h-Deformation of GL(1∣1)

An h-deformation of a (graded) Hopf algebra of functions on supergroup GL(11) is introduced via a contraction of GL _q (11). The deformation parameter h is odd (Grassmann). A related differential calculus on h-superplane is presented.     

On the integral construction of the universal R-matrix for quantum current superalgebraUq (oŝp(1,2))

We discuss the problems in the construction of the universal R-matrix for the the Drinfeld current realization of quantum affine superalgebrasUq (oŝp(1,2)), where we try to present the universal R-matrix for the corresponding “Drinfeld” comultiplication in the form of certain integrals over current operators with specially chosen contours. Presented at the 10th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 21–23 June 2001. Author’s research is partially supported by the Summer Research Fellowship and the Taft Foundation at the University of Cincinnati.     

∗-Representations of the Quantum Algebra U q(sl(3))

Abstract

Studied in this paper are real forms of the quantum algebra U _q(sl(3)). Integrable operator representations of ?-algebras are defined. Irreducible representations are classified up to a unitary equivalence.     

The Hamiltonian systems on the Poisson structure of the quasi-classical limit of GL q (∞)

We consider the Hamiltonian systems on the Poisson structure of GL() which is introduced from the quantum group GL _q () by the so-called quasi-classical limit of GL _q (). Furthermore, we show that the Toda lattice hierarchy is a Hamiltonian system of this structure.     

Realisations of quantum GL p,q(2) and Jordanian GLh,h′(2)

The quantum group GL _p,q(2) is known to be related to the Jordanian GL_h,h(2) via a contraction procedure. It can also be realised using the generators of the Hopf algebra G _r,s. We contract the G _r,s quantum group to obtain its Jordanian analogue G _m,k, which provides a realisation of GL_h,h(2) in a manner similar to the q-deformed case.     

Iterative Construction of U q (s ℓ(n+1)) Representations and Lax Matrix Factorisation

The iterative construction of a generic representation of g?(n + 1) or of the trigonomentric deformation of its enveloping algebra is conveniently formulated in terms of Lax matrices. The Lax matrix of the constructed representation factorises into parts determined by the Lax matrix of a generic representation of the algebra with reduced rank and others appearing in the factorised expression of the Lax matrix of the special Jordan–Schwinger representation.     

Fusion rules of U q(sl(2, ℂ)), q m = 1

The decomposition of multiple tensor products of finite-dimensional irreducible representations of U _q(sl(2, )), for q's which are roots of 1, is studied in detail.     

Cyclic Representation of the gl√q (n)-Covariant Oscillator Algebra

The cyclic representation ofgl_q (n)-covariant oscillator algebrais discussed by use of some noncommutating cyclicvariables.     

Three-dimensional representation of the double quantum algebrasu q(η(J)) and theq-deformation of the double complex ernst equation

A three-dimensional representation of the double quantum algebrasu _q((J)) is given. By the use of this representation and a Lax pair, we obtain a nonlinear Ernst equation system. By the harmonic function method, a solution of theq-deformed double complex Ernst equation is given.     

On two-point boundary correlations of the gl（1｜1） supersymmetric vertex model

The gl（1｜1） supersymmetric vertex model with domain wall boundary conditions （DWBC） on an N ×N square lattice is considered. We obtain the reduction formulae for the two-point boundary correlation functions of the model.     

Baxterization of ther-matrix for the adjoint representation of U q [D(2, 1;α)]

The Baxterization procedure is applied to the Braid group representation arising from the adjoint representation of the quantum supergroup U _q [D(2, 1;)]. This yields a two-parameter family of solutions to the (parameter dependent) Yang-Baxter equation of interest in supersymmetric lattice models.     

Two-Parameter Deformation of the Oscillator Algebra and (p,q)–Analog of Two-Dimensional Conformal Field Theory

Abstract

The two-parameter deformation of canonical commutation relations is discussed. The self-adjointness property of the (p, q)-deformed position Q and momentum P operators is investigated. The (p, q)-analog of two-dimensional conformal field theory based on the (p, q)-deformation of the su(1, 1) subalgebra of the Virasoro algebra is presented.     

On the Anderson localization and “universal degradation” of the critical temperature in superconductors

We give an analytical formula for the critical temperature of a superconductor taking into account the enhanced Coulomb repulsion which will be considered as frequency dependent.     

On the universal linear in-layer resistivity ϱ∝T 1of high-T c superconductors

Based on the Boltzmann equation an inversion formula for the resistivity is deduced which for a general coupling between two carrier bands yields, in dimensionD, T ^D–1. While being well known inD=3, this is a new result inD=2. It is then shown that in a 2-band model of strongly anisotropic masses as are found in the cuprate layer structures, the in-layer resistivity follows the law forD=2, T ¹, in agreement with the majority of experiments. Exceptions to the linear law and to the 2-band assumption are also discussed. Assuming the spin-flip term of the interaction to be dominant, the spin-flip relaxation time _sf is estimated by using the data on · _sf is found to be orders of magnitude shorter than the EPR period thus explaining the complete absence of intrinsic EPR signals in the cuprates. It is also shown that the spin-flip interaction gives rise to pairing, preferentially of the heavier carriers.     

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.     

Mirror supersymmetry in the space of (super) extended Poincaré algebras p(p,q)

We classify extended Poincaré Lie superalgebras and Lie algebras of any signature (p, q), i.e. Lie superalgebras and ₂-graded Lie algebras g = g₀ + g₁, where g₀ = s₀(V) + V is the (generalized) Poincaré Lie algebra of the pseudo Euclidean vector space V = ^{p, q} of signature (p, q) and g₁ is a spin 1/2 s₀(V)-module extended to a s₀-module with kernel V.As a result of the classification, we obtain, if g₁ = S is the spinor module, the numbers L ⁺(n, s) (resp. L ^–(n, s)) of independent such Lie super algebras (resp. Lie algebras), which are periodic functions of the dimension n=p+q (mod 8) and the signature s=p–q (mod 8) and satisfy: L ⁺(–n, s)=L ^– (n, s).Supported by Max-Planck-Institut für Mathematik (Bonn).Supported by the Alexander von Humboldt Foundation, MSRI (Berkeley) and SFB 256 (Bonn University).     

Relaxation of the analyzing power of the (p, τX) and (p, αX) reactions in the continuum

The observed continuous decrease of the analyzing power for the reactions ($\text{p}$, τX) and ($\text{p}$, αX) on ⁹⁰Zr and ²⁰⁹Bi at E_p = 72 MeV as a function of the excitation energy of the final nucleus is described by a phenomenological relaxation process.