Using the fermionic oscillators c (r); c y (r) of the previous Section, with the same normal ordering prescription Eq. (2.6), one denes the following anyons: a (r) = K (r)c (r) and ~a (r) = ~ K (r)c (r) ( 2N) (3.2) 4

6 where the disorder operators are now K (r) = ~ K y (r) = q? 2 K (r) = ~ K y (r) = q? 2 t2z 0 "(t?r)(:n (t):?:n 2N?+ (t):)?:n 2N?+ (r): for = ; : : : ; N t2z 0 "(t?r)(:n (t):?:n 2N?+ (t):)+:n 2N?+ (r): for = N + ; : : : ; 2N (3.3) Theorem 2 An anyonic realization of the simple generators of the quantum ane Lie algebra U q ( b C N ) with central charge equal to is given by ( = 0; ; : : : ; N) X X = (r) and E r2z = E (r) (3.4) r2z 0 0 where i (r) = h i (r) = : n i (r) :? : n i+ (r) : + : n 2N?i (r) :? : n 2N?i+ (r) : E + i (r) = a y i(r)a i+ (r) + a y 2N?i(r)a 2N?i+ (r) E i? (r) = ~a y i+(r)~a i (r) + ~a y 2N?i+(r)~a 2N?i (r) (3.5) are associated to the simple short roots of C N (i = ; : : : ; N? ), N (r) = h N (r) = : n N (r) :? : n N + (r) : E N + (r) = ay N(r)a N + (r) E N(r)? = ~a y N +(r)~a N (r) (3.6) are associated to the simple long root of C N, and 0 (r) = h 0 (r) = : n 2N (r) :? : n (r + ) : + r;?=2 E + 0 (r) = ay 2N(r)a (r + ) are associated to the ane root of C N. E? 0 (r) = ~a y (r + )~a 2N (r) (3.7) roof We have now to check that the realization Eqs. (3.5), (3.6), (3.7) indeed satisfy the quantum ane Lie algebra U q ( b C N ) in the Serre{Chevalley basis, i.e. the equations (2.a-2.c) for ; = 0; ; : : : ; N, together with the quantum Serre relations (2.2) hold, where a is now the Cartan matrix of b C N and q i = q for i = ; : : : ; N? (short roots) and q 0 = q N = q 2 (long roots). The proof is based on the strategy exposed in the previous section. First of all, let us dene for = ; : : : ; 2N? k (r) = n (r)? n + (r) f + (r) = cy (r)c +(r) f? (r) = cy +(r)c (r) = f + (r) y (3.8a) (3.8b) (3.8c) 5

7 and f + 0 (r) = cy 2N (r)c (r + ) f? 0 (r) = cy (r + )c 2N (r) = f + 0 (r) y (3.8d) (3.8e) As before, by using the denitions (3.2) of a; ~a and taking into account Eq. (3.3), the expressions (3.5), (3.6) and (3.7) simplify as ( = 0; ; : : : ; N) where E (r) = e (r) q 2 (3.9) t2z 0 "(t?r)h (r) e i (r) = f i (r)q 2 k 2N?i(r) + f 2N?i(r)q? 2 k i(r) e N (r) = f N (r) e 0 (r) = f 0 (r) for i = ; : : : ; N? (3.0a) (3.0b) (3.0c) The step is achieved by noticing that for any xed r 2 Z 0, the set fk i (r); f i (r); i = ; : : : ; N? g and fk 2N?i (r); f 2N?i(r); i = ; : : : ; N? g are representations of spin 0 and /2 of U q (A N? ); therefore the set fh i (r); e i (r); i = ; : : : ; N? g is a representation of U q (A N? ) corresponding to the coproduct of these two representations. Analogously, of spin 0 and /2 and thus of U q (A ). More- fh N (r); e N(r)g is a representation of A over, from the properties of the fermionic oscillators, one can directly check the quantum Serre relations and Eqs. (2.b-c) involving h N (r) and e N(r) and thus prove that the set fh (r); e (r); = ; : : : ; Ng is a representation of U q (C N ). Then Eqs. (3.4) and (3.9) show that also the set f ; E ; = ; : : : ; Ng is a representation of U q(c N ), obtained by iterated coproduct 3. For the step 2, one has to check that the equations (2.a-2.c) and (2.2) also hold when ; can take the value 0. Actually the only non trivial cases are for Eqs. (2.c) and (2.2) when = 0; = or = ; = 0. Thus we have just to show that the set f ; E ; = 0; g is a representation of U q(c 2 ); as h 0 (r) involves oscillators both of the site r and of the site r +, it is convenient to rewrite Eq. (3.4) rearranging the (convergent) series for and E : = X r2z 0 0 (r) E = X r2z 0 E 0 (r) (3.) where and 0 (r) = h 0 (r) = k (r + ) + k 2N? (r) (3.2) E 0 + = a y (r + )a 2 (r + ) + a y 2N?(r)a 2N (r) (3.3) E 0? = ~a y 2(r + )~a (r + ) + ~a y 2N(r)~a 2N? (r) (3.4) 3 Let us remark that this anyonic representation of U q (C N ) does not require any Gutzwiller projection on the fermionic oscillators and thus it is an improvement of the one given in ref. [4]. 6

8 Then using the denitions (3.2) of a; ~a and taking into account Eq. (3.3), one gets where E 0 (r) = e0 (r) q 2 t2z 0 "(t?r)h 0 (t) (3.5) e 0 (r) = f 2N? (r)q 2 k (r+) + f (r + )q? 2 k 2N?(r) (3.6) Let us show now that the set fh 0 (r); e 0 (r); h 0 (r); e 0 For a xed r 2 Z 0, the triple fh 0 (r); e 0 (r)g is a representation of U q (A ), corresponding to the coproduct of fk 2N? (r); f2n?(r)g and fk (r + ); f (r + )g. Therefore one has [e 0 + (r); e 0? (r)] = [h 0 (r)] q. Using elementary properties of fermionic oscillators, one checks the equations [e + 0 ; e 0? ] = [e? 0 ; e 0 + ] = 0 and the quantum Serre relations on the generators e 0 ; e 0. Then from Eq. (3.9) for = 0 and Eq. (3.5), it follows that the (r)g is a representation of U q (C 2 ). set f 0 ; E 0 ; ; E g is a representation of U q (C 2 ) and thus Eqs. (2.a-c) and the Serre relations (2.2) on the generators (3.4) also hold for ; 2 f0; g, which concludes the proof. As in the U q ( b A N? ) case, the central charge is because of the normal ordering prescription Eq. (2.6). 4 Conclusion We have presented here a method to get an anyonic realization of the quantum ane Lie algebras U q ( b A N? ) and U q ( b C N ) with central charge =. Let us emphasize the role of the denition of the ordering of anyons which is crucial in this construction. Representations with vanishing central could be built in the same way by using alternative ordering prescriptions. These representations with = 0 and = can be combined together to get representations with arbitrary positive integer central charges which are in general reducible. This composition of representations, with the correct deformed coproduct, is naturally achieved by considering anyons dened on a two-dimensional lattice [2]. It is worthwhile to notice that these anyonic constructions have nothing to do with q-deformed oscillators. Finally, let us mention that it is also possible to obtain a supersymmetric version of the construction (see ref. [3] for the case U q ( b A(m; n))). Acknowledgements This research was partially supported by MURST and EU, within the framework of the program "Gauge Theories, Applied Supersymmetry and Quantum Gravity" under contract SCI*-CT Two of us (L.F. and.s.) are also grateful to EU for partial nancial help (European Network ERBCRXTCXT ). References [] A.J. Feingold, I.B. Frenkel, Classical ane algebras, Adv. Math. 56 (985) 7. [2] L. Frappat, A. Sciarrino, S. Sciuto,. Sorba, Anyonic realization of the quantum ane Lie algebra U q ( b A N? ), hys. Lett. B369 (996) 33. 7

9 [3] L. Frappat, A. Sciarrino, S. Sciuto,. Sorba, Anyonic realization of the quantum ane Lie superalgebra U q ( b A(M; N)), preprint ENSLA-AL-603/96, DST-T-35/96 and DFTT-39/96. [4] M. Frau, M.A. R.-Monteiro and S. Sciuto, q-deformed classical Lie algebras and their anyonic realization, J. hys. A27 (994) 80. [5] A. Lerda, Anyons: Quantum Mechanics of articles with Fractional Statistics (Springer-Verlag, Berlin, Germany 992). [6] M. Frau, A. Lerda and S. Sciuto, Anyons and deformed Lie algebras, roceedings of the International School of hysics "E. Fermi", Course CXXVII, L. Castellani and J. Wess Eds, Amsterdam