Relations betweenGL p,q (2)'s

Quantum groups have some peculiar properties is two dimensions. We formulate conditions sufficient for the product of two quantum matrices (with not necessarily the same values of deformation parameters) to be a quantum matrix again. This is then used to study the powers and exponential form of matrices fromGL _p,q (2), generalising this way properties ofGL _q (2)-matrices.     

A (p,q)-Deformation of Quantum Mechanics

On a one-dimensional lattice with a geometric sequence spacing, the Hermitian conjugation of a (p, q)-derivative operator is discussed by means of (p, q)-integration. Then a (p, q)-deformation of both the Heisenberg algebra for the canonical coordinates and the Heisen berg-Weyl algebra for the harmonic oscillator is presented. It is shown that although in the algebraic aspect the (p, q)-deformation discussed here is identical with 9-deformation given by Truong, the (p, q)-deformed SchrGdinger picture is in fact different from the q-deformed one.     

Representation of Spin Group Spin（p, q）

The representation η（P, q） of spin group Spin（p, q） in any dimensional space is given by induction, and the relation between two representations, which are obtained in two kinds of inductions from Spin（p, q） to Spin（p ＋ 1, q ＋ 1） are studied.     

Unitarization of a singular representation ofSO(p,q)

A geometric construction of a certain singular unitary representation ofSO _e(p,q), withp+q even is given. The representation is realized geometrically as the kernel of aSO _e(p,q)-invariant operator on a space of sections over a homogeneous space forSO _e(p,q). TheK-structure of these representations is elucidated and we demonstrate their unitarity by explicitly writing down anso(p,q) positive definite hermitian form. Finally, we demonstrate that the annihilator inU[g] of this representation is the Joseph ideal, which is the maximal primitive ideal associated with the minimal coadjoint orbit.     

Intertwining relations,asymmetric face model,and algebraic Bethe ansatz forSU p,q (2) invariant spin chain

Intertwining relations for the quantumR-matrix of theSU _p,q (2) invariant spin chain are obtained and the corresponding face model is deduced. An important difference is seen to arise due to the asymmetry generated by the parametersp andq, which leads to a asymmetric face model. An algebraic Bethe ansatz is set up and solved with the help of these intertwining vectors.     

The dual (p,q)-Alexander-Conway Hopf algebras and the associated universal ℐ-matrix

The dually conjugate Hopf algebrasFun _p,q (R) andU _p,q (R) associated with the two-parametric (p,q)-Alexander-Conway solution (R) of the Yang-Baxter equation are studied. Using the Hopf duality construction, the full Hopf structure of the quasitriangular enveloping algebraU _p,q (R) is extracted. The universal ?-matrix forsFun _p,q (R) is derived. While expressing an arbitrary group element of the quantum group characterized by the noncommuting parameters in a representation independent way, the ?-matrix generalizes the familiar exponential relation between a Lie group and its Lie algebra. The universal ?-matrix and the FRT matrix generators,L ^(±), forU _p,q (R) are derived from the ?-matrix.     

Classification of irreducible super-unitary representations ofsu(p,q/n)

In this paper we classify all the irreducible super-unitary representations ofsu(p,q/n), which can be integrated up to a unitary representation ofS(U(p,q)×U(n)), a Lie group corresponding to the even part ofsu(p,q/n). Note that a real form of the Lie superalgebrasl(m/n;) which has non-trivial superunitary representations is of the formsu(p,q/n)(p+q=m) orsu(m/r,s)(r+s=n). Moreover, we give an explicit realization for each irreducible super-unitary representation, using the oscillator representation of an orthosymplectic Lie superalgebra.     

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.     

Integrable Equations of the Form q t =L 1(x,t,q,q x,q xx )q xxx +L 2(x,t,q,q x,q xx )

Integrable equations of the form q _t =L ₁(x,t,q,q _x ,q _xx )q _xxx +L ₂(x,t,q,q _x ,q _xx ) are considered using linearization. A new type of integrable equations which are the generalization of the integrable equations of Fokas and Ibragimov and Shabat are given.     

Unitons of Harmonic Maps from R2 to U(p,q)

We calculate a second cohomology class which determines a deformation quantization up to equivalence for a deformation quantization with separation of variables on a Kähler manifold, following P. Deligne.     

(p,q)-Analog of Two-Dimensional Conformal Field Theory. The Ward Identities and Correlation Functions

Abstract

A (p, q)-analog of two-dimensional conformally invariant field theory based on the quantum algebra U_pq (su(1, 1)) is proposed. The representation of the algebra U_pq (su(1, 1)) on the space of quasi-primary fields is given. The (p, q)-deformed Ward identities of conformal field theory are defined. The two- and three-point correlation functions of quasi-primary fields are calculated.     

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.     

Representations ofSO(p+q) andO(p) with an application to the shapes of8,9Li

It is easy to show that the symmetry groups governing a system ofZ protons andN neutrons areSO(p+q) andO(p), wherep, q are related toZ, N and the symmetry groups are transitive on a Grassmann manifoldG _p,q. In this paper the general representations ofSO(p+q) andO(p) are found and used to describe the geodesics onG _p,q for the nuclear manifolds of the neutron rich-elements^8,9Li.     

利用SU（2）_（q，s）量子代数的两参数变形振子实现讨论SU（2）_（q，s）相干态

利用ＳＵ（２）ｑ，ｓ量子代数的两参数变形振子实现构造出与Ｐｅｒｅｌｏｍｏｖ相干态形式不同的ＳＵ（２）ｑ，ｓ相干态．证明了ＳＵ（２）ｑ，ｓ，量子代数的表示基是正交的，并讨论了它的相干态的归一性和完备性．指出ＳＵ（２）ｑ，ｓ相干态的相干性受参数ｑ，ｓ的影响，它比单参数变形ＳＵ（２）ｑ相干态更具一般性．     

The (p, 2p) reaction for slightly deformed nuclei

A perturbation approach for the calculation of (p, 2p) angular correlations for slightly deformed nuclei is developed using the formalism of the distorted wave t-matrix approximation (DWTA) in the static limit where approximations that incorporate absorption and focussing effects are used for the spin-orbit independent optical-model waves. Using a finite-range off-shell t-matrix that fits p-p scattering at 90° to approximate the p-p interaction inside the nucleus, a zero-order calculation is performed and normalized to ¹⁹F(p, 2p)¹⁸O angular correlation data at 42.7 MeV. A rms study is undertaken in order to parametrize the bound-state and the sensitivity of the shape of the angular correlation to the rms value is observed. The deformation of the bound-state is introduced using normalized perturbation theory and calculations with various values of the deformation parameter β for prolate and oblate deformations are performed and the results are compared. The shape of the (p, 2p) angular correlation is shown to be highly sensitive to the degree of deformation and is consistent with the shape dependence of the correlation to the rms value of the bound-state wave function.     

Nucleon-nucleon final-state interactions in the reactions He(p, d)2p, He(p, t)2p and He(p, He)pn

The energy spectra of deuterons, tritons and ³He particles from the reactions ³He(p, d)2p, ⁴He(p, t)2p and ⁴He(p, ³He)pn have been measured at angles between 6° and 60° lab. The ³He(p, d)2p reaction was studied at both 30.5 and 49.5 MeV incident proton energies, while the other two reactions were studied at 49.5 MeV only. The energy spectra are compared with calculations based on the Watson-Migdal model of final-state interactions.     

The Ar(p, d)Ar reaction

The ³He spin analysing power of hydrogen (protons) has been measured at a c.m. energy of 20 MeV at 14 c.m. angles from 40° to 160°. The measurement of ³He target polarization was calibrated by measuring the ³He spin analysing power of ⁴He at an energy and angle where it was ?1.00. The proton-³He differential elastic scattering cross section is also reported at a c.m. energy of 20.0 MeV for 36 c.m. angles from 20° to 170°.