On a possible algebra morphism of U q [osp(1/2n)] onto the deformed oscillator algebra W q (n)

We formulate a conjecture stating that the algebra ofn pairs of deformed Bose creation and annihilation operators is a factor algebra of U _q [osp(1/2n)], considered as a Hopf algebra, and prove it for then = 2 case. To this end, we show that for any value ofq, U _q [osp(1/4)] can be viewed as a superalgebra freely generated by two pairsB ₁ ^± ,B ₂ ^± of deformed para-Bose operators. We write down all Hopf algebra relations, an analogue of the Cartan-Weyl basis, the commutation relations between the generators and a basis in U _q [osp(1/2n)] entirely in terms ofB ₁ ^± ,B ₂ ^± .     

Finite-dimensional representations of U q (osp(1/2n)) and its connection with quantum so(2n+1)

It is shown that the quantum supergroup U _q (osp(1/2_n)) is essentially isomorphic to the quantum group U _-q (so(2n+1)) restricted to tensorial representations. This renders it straightforward to classify all the finite-dimensional irreducible representations of U _q (osp(1/2n)) at generic q. In particular, it is proved that at generic q, every-dimensional irrep of this quantum supergroup is a deformation of an osp(1/2n) irrep, and all the finite-dimensional representations are completely reducible.     

Representation theory of osp(1,2) q

We investigate the representations of the osp(1, 2) _q algebra. We derive all the finite-dimensional irreducible representations, whenq is not a root of unity. We also discuss the connection between those of osp(1, 2) _q and sl(2) _q .     

A Two-Parameter Deformed N = 2 SUSY Algebra

A two-parameter deformed N = 2 SUSY algebra is constructed by using the q-deformed bosonic and fermionic Newton oscillator algebras. The Fock space representation of the (q ₁,q ₂)-deformed N = 2 SUSY algebra is analyzed. The comparison between the algebra constructed and earlier versions of deformed N = 2 SUSY algebras is also made.     

Cartan-Weyl Basis For Quantum Affine SuperalgebraU q (\widehat{osp}(1|2))(1|2))

A Cartan-Weyl basis for the quantum affine superalgebraU _q (osp(1|2)) is constructed in an explicit form. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997. Supported by KNB grant No. 2P 30208706. Supported by Russian Foundation for Fundamental Research, grant No. 96-01-01421.     

Representation of osp (2, 2) q (2) andS-matrix of super sine-Gordon theory

We investigate the representations of the osp (2, 2) _q ⁽²⁾ algebra, which leads to theS-matrix of super sine-Gordon theory. TheS-matrix has been derived from supersymmetric conformal field theory with some assumptions. We show that the conjecturedS-matrix can be derived from the representation theory using a correspondence between the representations of osp (1, 2) _q and those of sl(2) _q .     

Quantization of U q [so(2n + 1)] with deformed para-Fermi operators

The observation thatn pairs of para-Fermi (pF) operators generate the universal enveloping algebra of the orthogonal Lie algebra so(2n + 1) is used in order to define deformed pF operators. It is shown that these operators are an alternative to the Chevalley generators. With this background U _q [so(2n + 1)] and its Cartan-Weyl generators are written down entirely in terms of deformed para-Fermi operators.     

Unitary Operator of su q (n)-Covariant Oscillator Algebra

The unitary operator of su _q (n)-covariant oscillator algebra is constructed and two types of q-coherent states are obtained explicitly.     

gl q (n) and quantum monodromy

A generalized Toda lattice based on gl(n) is considered. The Poisson brackets are expressed in terms of a Lax connection, L=–() and a classical r-matrix, {₁,₂}=[r,₁+₂}. The essential point is that the local lattice transfer matrix is taken to be the ordinary exponential, T=e; this assures the intepretation of the local and the global transfer matrices in terms of monodromy, which is not true of the T-matrix used for the sl(n) Toda lattice. To relate this exponential transfer matrix to the more manageable and traditional factorized form, it is necessary to make specific assumptions about the equal time operator product expansions. The simplest possible assumptions lead to an equivalent, factorized expression for T, in terms of operators in (an extension of) the enveloping algebra of gl(n). Restricted to sl(n), and to multiplicity-free representations, these operators satisfy the commutation relations of sl _q (n), which provides a very simple injection of sl _q (n) into the enveloping algebra of sl(n). A deformed coproduct, similar in form to the familiar coproduct on sl _q (n), turns gl(n) into a deformed Hopf algebra gl _q (n). It contains sl _q (n) as a subalgebra, but not as a sub-Hopf algebra.     

gl q,q* (n)-Covariant multimode oscillators and q-symmetric states

In this paper theql _q (n) oscillator algebra is extended to the complex deformation parameter case [gl _q,q* (n) algebra], and q-symmetric states forgl _q,q* (n)-covariant multimode oscillator system are investigated.     

Propagator for the Chebyshev q object

We propose an alternative role of the harmonic oscillator algebra. Observing that the q-deformed harmonic oscillator algebra defines the Chebyshev q object, we show that the q-free particle and the pulsed oscillator are special cases of the Chebyshev q object, characterized by a common deformation parameter q and reduced to a usual free particle as q tends to unity. For the deformed free particle, q is a real number, whereas for the pulsed oscillator it belongs to S ¹. Then, we derive the propagator for the Chebyshev q object, from which we obtain the propagators for the deformed free particle and the pulsed oscillator.     

Realization ofU q (so(N)) within the differential algebra onR q N

We realize the Hopf algebraU _q–1 (so(N)) as an algebra of differential operators on the quantum Euclidean spaceR _q ^N . The generators are suitableq-deformed analogs of the angular momentum components on ordinaryR ^N . The algebra Fun(R _q ^N ) of functions onR _q ^N splits into a direct sum of irreducible vector representations ofU _q–1 (so(N)); the latter are explicitly constructed as highest weight representations.     

Decomposition of the step operators of theλ √q (n)-covariant oscillator algebra withq as a root of unity

In this paper the decomposition of the creation and annihilation operators ofgl _√q (n)-covariant oscillator algebra is discussed when the deformation parameterq is a (s+1)-th primitive root of unity.     

The cyclic representations of the quantum superalgebra U q osp(2, 1) withq a root of unity

By generalizing De Concini and Kac's cyclic representation theory of quantum groups at roots of unity, the cyclic representations of the quantum superalgebra U _q osp(2, 1) are constructed in three classes: irreducible representations with single multiplicities, irreducible representations with the multiplicities larger than one, and indecomposable representations.This work is supported in part by the National Sciene Foundation in China.     

Differential calculus on the inhomogeneous quantum groups IGL q (n)

We obtain the inhomogeneousq-groups IGL _q (n) via a projection from GL _q (n + 1). The bicovariant differential calculus of IGL _q (n) is constructed, and the corresponding quantum Lie algebra is given explicitly.     

Heisenberg realization for U q (sl n ) on the flag manifold

We give the Heisenberg realization for the quantum algebra U _q (sl _n ), which is written by theq-difference operator on the flag manifold. We construct it from the action of U _q (sl _n ) on theq-symmetric algebraA _q (Mat _n ) by the Borel-Weil-like approach. Our realization is applicable to the construction of the free field realization for U _q [2].     

The q-boson realizations of the quantum groups U q(B n)

We give explicit realization for the quantum enveloping algebras U _q(B _n). In these formulae the generators of the algebra are expressed by means of 2n–1 canonical q-boson pairs and one auxiliary representation of U _q(B _n–1)     

Finite-dimensional irreducible representations of the quantum superalgebraU q [gl(n/1)]

It is shown that every finite-dimensional irreducible module over the general linear Lie superalgebragl(n/1) can be deformed to an irreducible module ofU _q [gl(n/1)], aq-analogue of the universal enveloping algebra ofgl(n/1). The results are extended also to all Kac modules, which in the atypical cases remain indecomposible. Within each module expressions for the transformations of the Gel'fand-Zetlin basis under the action of the algebra generators are written down. An analogoue of the Poincaré-Birkhoff-Witt theorem is formulated.     

Quantized universal enveloping superalgebra of type Q and a super-extension of the Hecke algebra

The quantized universal enveloping algebra U _q(q(n)) of the strange Lie superalgebra q(n) and a super-analogue HC _q (N) of the Hecke algebra H _q (N) are constructed. These objects are in a duality similar to the known duality between U _q (gl(n)) and H _q (N).     

su q (2)-invariant harmonic oscillator

We propose a q-deformation of the su(2)-invariant Schrödinger equation of a spinless particle in a central potential, which allows us not only to determine a deformed spectrum and the corresponding eigenstates, as in other approaches, but also to calculate the expectation values of some physically-relevant operators. Here we consider the case of the isotropic harmonic oscillator and of the quadrupole operator governing its interaction with an external field. We obtain the spectrum and wave functions both for q R⁺ and generic q S ¹, and study the effects of the q-value range and of the arbitrariness in the su _q (2) Casimir operator choice. We then show that the quadrupole operator in l = 0 states provides a good measure of the deformation influence on the wave functions and on the Hilbert space spanned by them.