Searcharxiv⌕ Search

arXiv subjects

C. Quesne

Publications and source records attributed to C. Quesne.

144 records · Page 8Linked to original sources
arXiv1997

Coloured quantum universal enveloping algebras

We define some new algebraic structures, termed coloured Hopf algebras, by combining the coalgebra structures and antipodes of a standard Hopf algebra set $\cal H$, corresponding to some parameter set $\cal Q$, with the transformations of an algebra isomorphism group $\cal G$, herein called colour group. Such transformations are labelled by some colour parameters, taking values in a colour set $\cal C$. We show that various classes of Hopf algebras, such as almost cocommutative, coboundary, quasitriangular, and triangular ones, can be extended into corresponding coloured algebraic structures, and that coloured quasitriangular Hopf algebras, in particular, are characterized by the existence of a coloured universal $\cal R$-matrix, satisfying the coloured Yang-Baxter equation. The present definitions extend those previously introduced by Ohtsuki, which correspond to some substructures in those cases where the colour group is abelian. We apply the new concepts to construct coloured quantum universal enveloping algebras of both semisimple and nonsemisimple Lie algebras, considering several examples with fixed or varying parameters. As a by-product, some of the matrix representations of coloured universal $\cal R$-matrices, derived in the present paper, provide new solutions of the coloured Yang-Baxter equation, which might be of interest in the context of integrable models.

q-alg↗
arXiv1997

Coloured Hopf algebras

Quite recently, a ``coloured'' extension of the Yang-Baxter equation has appeared in the literature and various solutions of it have been proposed. In the present contribution, we introduce a generalization of Hopf algebras, to be referred to as coloured Hopf algebras, wherein the comultiplication, counit, and antipode maps are labelled by some colour parameters. The latter may take values in any finite, countably infinite, or uncountably infinite set. A straightforward extension of the quasitriangularity property involves a coloured universal ${\cal R}$-matrix, satisfying the coloured Yang-Baxter equation. We show how coloured Hopf algebras can be constructed from standard ones by using an algebra isomorphism group, called colour group. Finally, we present two examples of coloured quantum universal enveloping algebras.

q-alg↗
arXiv1997

Coloured Hopf algebras

Some new algebraic structures related to the coloured Yang-Baxter equation, and termed coloured Hopf algebras, are reviewed. Coloured quantum universal enveloping algebras of Lie algebras are defined in this context. An extension to the coloured graded Yang-Baxter equation and coloured Hopf superalgebras is also presented. The coloured two-parameter quantum universal enveloping algebra of $gl(1/1)$ is considered as an example.

q-alg↗
arXiv1997

An exactly solvable three-particle problem with three-body interaction

The energy spectrum of the three-particle Hamiltonian obtained by replacing the two-body trigonometric potential of the Sutherland problem by a three-body one of a similar form is derived. When expressed in appropriate variables, the corresponding wave functions are shown to be expressible in terms of Jack polynomials. The exact solvability of the problem with three-body interaction is explained by a hidden sl(3,\R) symmetry.

hep-th↗
arXiv1997

Zero-energy states for a class of quasi-exactly solvable rational potentials

Quasi-exactly solvable rational potentials with known zero-energy solutions of the Schro\" odinger equation are constructed by starting from exactly solvable potentials for which the Schr\" odinger equation admits an so(2,1) potential algebra. For some of them, the zero-energy wave function is shown to be normalizable and to describe a bound state.

quant-ph↗
arXiv1997

Nonlinear deformed su(2) algebras involving two deforming functions

The most common nonlinear deformations of the su(2) Lie algebra, introduced by Polychronakos and Ro\v cek, involve a single arbitrary function of J_0 and include the quantum algebra su_q(2) as a special case. In the present contribution, less common nonlinear deformations of su(2), introduced by Delbecq and Quesne and involving two deforming functions of J_0, are reviewed. Such algebras include Witten's quadratic deformation of su(2) as a special case. Contrary to the former deformations, for which the spectrum of J_0 is linear as for su(2), the latter give rise to exponential spectra, a property that has aroused much interest in connection with some physical problems. Another interesting algebra of this type, denoted by ${\cal A}^+_q(1)$, has two series of (N+1)-dimensional unitary irreducible representations, where N=0, 1, 2, .... To allow the coupling of any two such representations, a generalization of the standard Hopf axioms is proposed. The resulting algebraic structure, referred to as a two-colour quasitriangular Hopf algebra, is described.

q-alg↗
arXiv1997

Representation Theory of Generalized Deformed Oscillator Algebras

The representation theory of the generalized deformed oscillator algebras (GDOA's) is developed. GDOA's are generated by the four operators ${1,a,a^†,N}$. Their commutators and Hermiticity properties are those of the boson oscillator algebra, except for $[a, a^†]_q = G(N)$, where $[a,b]_q = a b - q b a$ and $G(N)$ is a Hermitian, analytic function. The unitary irreductible representations are obtained by means of a Casimir operator $C$ and the semi-positive operator $a^† a$. They may belong to one out of four classes: bounded from below (BFB), bounded from above (BFA), finite-dimentional (FD), unbounded (UB). Some examples of these different types of unirreps are given.

q-alg↗
arXiv1997

A nonlinear deformed su(2) algebra with a two-colour quasitriangular Hopf structure

Nonlinear deformations of the enveloping algebra of su(2), involving two arbitrary functions of J_0 and generalizing the Witten algebra, were introduced some time ago by Delbecq and Quesne. In the present paper, the problem of endowing some of them with a Hopf algebraic structure is addressed by studying in detail a specific example, referred to as ${\cal A}^+_q(1)$. This algebra is shown to possess two series of (N+1)-dimensional unitary irreducible representations, where N=0, 1, 2, .... To allow the coupling of any two such representations, a generalization of the standard Hopf axioms is proposed by proceeding in two steps. In the first one, a variant and extension of the deforming functional technique is introduced: variant because a map between two deformed algebras, su_q(2) and ${\cal A}^+_q(1)$, is considered instead of a map between a Lie algebra and a deformed one, and extension because use is made of a two-valued functional, whose inverse is singular. As a result, the Hopf structure of su_q(2) is carried over to ${\cal A}^+_q(1)$, thereby endowing the latter with a double Hopf structure. In the second step, the definition of the coproduct, counit, antipode, and R-matrix is extended so that the double Hopf algebra is enlarged into a new algebraic structure. The latter is referred to as a two-colour quasitriangular Hopf algebra because the corresponding R-matrix is a solution of the coloured Yang-Baxter equation, where the `colour' parameters take two discrete values associated with the two series of finite-dimensional representations.

q-alg↗
arXiv1996

Representation theory of deformed oscillator algebras

The representation theory of deformed oscillator algebras, defined in terms of an arbitrary function of the number operator~$N$, is developed in terms of the eigenvalues of a Casimir operator~$C$. It is shown that according to the nature of the $N$ spectrum, their unitary irreducible representations may fall into one out of four classes, some of which contain bosonic, fermionic or parafermionic Fock-space representations as special cases. The general theory is illustrated by classifying the unitary irreducible representations of the Arik-Coon, Chaturvedi-Srinivasan, and Tamm-Dancoff oscillator algebras, which may be derived from the boson one by the recursive minimal-deformation procedure of Katriel and Quesne. The effects on non-Fock-space representations of the minimal deformation and of the quommutator-commutator transformation, considered in such a procedure, are studied in detail.

q-alg↗
arXiv1996

Parasupersymmetric Quantum Mechanics with Generalized Deformed Parafermions

A superposition of bosons and generalized deformed parafermions corresponding to an arbitrary paraquantization order $p$ is considered to provide deformations of parasupersymmetric quantum mechanics. New families of parasupersymmetric Hamiltonians are constructed in connection with two examples of su(2) nonlinear deformations such as introduced by Polychronakos and Ro\v cek.

hep-th↗
arXiv1995

Recursively minimally-deformed oscillators

A recursive deformation of the boson commutation relation is introduced. Each step consists of a minimal deformation of a commutator $[a,\ad]=f_k(\cdots;\no)$ into $[a,\ad]_{q_{k+1}}=f_k(\cdots;\no)$, where $\cdots$ stands for the set of deformation parameters that $f_k$ depends on, followed by a transformation into the commutator $[a,\ad]=f_{k+1}(\cdots,\, q_{k+1};\no)$ to which the deformed commutator is equivalent within the Fock space. Starting from the harmonic oscillator commutation relation $[a,\ad]=1$ we obtain the Arik-Coon and the Macfarlane-Biedenharn oscillators at the first and second steps, respectively, followed by a sequence of multiparameter generalizations. Several other types of deformed commutation relations related to the treatment of integrable models and to parastatistics are also obtained. The ``generic'' form consists of a linear combination of exponentials of the number operator, and the various recursive families can be classified according to the number of free linear parameters involved, that depends on the form of the initial commutator.

q-alg↗
arXiv1995

Supersymmetry and superalgebra for the two-body system with a Dirac oscillator interaction

Some years ago, one of the authors~(MM) revived a concept to which he gave the name of single-particle Dirac oscillator, while another~(CQ) showed that it corresponds to a realization of supersymmetric quantum mechanics. The Dirac oscillator in its one- and many-body versions has had a great number of applications. Recently, it included the analytic expression for the eigenstates and eigenvalues of a two-particle system with a new type of Dirac oscillator interaction of frequency~$ω$. By considering the latter together with its partner corresponding to the replacement of~$ω$ by~$-ω$, we are able to get a supersymmetric formulation of the problem and find the superalgebra that explains its degeneracy.

hep-th↗
arXiv1995

Comment on ``Generalized $q$-oscillators and their Hopf structures''

In a recent paper (1994 {\sl J.\ Phys.\ A: Math.\ Gen.\ }{\bf 27} 5907), Oh and Singh determined a Hopf structure for a generalized $q$-oscillator algebra. We prove that under some general assumptions, the latter is, apart from some algebras isomorphic to su$_q$(2), su$_q$(1,1), or their undeformed counterparts, the only generalized deformed oscillator algebra that supports a Hopf structure. We show in addition that the latter can be equipped with a universal $\cR$-matrix, thereby making it into a quasitriangular Hopf algebra.

q-alg↗
arXiv1995

Generalization of a result of Matsuo and Cherednik to the Calogero-Sutherland- Moser integrable models with exchange terms

A few years ago, Matsuo and Cherednik proved that from some solutions of the Knizhnik-Zamolodchikov (KZ) equations, which first appeared in conformal field theory, one can obtain wave functions for the Calogero integrable system. In the present communication, it is shown that from some solutions of generalized KZ equations, one can construct wave functions, characterized by any given permutational symmetry, for some Calogero-Sutherland-Moser integrable models with exchange terms. Such models include the spin generalizations of the original Calogero and Sutherland ones, as well as that with $δ$-function interaction.

hep-th↗
arXiv1995

Exchange operators and extended Heisenberg algebra for the three-body Calogero-Marchioro-Wolfes problem

The exchange operator formalism previously introduced for the Calogero problem is extended to the three-body Calogero-Marchioro-Wolfes one. In the absence of oscillator potential, the Hamiltonian of the latter is interpreted as a free particle Hamiltonian, expressed in terms of generalized momenta. In the presence of oscillator potential, it is regarded as a free modified boson Hamiltonian. The modified boson operators are shown to belong to a $D_6$-extended Heisenberg algebra. A proof of complete integrability is also provided.

hep-th↗
arXiv1995

Knizhnik-Zamolodchikov equations and the Calogero-Sutherland-Moser integrable models with exchange terms

It is shown that from some solutions of generalized Knizhnik-Zamolodchikov equations one can construct eigenfunctions of the Calogero-Sutherland-Moser Hamiltonians with exchange terms, which are characterized by any given permutational symmetry under particle exchange. This generalizes some results previously derived by Matsuo and Cherednik for the ordinary Calogero-Sutherland-Moser Hamiltonians.

hep-th↗
arXiv1995

On some nonlinear extensions of the angular momentum algebra

Deformations of the Lie algebras so(4), so(3,1), and e(3) that leave their so(3) subalgebra undeformed and preserve their coset structure are considered. It is shown that such deformed algebras are associative for any choice of the deformation parameters. Their Casimir operators are obtained and some of their unitary irreducible representations are constructed. For vanishing deformation, the latter go over into those of the corresponding Lie algebras that contain each of the so(3) unitary irreducible representations at most once. It is also proved that similar deformations of the Lie algebras su(3), sl(3,R), and of the semidirect sum of an abelian algebra t(5) and so(3) do not lead to associative algebras.

q-alg↗