Yangian Realization for Dirac Oscillator

We investigate the realizations of Yangian algebra for a Dirac oscillator. Applying the representation theory of Y(sl(2)) to Dirac oscillator, shift operaors for different energy levelsfor this system are obtained.     

Yangian Realization for Dirac Oscillator

We investigate the realizations of Yangian algebra for a Dirac oscillator. Applying the representation theory of Y(sl(2)) to Dirac oscillator, shift operators for different energy levels for this system are obtained.     

Boson realization of YangiansY(sl(2)) andY(sl(3))

Studying the algebraic structure of the YangiansY(sl(2)), andY(sl(3)) we present their boson realizations. In the case ofY(sl(2)) we give the realization by using 1-canonical boson pair and two parameters and in theY(sl(3)) with 2-canonical boson pairs, two parameters and subalgebrasl(2). Substituting the realization ofU(sl(2)), we can obtain the pure boson realization by 3-canonical pairs and three parameters.Presented at the 4th Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.     

Deformed yangians and integrable models

Twisted Hopf algebraslξ(2) gives rise to a deformation of the Yangiany(sl(2)). The corresponding deformations of the integrableXXX-spin chain and the Gaudin model are discussed. Presented at 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrables Systems”, Prague, 19–21 June 1997. This research was supported by the Swedish National Research Council and RFFI grant 96-01-00851.     

Operator coproduct-realization of quantum group transformations in two-dimensional gravity I

A simple connection between the universalR matrix ofU _q(sl(2)) (for spins 1/2 andJ) and the required form of the coproduct action of the Hilbert space generators of the quantum group symmetry is put forward. This leads us to an explicit operator realization of the coproduct action on the covariant operators. It allows us to derive the expected quantum group covariance of the fusion and braiding matrices, although it is of a new type: the generators depend upon worldsheet variables, and obey a new central extension of theU _q(sl(2)) algebra realized by (what we call) fixed point commutation relations. This is explained by showing on a general ground that the link between the algebra of field transformations and that of the coproduct generators is much weaker than previously thought. The central charges of our extendedU _q(sl(2)) algebra, which includes the Liouville zero-mode momentum in a non-trivial way, are related to Virasoro-descendants of unity. We also show how our approach can be used to derive the Hopf algebra structure of the extended quantum-group symmetry related to the presence of both of the screening charges of 2D gravity.Partially supported by the EC contracts CHRXCT920069 and CHRXCT920035.Unité Propre du Centre National de la Recherche Scientifique, associée à l'École Normale Supérieure et à l'Université de Paris-Sud.     

Quasi-exact solvability beyond the <Emphasis Type="Italic">sl</Emphasis>(2) algebraization

We present evidence to suggest that the study of one-dimensional quasi-exactly solvable (QES) models in quantum mechanics should be extended beyond the usual sl(2) approach. The motivation is twofold: We first show that certain quasi-exactly solvable potentials constructed with the sl(2) Liealgebraic method allow for a new larger portion of the spectrum to be obtained algebraically. This is done via another algebraization in which the algebraic Hamiltonian cannot be expressed as a polynomial in the generators of sl(2). We then show an example of a new quasi-exactly solvable potential which cannot be obtained within the Lie algebraic approach. The text was submitted by the authors in English.     

On factorizing twists for the Yangian and the quantum affine algebra

We calculate explicit expressions for factorizing Drinfel’d twists in evaluation representations of the YangianY(sl₂) and of the quantum affine algebraUq . From the twists we derive a closed and representation-independent form of theR-matrices in these representations. Employing a carefully chosen basis it is possible to recover the results for the Yangian (rational case) as theq → 1 limit of the expressions for the quantum affine algebra (trigonometric case). Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001.     

sl(2) variational schemes for solving one class of nonlinear quantum models

Hamiltonians of a wide-spread class of G_inv invariant nonlinear quantum models, including multiboson and frequency conversion ones, are expressed as nonlinear functions of sl(2) generators. This enables us to use variational schemes, based on sl(2) generalized coherent states as trial functions, to obtain new (sl(2) cluster) quasi-classical solutions of spectral and evolution tasks which smoothly approximate exact ones and catch explicitly quantum cooperative features of the models under study (in contrast with standard independent-mode quasi-classical approximations). In such a way new analytical expressions are found for the energy spectra which are beyond both equidistant and quasi-equidistant ones obtained earlier.     

Unitary representations of the quantum group SU q (1,1): Structure of the dual space ofU q (sl(2))

Real forms of the quantum universal enveloping algebraU _q (sl(2)) and a topological quantum group associated with this algebra are discussed.     

q-deformed conformal and Poincaré algebras on quantum 4-spinors

We investigate quantum deformation of conformal algebras by constructing the quantum space forsl _q (4). The differential calculus on the quantum space and the action of the quantum generators are studied. We derive deformedsu(2,2) algebra from the deformedsl(4) algebra using the quantum 4-spinor and its conjugate spinor. The quantum 6-vector inso _q (4,2) is constructed as a tensor product of two sets of 4-spinors. We obtain theq-deformed conformal algebra with the suitable assignment of the generators which satisfy the reality condition. The deformed Poincaré algebra is derived through a contraction procedure.Work partially supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (#030083)     

Drinfel'd Twists and Functional Bethe Ansatz

Using the Functional Bethe Ansatz technique, factorizing Drinfel'd twists for any finite dimensional irreducible representations of the Yangian Y(sl₂) are constructed.     

Embedding Uq(sl(2)) and Sine Algebras in Generalized Clifford Algebras

We establish the connection between certain quantum algebras and generalizedClifford algebras (GCA). To be precise, we embed the quantum tori Lie algebraand U _q (sl(2)) in GCA.     

Composition of kinetic momenta: The(sl(2)) casecase

The tensor products of (restricted and unrestricted) finite dimensional irreducible representations of(sl(2)) are considered forq a root of unity. They are decomposed into direct sums of irreducible and/or indecomposable representations.     

Coherent state operators and n-point invariants for U q ((sl(2))

We write down a complete set of n-point U_q(sl(2)) invariants (using a polynomial basis for the irreducible finite dimensional U _q -modules) that are regular for all nonzero values of the deformation parameter q.     

On representations of the elliptic quantum groupE τ,η(sl 2)

We describe representation theory of the elliptic quantum groupE _,(sl ₂). It turns out that the representation theory is parallel to the representation theory of the YangianY(sl ₂) and the quantum loop group .We introduce basic notions of representation theory of the elliptic quantum groupE _,(sl ₂) and construct three families of modules: evaluation modules, cyclic modules, one-dimensional modules. We show that under certain conditions any irreducible highest weight module of finite type is isomorphic to a tensor product of evaluation modules and a one-dimensional module. We describe fusion of finite dimensional evaluation modules. In particular, we show that under certain conditions the tensor product of two evaluation modules becomes reducible and contains an evaluation module, in this case the imbedding of the evaluation module into the tensor product is given in terms of elliptic binomial coefficients. We describe the determinant element of the elliptic quantum group. Representation theory becomes special ifN=m+l, whereN,m,l are integers. We indicate some new features in this case.The authors were supported in part by NSF grants DMS-9400841 and DMS-9501290.     

Universal R operator with Jordanian deformation of conformal symmetry

The Jordanian deformation of sl(2) bialgebra structure is studied in view of physical applications to breaking of conformal symmetry in the high energy asymptotics of scattering. Representations are formulated in terms of polynomials, generators in terms of differential operators. The deformed R operator with generic representations is analyzed in spectral and integral forms.     

Centre and representations of small quantum superalgebras at roots of unity

When the deformation parameter is a root of unity, the centre of a quantum group can be described by a set of generators and non trivial relations. In the case ofU _q (sl(N)), these relations simply derive from the expressions of the deformed Casimir operators. In the case ofU _q (osp(1|2)), the relation is simple if we use an operator which anticommutes with the fermionic generators and whose square is the quadratic Casimir. This operator also simplifies the classification of finite dimensional irreducible representations. In the case ofU _q (sl(1|2)), the relations derive from the (infinite set of) standard Casimir operators.Presented at the 5th International Colloquium on Quantum Groups: Quantum Groups and Integrable Systems, Prague, 20–22 June 1996.     

Drinfeld twist for two-parametric deformation of gl(2) and sl(1/2)

Drinfeld twist is applied to the Lie algebra gl(2) so that a two-parametric deformation of it is obtained, which is identical to the Jordanian deformation of the gl(2) obtained by Aneva et al. The same twist element is applied to deform the Lie superalgebra sl(1/2), since the gl(2) is embedded into the sl(1/2). By making use of the FRT-formalism, we construct a deformation of the Lie supergroup SL(1/2).     

The jordanian deformation of Su(2) and clebsch-gordan coefficients

Representation theory for the Jordanian quantum algebraU _h (sl(2)) is developed using a nonlinear relation between its generators and those of sl(2). Closed form expressions are given for the action of the generators ofU _h (sl(2)) on the basis vectors of finite dimensional irreducible representations. In the tensor product of two such representations, a new basis is constructed on which the generators ofU _h (sl(2)) have a simple action. Using this basis, a general formula is obtained for the Clebsch-Gordan coefficients ofU _h (sl(2)). Some remarkable properties of these Clebsch-Gordan coefficients are derived. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.