GLq(N)-covariant braided differential bialgebras

We study the possibility of defining the (braided) comultiplication for the GL _q (N)-covariant differential complexes on some quantum spaces. We discover suchdifferential bialgebras (and Hopf algebras) on the bosonic and fermionic quantum hyperplanes (with additive coproduct) and on the braided matrix algebra BM _q (N) with both multiplicative and additive coproducts. The latter case is related (forN = 2) to theq-Minkowski space andq-Poincaré algebra.     

Schrödinger equations for su q (2)-invariant quantum systems

By deforming the Hamiltonian of a spinless particle in a central potential we set up su _q (2)-invariant Schrödinger equations within the usual framework of quantum mechanics. Different deformations correspond to a given Hamiltonian. We explicitly solve different stationary Schrödinger equations for the free particle and for the hydrogen atom, and compare the associated energy spectra.     

Braid group action on theq-Weyl algebra

We provide a braid group action on theq-deformed Weyl algebraW _q (n). The restriction of this action to the representations ofU _q (A _n–1 ) andU _q (C _n ) inW _q (n) is seen to agree with the braid group action introduced by Lusztig on these quantum algebras.Supported in part by the National Sciences and Engineering Research Council (NSERC) of Canada.     

q-transformation theory

We obtain the complete set of states of theq-oscillator in both configuration space and momentum space as well as the transformation between these spaces. The states as well as the matrix elements lie in the SU _q (2) algebra. To obtain transition probabilities, one must take the Woronowicz square.     

Inhomogeneous quantum groups and their quantized universal enveloping algebras

The quantum group IGL _q (N), the inhomogenization of GL _q (N), is formulated with -matrices. Theq-deformed universal enveloping algebra is constructed as the algebra of regular functionals in this formulation and contains the partial derivatives of the covariant differential calculus on the quantum space.     

SO q (n+1,n−1) as a real form of SO q (2n,ℂ)

Quantum pseudo-orthogonal groups SO _q (n+1,n–1) are defined as real forms of quantum orthogonal groups SO _q (n+1,n–1) by means of a suitable antilinear involution. In particular, the casen=2 gives a quantized Lorentz group.     

A q-analog of the sixth Painlevé equation

A q-difference analog of the sixth Painlevé equation is presented. It arises as the condition for preserving the connection matrix of linear q-difference equations, in close analogy with the monodromy-preserving deformation of linear differential equations. The continuous limit and special solutions in terms of q-hypergeometric functions are also discussed.     

Representations of the coordinate ring of GL q (3)

It is shown that finite-dimensional irreducible representations of the quantum matrix algebraM _q (3) (the coordinate ring of GL _q (3)) exist only whenq is a root of unity (q ^p = 1). The dimensions of these representations can only be one of the following values:p ³,p ³/2,p ³/4, orp ³/8. The topology of the space of states ranges between two extremes, from a three-dimensional torusS ¹ ×S ¹ ×S ¹ (which may be thought of as a generalization of the cyclic representation) to a three-dimensional cube [0, 1] × [0, 1] × [0, 1].     

Topological representations of U q (sl2(ℂ)) on the torus and the mapping class group

We compute the mapping class group action on cycles on the configuration space of the torus with one puncture, with coefficients in a local system arising in conformal field theory. This action commutes with the topological action of the quantum group U _q (sl₂()), and is given in vertex form.     

On the universalR-matrix for Uq(241-1241-1241-1)

A compact form for the universalR-matrix of U _q (sl _n ) is derived and illustrated by simple applications.     

Bicovariant differential geometry of the quantum group SL h (2)

There are only two quantum group structures on the space of two by two unimodular matrices, these are the SL _q (2) and the SL _h (2) quantum groups. The differential geometry of SL _q (2) is well known. In this Letter, we develop the differential geometry of SL _h (2), and show that the space of left invariant vector fields is three-dimensional.     

On the hopf structure of U p,q (gl(1∣1)) and the universal T-matrix of fun p,q (GL(1∣1))

The dually conjugate Hopf superalgebras Fun _p,q (GL(11)) and U _p,q (gl(11)) are studied using the Frønsdal-Galindo approach and the full Hopf structure of U _p,q (gl(11)) is extracted. A finite expression for the universal T-matrix, identified with the dual form and expressing the generalization of the exponential map of the classical groups, is obtained for Fun _p,q (GL(11)). In a representation with a colour index, the T-matrix assumes a form that satisfies a coloured graded Yang-Baxter equation.     

New boson representations of the sl q (2) with multiplicity two and new solutions to the Yang-Baxter equation atq p =1

Whenq is a root of unity, the representations of the quantum universal enveloping algebra sl _q (2) with multiplicity two are constructed from theq-deformed boson realization with an arbitrary parameter which is in a very general form and is first presented in this Letter. The new solutions to the Yang-Baxter equation are obtained from these representations through the universalR-matrix.This work is supported in part by the National Foundation of Natural Science of China.     

Left regular representation and contraction of sl q (2) to e q (2)

The left regular representation of the quantum algebras sl _q (2) and e _q (2) are discussed and shown to be related by contraction. The reducibility is studied andq-difference intertwining operators are constructed.     

SU q (2) covariant\hat R-matrices for reducible representations

We consider SU _q (2) covariant -matrices for the reducible3 1 representation. There are three solutions to the Yang-Baxter equation. They coincide with the previously known -matrices for SO _q (3) and SO _q (3, 1). Also, they are the three -matrices which can be constructed by using four different SU _q (2) doublets. Only two of the three -matrices allow a differential structure on the reducible four-dimensional quantum space.     

Coordinates of the quantum plane as q-tensor operators in (\mathfrak{s}\mathfrak{u}(2)*\mathfrak{s}\mathfrak{u}(2))

The relation between the set of transformations of the quantum plane and the quantum universal enveloping algebra U _q (u(2)) is investigated by constructing representations of the factor algebra U _q (u(2))* . The noncommuting coordinates of , on which U _q (2) * U _q (2) acts, are realized as q-spinors with respect to each U _q (u(2)) algebra. The representation matrices of U _q (2) are constructed as polynomials in these spinor components. This construction allows a derivation of the commutation relations of the noncommuting coordinates of directly from properties of U _q (u(2)). The generalization of these results to U _q (u(n)) and is also discussed.     

Self-similar potentials and theq-oscillator algebra at roots of unity

Properties of the simplest class of self-similar potentials are analyzed. Wave functions of the corresponding Schrödinger equation provide bases of representations of theq-deformed Heisenberg-Weyl algebra. When the parameterq is a root of unity, the functional form of the potentials can be found explicitly. The generalq ³ = 1 and the particularq ⁴ = 1 potentials are given by the equi-anharmonic and (pseudo) lemniscatic Weierstrass functions, respectively.     

On a possible origin of quantum groups

A Poisson bracket structure having the commutation relations of the quantum group SL _q (2) is quantized by means of the Moyal star-product on C (²), showing that quantum groups are not exactly quantizations, but require a quantization (with another parameter) in the background. The resulting associative algebra is a strongly invariant nonlinear star-product realization of the q-algebra U _q (sl(2)). The principle of strong invariance (the requirement that the star-commutator is star-expressed, up to a phase, by the same function as its classical limit) implies essentially the uniqueness of the commutation relations of U _q (sl(2)).     

The uniqueness theorem for the universal R-matrix

Up to now, the universal R-matrix for quantized Kac-Moody algebras is believed to be uniquely determined (for some ansatz) by properties of a quasi-cocommutativity and a quasi-triangularity. We prove here that the universal R-matrix (for the same ansatz) is uniquely determined by the property of the quasi-cocommutativity only. Thus, the quasi-triangular property (and the Yang-Baxter equation!) for the universal R-matrix is a consequence of the linear equation of the quasi-cocommutativity. The proof is based on properties of singular vectors in the tensor product of the Verma modules and the structure of extremal projector for quantized algebras. Explicit expressions of the universal R-matrix for quantized algebras U _q (A _inf1 ^sup(1) ) and U _q (A _inf2 ^sup(2) ) are given.

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On the representations of quantum oscillator algebra

It is shown that for q<1, the quantum oscillator algebra has a supplementary family of representations inequivalent to the usual q-Fock representation, with no counterpart at the limit q=1. They are used to build representations of SU _q (1,1) and E(2) in Schwinger's way.