Nonclassical type representations of the q-deformed algebra U′q(son)

The nonstandard q-deformation U_q(so_n) of the universal enveloping algebra U(so _n ) has irreducible finite dimensional representations which are a q-deformation of the well-known irreducible finite dimensional representations of U(so _n ). But U_q(so_n) also has irreducible finite dimensional representations which have no classical analogue. The aim of this paper is to give these representations which are called nonclassical type representations. They are given by explicit formulas for operators of the representations corresponding to the generators of U_q(so_n).     

Representations of the Quantum Algebra Uq(un, 1)

The main aim of the paper is to study infinite-dimensional representations of the real form U _q (u _{n, 1}) of the quantized universal enveloping algebra U _q (gl _{n + 1}). We investigate the principal series of representations of U _q (u _{n, 1}) and calculate the intertwining operators for pairs of these representations. Some of the principal series representations are reducible. The structure of these representations is determined. Then we classify irreducible representations of U _q (u _{n, 1}) obtained from irreducible and reducible principal series representations. All *-representations in this set of irreducible representations are separated. Unlike the classical case, the algebra U _q (u _{n, 1}) has finite-dimensional irreducible *-representations.     

Representations of the nonstandard (twisted) deformationU′ q(so n ) forq a root of unity

Irreducible representations of the algebrasU′ _q(so _n ) forq a root of unityq ^p=1 are given. The main class of these representations act onp ^N-dimensional linear space (whereN is a number of positive roots of the Lie algebra so _n ) and are given byr = dim so _n complex parameters. Some classes of degenerate irreducible representations are also described. Presented at the 9th Colloquium “Quantum Groups and Integrable Systems”, Prague, 22–24 June 2000. The research described in this publication was made possible in part by Grant UP1-2115 of the U.S. Civilian Research and Development Foundation for the Independent States of the Former Soviet Union (CRDF).     

Painlevé Functions and Conformal Blocks

We outline recent developments relating Painlevé equations and 2D conformal field theory. Generic tau functions of Painlevé VI and Painlevé III₃ are written as linear combinations of c=1 conformal blocks and their irregular limits. This provides explicit combinatorial series representations of the tau functions, and helps to establish a connection formula for the tau function in the Painlevé VI case.     

On q-Laplace operator and q-harmonic polynomials

A q-Laplace operator and q-harmonic polynomials on the quantum vector space are studied. A q analog of associated spherical harmonics are constructed. They constitute an orthonormal basis of the space of q-harmonic polynomials. A q analog of separation of variables is given.     

Ising Correlations and Elliptic Determinants

Correlation functions of the two-dimensional Ising model on the periodic lattice can be expressed in terms of form factors—matrix elements of the spin operator in the basis of common eigenstates of the transfer matrix and translation operator. Free-fermion structure of the model implies that any multiparticle form factor is given by the pfaffian of a matrix constructed from the two-particle ones. Crossed two-particle form factors can be obtained by inverting a block of the matrix of linear transformation induced on fermions by the spin conjugation. We show that the corresponding matrix is of elliptic Cauchy type and use this observation to solve the inversion problem explicitly. Non-crossed two-particle form factors are then obtained using theta functional interpolation formulas. This gives a new simple proof of the factorized formulas for periodic Ising form factors, conjectured by A. Bugrij and one of the authors.     

Magnetothermal investigation of crystallization of Ni–P amorphous alloys

A new magnetic phase analysis method is presented and its application to a certain class of amorphous alloys is thoroughly discussed. Special attention is paid to the size of clusters with strong magnetic properties that form the separating phase. This magnetothermal (MT) method reveals the formation or disappearance of a phase with strong magnetic properties, and in certain cases it can give valuable information, even in the temperature range where the sample is paramagnetic. The accuracy and applicability of the MT method are discussed. Combining this method with differential scanning calorimetry (DSC), the transformations in melt-quenched (MQ) and electrolessly deposited (ED) amorphous Ni–P alloys with hypoeutectic and close to eutectic composition are investigated under the condition of a constant heating rate. Up to four stages of Ni precipitation in the alloys are revealed, depending on the phosphorous content and the method of sample preparation.     

Two-particle correlations from the q-Boson viewpoint

We propose and develop to some extent a novel approach, which allows to effectively describe, for relativistic heavy-ion collisions, the empirically observed deviation from unity of the intercept λ (i.e. the measured value corresponding to zero relative momentum p of two registered identical pions or kaons) of the two-particle correlation function C(p,K). The approach is based on the use of two versions of the so-called q-deformed oscillators and the corresponding picture of ideal gases of q-bosons. By these techniques the intercept λ is put into direct correspondence with the deformation parameter q. For fixed deformation strength, the model predicts particular dependence of the intercept λ on the pion pair mean momentum K. Received: 2 July 1999 / Revised version: 11 November 1999