29 April 2002

Physics Letters A 296 (2002) 259–264

www.elsevier.com/locate/pla

New algebraic structures in theCλ-extended Hamiltonian system

E.H. El Kinani a,b,c,d,1

a The Abdus Salam International Centre for Theoretical Physics, ICTP, Strada costera 11, 34100 Trieste, Italyb Groupe de Mathématique Physique, Département de Mathématiques, Faculté des Sciences et Techniques, Boutalamine B.P. 509,

Errachidia, Morocco 2

c UFR de Géométrie Différentielle et Applications, Faculté des Sciences Ben M’sik, Casa, Moroccod UFR de Physique Théorique, Faculté des Sciences, Rabat, Morocco

Received 20 August 2001; received in revised form 21 March 2002; accepted 21 March 2002

Communicated by P.R. Holland

Abstract

A realization of various algebraic structures in terms of theCλ-extended oscillator algebras is introduced. In particular, theCλ-extended oscillator algebras realization of the Fairlie–Fletcher–Zachos (FFZ) algebra is given. This latter lead easily to therealization of the quantumUt(sl(2)) algebra. The new deformed Virasoro algebra is also presented. 2002 Elsevier ScienceB.V. All rights reserved.

1. Introduction

Deformations of different groups and algebras haveattracted great attention during the last few years.These new mathematical objects called quantum alge-bras or quantum groups have found a lot of interestingphysical applications. On the other hand, recently var-ious extensions and deformations of the oscillator al-gebra have indeed been applied in the description ofsystems with nonstandard statistics, with violation ofthe Pauli principal, in the construction of integrablelattice models, as well as in the algebraic treatment ofn-particle integrable models. Among these various de-formations and extensions, we mention the following:

E-mail addresses: kinhass@hotmail.com,hkinani@ictp.trieste.it, el-kinani@fste.ac.ma (E.H. El Kinani).

1 Junior Associate at the Abdus Salam ICTP.2 Permanent address.

(i) The generalized deformed oscillator algebras(GDOA’s) [1–3], generated by the unit, creation, anni-hilation, and number operators(I, a†, a,N) satisfyingthe Hermiticity conditions(a†)† = a,N† =N , and thecommutation relations[N,a†]= a†, [N,a] = −a,

(1)[a, a†]

q= aa† − qa†a = F(N),

where q is some real number andF(N) is someHermitian, analytic function.

(ii) The G-extended oscillator algebras, whereGis some finite group, appeared in connection withn-particle integrable systems. For example, in thecase of Calogero model [4–6]G is the symmetricgroup Sn. For two particles,S2 is nothing but thecyclic group of order 2;C2 = {I,K |K2 = I } and theobtainingS2-extended oscillator algebra is generatedby the operators(I, a, a†,N andK) subject to theHermiticity conditions(a†)† = a, N† = N , K† =

0375-9601/02/$ – see front matter 2002 Elsevier Science B.V. All rights reserved.PII: S0375-9601(02)00291-8

260 E.H. El Kinani / Physics Letters A 296 (2002) 259–264

K−1, and the relations

(2)[N,a†]= a†, [N,K] = 0, K2 = I,

(3)[a, a†]= I + κK (κ ∈R), a†K = −Ka†,

together with their Hermitian conjugates.In this situation the Abelian groupS2 can be real-

ized in terms of Klein operatorK = (−1)N , whereNdenotes the number operators. Hence theS2-extendedoscillator algebra becomes a generalized deformedoscillator algebras, whereG(N) = I + κ(−1)N andq = 1, and known as the Calogero–Vasiliev algebraor also modified oscillator algebra [7,8]. By replac-ing C2 by the cyclic group of orderλ, i.e., Cλ ={I,K,K2, . . . ,Kλ−1}, one gets a new class ofG-extended oscillator algebras, generalizing the one de-scribing the two-particle Calogero model.

The Letter is organized as follows: in Section 2we review some basic notions concerning theCλ-extended oscillator algebras. Section 3 is devoted tothe construction of the FFZ and the quantumUt(sl(2))algebras. We propose the new deformed Virasoroalgebra in Section 4. Concluding remarks are given inthe last section.

2. Review of the Cλ-extended oscillator algebrasand its properties

In this section we briefly review the relevant defi-nitions and results regarding theCλ-extended oscilla-tor algebras (for more details see Ref. [9] and refer-ences quoted therein). TheCλ-extended oscillator al-gebrasAλ, whereλ take any value in the set{2,3, . . .},is defined as an algebra generated by the operators(I, a, a†,N andK) subject to the Hermiticity condi-tions (a†)† = a, N† = N , K† = K−1, and the rela-tions

(4)[N,a†]= a†, [N,K] = 0, Kλ = I,

[a, a†]= I +

λ−1∑µ=1

κµKµ,

(5)a†K = e−2πi/λKa†,

together with their Hermitian conjugates, whereκµ aresome complex parameters restricted by the conditionsκ∗µ = κλ−µ, andK is the generator of cyclic groupCλ.

For λ = 2 we obtain the Calogero–Vasiliev algebra,characterized by the commutation relations (2), (3).

Now let us examine the connection between theCλ-extended oscillator algebras and the generalizeddeformed algebras (GDOA’s). To begin, note that thecyclic groupCλ hasλ inequivalent unitary irreduciblematrix representationsΓ ν (ν = 0,1,2, . . . , λ − 1),which are one dimensional and such thatΓ ν(Kµ) =exp(2πiνµ/λ), for µ = 0,1,2, . . . , λ − 1. Hence theprojection operators on the carrier space ofΓ ν may bewritten as

(6)

Pµ = 1

λ

λ−1∑ν=0

(Γ µ

(Kν))∗Kν = 1

λ

λ−1∑ν=0

e−2πiµν/λKν,

and conversely we have

(7)Kν =λ−1∑µ=0

e2iπνµ/λPµ.

Then algebraAλ (Eqs. (4), (5)) can be rewritten interms ofI , a, a†, N andPµ = P †

µ as follows:[N,a†]= a†, [N,Pµ] = 0,

(8)λ−1∑µ=0

Pµ = I,

[a, a†]= I +

λ−1∑µ=0

αµPµ, a†Pµ = Pµ+1a†,

(9)PµPν = δµ,νPν,with the conventionsBµ′ = Bµ if µ′ = µ mod λ(whereB represents operators or parameters indexedbyµ′, µ). The parametersαµ are given by

(10)αµ =λ−1∑ν=1

exp(2iπµν/λ)κν,

restricted by the condition∑λ−1µ=0αµ = 0. Hence, we

may eliminate one of them, for instance,αλ−1. Inthis situation the cyclic group generatorK and theprojection operatorsPµ can be realized in terms ofNas

K = e2πiN/λ,

(11)

Pµ = 1

λ

λ−1∑ν=0

e2πiν(N−µ)/λ (µ= 0,1,2, . . . , λ− 1),

E.H. El Kinani / Physics Letters A 296 (2002) 259–264 261

respectively. With such a choice, theCλ-extended os-cillator algebras becomes a (GDOA’s), characterizedby q = 1 andG(N)= I +∑λ−1

µ=0αµPµ, wherePµ isgiven by the above equation.

In the bosonic Fock space representation [9], wemay consider the bosonic oscillator Hamiltonian, de-fined as usual by

(12)H0 = 1

2

{a, a†},

which can be rewritten in terms of the projection op-erators as

(13)H0 =N + 1

2I +

λ−1∑µ=0

γµPµ,

whereγ0 = (1/2)α0 andγµ =∑µ−1ν=0 αν+(1/2)αµ for

all µ= 1,2, . . . , λ− 1.The eigenvectors ofH0 are the states|n〉 = |kλ +

µ〉, obtained from the vacuum state|0〉, by succes-sive application of the creation operatora†. The corre-sponding eigenvalues are given by

Ekλ+µ = kλ+µ+ γµ + 1

2,

(14)k = 0,1, . . . , µ= 0,1, . . . , λ− 1.

In each subspace of theZλ-graded Fock space, thespectrum ofH0 is therefore harmonic, but theλ infi-nite sets of equally spaced energy levels, correspond-ing to µ = 0,1,2, . . . , λ− 1, may be shifted with re-spect to each other by some amounts depending uponthe algebra parametersα0, . . . , αλ−2, through their lin-ear combinationsγµ, µ= 0,1, . . . , λ− 1.

In the case of Calogero–Vasiliev oscillator, thesituation becomes very simple and coincides with thatof the modified harmonic oscillator.

3. Cλ-extended oscillator algebras realization ofthe FFZ and the Ut(sl(2)) symmetries

3.1. Cλ-extended oscillator algebras realization ofthe FFZ symmetry

Before going on, we would like to give a short re-view concerning thesdiff (X2n); algebra of volume-preserving diffeomorphisms on smooth manifoldX2n.Let X2n be a 2n-dimensional symplectic manifold

with a symplectic structureωab, which can be repre-sented in terms of the canonical constant antisymmet-ric 2n× 2n matrix. Thensdiff (X2n) is defined as

sdiff(X2n)

(15)

={φ(σ) ∈ C∞(M) |[φ1(σ ),φ2(σ )

]= ωab ∂φ1

∂σa

∂φ2

∂σb

},

where σ = (σ1, . . . , σ2n) denotes the correspondinglocal coordinates onX2n. In the simplest caseX2n =T 2 = S1 × S1, the Lie algebra elements ofsdiff (S1 ×S1) are given by

(16)φn(σ)= exp(n × σ),wheren × σ = n1σ1 + n2σ2, thensdiff (T 2) takes thefollowing form:

(17)[φm, φn] = (m-n)φm+n,

wherem-n =m1n2 − n1m2.This algebra has been studied first by Arnold [10]

and investigated by many other authors in the theoryof relativistic surfaces (see [11] for more details).

The FFZ algebra or trigonometric sine algebra isdefined as the quantum deformation of the Lie algebrasdiff (S1 × S1), which is generated by the generatorsTm satisfying the following commutation relations:

(18)

[Tm, Tn] = −2i sin

(2π

h(m-n)

)Tm+n, h ∈ C∗.

Note that the limith→ 0 reproduces the algebraof area preserving algebra on the torus (Eq. (17)). An-other approach to the definition of the above algebrais based on the ideas of noncommutative geometry[12]. Precisely, on the quantum two torus which is de-fined as an associativeC∗-algebra generated by twounitary generatorsU1 andU2 satisfying the relationsU1U2 = qU2U1, whereq = eih.

Now we turn to present theCλ-extended oscillatoralgebras realization of the FFZ algebra. To begin with,let us define the following operators depending on thepair (m1,m2) and the operatorsa† andK:

(19)Tm = eiπm1m2/λ(a†)m1(K)m2.

Before going on, let us discuss the problem of thenegative powers of the creation operatorsa†, which

262 E.H. El Kinani / Physics Letters A 296 (2002) 259–264

makes this construction formal. Indeed, to overcomethis difficulty one consider the Bargmann represen-tation of theCλ-extended oscillator algebras givenin [13]. With such representation the generators ofCλ-extended oscillator algebras are identified with differ-ential operators.

Using relations (4), (5), one obtains

(20)TmTn = e−iπm-n/λTm+n.

From the above equation, one easily gets the fol-lowing relations:

(21)[Tm, Tn] = −2i sin

(π

λ(m-n)

)Tm+n.

So, theT ’s satisfy the FFZ algebra (18), wherewe have used the following change 2λ = h. In whatfollows, we will generalize this construction for thequantumUt(sl(2)) algebra.

3.2. Cλ-extended oscillator algebras realization ofthe Ut (sl(2)) symmetry

First let us recall that theUt(sl(2)) algebra emergesin several contexts, e.g., in sine-Gordon theory [14],in Chern–Simon theory [15], and recently it wasuncovered in the Landau problem which is intimatelyconnected to the problem of fractional quantum Halleffect [16]. It is well known [15,17] that the FFZalgebra lead to the quantumUt (sl(2)) algebra. Relyingon this fact, we present theCλ-extended oscillatorrealization of this latter. To start, let us recall that theUt (sl(2)) is defined as a complex unital associativealgebra overC(q), the field of fraction for the ringof formal power series in the indeterminateq (q �=0,1), generated by the generatorsX±, H andH−1

satisfying the following commutation relations:

H−1H =HH−1 = 1, HX±H−1 = t±2X±,

(22)[X+,X−] = H −H−1

t − t−1 .

Let us present the following construction depend-ing on the pair (m,n) and the generatorsT , X±, HandH−1:

X+ = 1

t − t−1(Sm + Sn),

X− = 1

t − t−1(S−m + S−n),

(23)H = Sm−n, H−1 = Sn−m,

where the deformation parametert = exp(−iπ ×(m-n)). Calculating the commutation relations forX±,H andH−1 using Eqs. (19)–(21), one gets easilythe commutation relations forUt(sl(2)).

4. Cλ-deformed Virasoro algebra

In this section, we introduce the new deformedVirasoro algebra using theCλ-extended oscillatorgenerators. To begin with, recall that the Virasoroalgebra termed Witt algebra or also conformal algebrawas first introduced in the context of string theoriesand it is relevant to any theory in 2-dimensional space–times which possesses conformal invariance. The Wittalgebra is the complexification of the Lie algebraVect(S1). An element ofW is a linear combination ofthe elements of the formeinθ (d/dθ), whereθ is a realparameter and the Lie bracket onW is given by

(24)

[eimθ

d

dθ, einθ

d

dθ

]= i(m− n)ei(n+m)θ d

dθ.

It is rather convenient to consider an embedding ofthe circle into complex planeC with the coordinatesz,so thatz= eiθ and the element of the basisem (m ∈ Z)are expressed asem = −zk+1∂z. In this basis thecommutation relations have the following form:

(25)[em, en] = (m− n)em+n.

On the other hand, the deformation (q-deformation)of this algebra was first introduced by Curtright andZachos [18] and investigated on many occasions bymany authors [19–21], and defined by the followingq-commutation relations:

[em, en]q = qm−nemen − qn−menem(26)= (qm−n − qn−m)

q − q−1 em+n.

Turn now to the construction of theCλ-deformedVirasoro algebra. To do this, we will adopt the ap-proach for undeformed case [22] where the genera-torsem are constructed from one undeformed oscilla-tor paira† anda as the infinite-dimensional extensionof the following realization ofsp(2)∼ o(2,1):

(27)e−1 = a, e0 = (a†)a, e1 = (

a†)2a.

E.H. El Kinani / Physics Letters A 296 (2002) 259–264 263

The extension to positive indicesm is straightfor-ward:

(28)em(κ)=(a†)m+1

a,

and for the negative values(m < −1) the generatorsem(κ) are described by the nonanalytic dependence(monomials ofa† with negatives powers), which actsas the differential operators in the Bargmann represen-tation [13].

From the following commutation relation betweenthe generatorsa†, a andK:

[a, a†]= I +

λ−1∑r=1

κrKr,

(29)a†K = e−2πi/λKa†,

one obtain easily after algebraic manipulation thefollowing relations:

(30)[a,(a†)m]=

(m+

λ−1∑r=1

frκrKr

)(a†)m−1

,

where thefr are given by

fr = 1+ e−r(−2πi/λ)+ e−2r(−2πi/λ)+ · · ·+ e−(m−1)r(−2πi/λ),

(31)f1 = 1.

Then from the previous equations, one gets the com-mutation relations for the generatorsem(κ):[em(κ), en(κ)

]= (m− n)em+n(κ)

(32)

+λ−1∑r=1

(e−(n+1)r(−2πi/λ)− e−(m+1)r(−2πi/λ))× κrKrem+n(κ),

which goes to the ordinary Virasoro algebra forκr → 0. However, whenκr �= 0 is something new, oneasks about the commutation relations betweenK andthe generatorsem(κ); thanks to Eqs. (5), (30), oneeasily finds

(33)[em(κ),K

]= g(κ)em(κ)K,where g(κ) = 1 − exp(2πi(m + 1)/λ). In the caseof the Calogero–Vasiliev case,λ = 2, relations (30)

become

(34)

[a,(a†)m]=

(m+ 1

2

(1− (−1)m

)κ1K

)(a†)m−1

,

and the commutation relations between the generatorsem(κ) are[em(κ), en(κ)

]= (m− n)em+n(κ)

(35)

+ 1

2

((−1)n − (−1)m

)κ1Kem+n(κ),

which is theK-deformed Virasoro algebra introducedin [23]. In this case, we haveg(2) = (1 + (−1)m);hence form odd, the generatorsem commute with theoperatorK, form even we haveg(2)= 2.

5. Conclusion

In this Letter, we have presented the realization ofFFZ algebra in terms of theCλ-extended oscillator al-gebras, we have shown how this realization lead to ob-tain the realization of th quantumUt(sl(2)) algebra.Otherwise, we have presented the new deformed Vira-soro algebra, which we call theCλ-deformed Virasoroalgebra. It is interesting to investigated these new al-gebraic structures in the Calogero–Vasiliev model. Fi-nally, note that in the same way one can construct theCλ-deformedW -algebras.

Acknowledgements

I would like to thank Prof. Randjbar-Daemi for hisinvitation to the High Energy Section of the AbdusSalam Centre for Theoretical Physics, Trieste, Italy,where this work was done. The author would liketo thank the referees for their useful suggestions andremarks.

References

[1] L.C. Biedenhrn, J. Phys. A 22 (1989) L873;A.J. Macfarlane, J. Phys. A 22 (1982) 4581.

[2] A. Junussis, J. Phys. A 26 (1982) L233.[3] C. Quesne, N. Vansteenkiste, J. Phys. A 28 (1995) 7019;

C. Quesne, N. Vansteenkiste, Helv. Phys. Acta 69 (1996) 141;

264 E.H. El Kinani / Physics Letters A 296 (2002) 259–264

C. Quesne, N. Vansteenkiste, Czech. J. Phys. 47 (1997) 115,and references quoted therein.

[4] A.P. Polychronakos, Phys. Rev. Lett. 69 (1992) 703.[5] F. Calogero, J. Math. Phys. 10 (1969) 2191;

F. Calogero, J. Math. Phys. 12 (1971) 419.[6] L. Brink, T.H. Hansson, M.A. Vasiliev, Phys. Lett. B 286

(1992) 109.[7] M.A. Vasiliev, Int. J. Mod. Phys. A 6 (1991) 1115.[8] T. Brzezinski, I.L. Egusquiza, A.J. Macfarlane, Phys. Lett.

B 311 (1993) 202.[9] C. Quesne, N. Vansteenkiste, Phys. Lett. A 240 (1998) 21;

C. Quesne, N. Vansteenkiste, Int. J. Theor. Phys. Lett. 39(2000) 1175.

[10] V.I. Arnold, Mathematical Methods of Classical Mechanics,Springer, 1979;V.I. Arnold, Ann. Inst. Fourier XVI (1969) 319.

[11] I. Bars, C.N. Pope, E. Sezgin, Phys. Lett. B 210 (1989) 85;E. Bergshoeff, C.N. Pope, E. Sezgin, X. Shen, Phys. Lett.B 240 (1990) 105.

[12] E.H. El Kinani, Mod. Phys. Lett. A 12 (1997) 1589.

[13] C. Quesne, quant-ph/0108126.[14] D. Bernard, A. Le Clair, Comm. Math. Phys. 142 (1991) 99.[15] I.I. Kogan, Int. J. Mod. Phys. A 9 (1994) 3887;

H. Sato, Mod. Phys. Lett. A 9 (1994) 451.[16] R.B. Laughlin, Phys. Rev. Lett. 50 (1988) 2677.[17] E.H. El Kinani, Phys. Lett. B 383 (1996) 403;

E.H. El Kinani, Mod. Phys. Lett. A 14 (1999) 1609;E.H. El Kinani, Mod. Phys. Lett. A 15 (2000) 2139;E.H. El Kinani, Rep. Math. Phys. 47 (2001) 49.

[18] T. Curtright, C. Zachos, Phys. Lett. B 243 (1990) 237.[19] M. Chaichian, D.E. Ilinas, Z. Popowicz, Phys. Lett. B 248

(1990) 95;M. Chaichian, Z. Popowicz, P. Presnajder, Phys. Lett. B 249(1990) 185.

[20] N. Aizawa, H. Sato, Phys. Lett. B 256 (1991) 185.[21] E.H. El Kinani, M. Zakkari, Phys. Lett. B 357 (1995) 105.[22] M. Chaichian, P. Kulish, J. Lukierski, Phys. Lett. B 237 (1990)

401.[23] E.H. El Kinani, Int. J. Theor. Phys. 39 (2000) 1457.