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.     

The Embedding of Quantum Algebra U(gl(n+1)) in Irreducible Representations of U(gl(n+1))

Using a formula derived by Jimbo, we express the generators of U(gl(n+1)), the q-analogue of U(gl(n+1)) in an irreducible representation with Gel'fand basis, as functions of the generators of U(gl(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.     

The exponential map for representations ofU p,q (gl(2))

For the quantum groupGL _p,q (2) and the corresponding quantum algebraU _p,q (gl(2)) Fronsdal and Galindo [Lett. Math. Phys.27 (1993) 59] explicitly constructed the so-called universalT-matrix. In a previous paper [J. Phys. A28 (1995) 2819] we showed how this universalT-matrix can be used to exponentiate representations from the quantum algebra to get representations (left comodules) for the quantum group. Here, further properties of the universalT-matrix are illustrated. In particular, it is shown how to obtain comodules of the quantum algebra by exponentiating modules of the quantum group. Also the relation with the universalR-matrix is discussed.Presented at the 4th International Colloquium Quantum Groups and Integrable Systems, Prague, 22–24 June 1995.     

Finite-dimensional representations of the quantum superalgebraU q[gl(n/m)] and relatedq-identities

Explicit expressions for the generators of the quantum superalgebraU _q [gl(n/m)] acting on a class of irreducible representations are given. The class under consideration consists of all essentially typical representations: for these a Gel'fand-Zetlin basis is known. The verification of the quantum superalgebra relations to be satisfied is shown to reduce to a set ofq-number identities.     

Characters of U q (gl(n))-Reflection Equation Algebra

We list characters (one-dimensional representations) of the reflection equation algebra associated with the fundamental vector representation of the Drinfeld–Jimbo quantum group _q (gl(n)).     

On a nonstandard two-parametric quantum algebra and its connections withU p,q (gl(2)) andU p,q(gl(1/1))

A quantum algebraU _{p, q} (,H,X _±) associated with a nonstandardR-matrix with two deformation parameters (p, q) is studied and, in particular, its universal -matrix is derived using Reshetikhin's method. Explicit construction of the (p, q)-dependent nonstandardR-matrix is obtained through a coloured generalized boson realization of the universal -matrix of the standardU _{p, q}(gl(2)) corresponding to a nongeneric case. General finite dimensional coloured representation of the universal -matrix ofU _{p, q}(gl(2)) is also derived. This representation, in nongeneric cases, becomes a source for various (p, q)-dependent nonstandardR-matrices. Superization ofU _{p, q}(,H,X _±) leads to the super-Hopf algebraU _{p, q}(gl(1/1)). A contraction procedure then yields a (p, q)-deformed super-Heisenberg algebraU _{p, q}(sh(1)) and its universal -matrix.     

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.     

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.     

Quantum Algebra U q(gl(3)) and Nonlinear Optics

Indecomposable representations are investigated for the U _q(gl(3)) quantum algebra. The matrix elements are explicitly determined for the elementary representations, and the extremal vectors which characterize invariant subspaces are given in explicit form. Quotient spaces are used to derive other representations from the elementary representations, including the finite-dimensional irreducible representations and infinite-dimensional representations which are bounded above. Applications to nonlinear-optical phenomena are discussed.     

Characters of Finite-Dimensional Irreducible q(n) -Modules

The solution of the Kac character problem for thequeer series of Lie superalgebras q(n) is announced. An explicit algorithm which computes the character of an arbitrary finite-dimensional irreducible q(n)-module is presented. As an illustration, the correction terms to the generic character formula of Penkov (Monatsh. Math.118 (1994), 419) are written down for all finite-dimensional irreducible representations of q(n), for n4, with nongeneric character.     

On Gelfand-Zetlin modules over U q(gl n)

We construct and investigate a new large family of simple modules over U _q(gl _n).     

Irreducible unitary representations of quantum Lorentz group

A complete classification of irreducible unitary representations of a one parameter deformationS _q L(2,C) (0<q<1) ofSL(2,C) is given. It shows that in spite of a popular belief the representation theory forS _q L(2,C) is not a smooth deformation of the one forSL(2,C).     

Drinfeld Twists and Algebraic Bethe Ansatz of the Supersymmetric Model Associated with U q (gl(m|n))

We construct the Drinfeld twists (or factorizing F-matrices) of the super-symmetric model associated with quantum superalgebra U_q(gl(m|n)), and obtain the completely symmetric representations of the creation operators of the model in the F-basis provided by the F-matrix. As an application of our general results, we present the explicit expressions of the Bethe vectors in the F-basis for the U_q(gl(2|1))-model (the quantum t-J model).     

Irreducible representations of the quantum analogue of SU(2)

We give the complete set of irreducible representations of U(SU(2))_q when q is a mth root of unity. In particular, we show that their dimensions are less or equal to m. Some of them are not highest-weight representations.     

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-Oscillator realizations of the quantum superalgebrassl q (m,n) andosp q (m, 2n)

Realizations of the quantum superalgebras corresponding to theA(m, n), B(m, n), C(n+1), andD(m, n) series are given in terms of the creation and annihilation operators ofq-deformed Bose and Fermi oscillators.     

The structure of U q (sl(2, C))

The structure of the deformation U _q (sl(2, C)) is discussed. The comultiplication, all commutation relations, and a conjugation follow in a clear way form the simple SL _q (2) structure. Fundamental and spin representation are given.