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We obtain the time-dependent correlation function describing the evolution of a single spin excitation state in a linear spin chain with isotropic nearestneighbour XY coupling, where the Hamiltonian is related to the Jacobi matrix of a set of orthogonal polynomials. For the Krawtchouk polynomial case, an arbitrary element of the correlation function is expressed in a simple closed form. Its asymptotic limit corresponds to the Jacobi matrix of the Charlier polynomial, and may be understood as a unitary evolution resulting from a Heisenberg group element. Correlation functions for Hamiltonians corresponding to Jacobi matrices for the Hahn, dual Hahn and Racah polynomials are also studied. For the Hahn polynomials we obtain the general correlation function, some of its special cases and the limit related to the Meixner polynomials, where the su(1, 1) algebra describes the underlying symmetry. For the cases of dual Hahn and Racah polynomials, the general expressions of the correlation functions contain summations which are not of hypergeometric type. Simplifications, however, occur in special cases.

The generators of the Jordanian quantum algebra U h (sl(2)) are expressed as nonlinear invertible functions of the classical sl(2) generators. This permits immediate explicit construction of the finite dimensional irreducible representations of the algebra U h (sl(2)). Using this construction, new finite dimensional solutions of the Yang-Baxter equation may be obtained.

A class of transformations of R q -matrices is introduced such that the q → 1 limit gives explicit nonstandard R h -matrices. The transformation matrix is singular itself at q → 1 limit. For the transformed matrix, the singularities, however, cancel yielding a well-defined construction. Our method can be implemented systematically for R-matrices of all dimensions and not only for sl(2) but also for algebras of higher dimensions. Explicit constructions are presented starting with U q (sl(2)) and U q (sl(3)), while choosing R q for (fund. rep.)⊗(arbitrary irrep.). The treatment for the general case and various perspectives are indicated. Our method yields nonstandard deformations along with a nonlinear map of the h-Borel subalgebra on the corresponding classical Borel subalgebra. For U h (sl(2)) this map is extended to the whole algebra and compared with another one proposed by us previously.

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We obtain an analytic expression for the specific heat of a system of N rigid rotators exactly in the high temperature limit, and via a pertubative approach in the low temperature limit. We then evaluate the specific heat of a diatomic gas with both translational and rotational degrees of freedom, and conclude that there is a mixing between the translational and rotational degrees of freedom in nonextensive statistics.Comment: 12 page

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A general construction is given for a class of invertible maps between the classical U(sl(2)) and the Jordanian U h (sl(2)) algebras. Here the role of the maps is studied in the context of construction of twist operators relating the cocommutative and non-cocommutative coproducts of the U(sl(2)) and U h (sl(2)) algebras respectively. It is shown that a particular map called the "minimal twist map" implements the simplest twist given directly by the factorized form of the R h matrix of Ballesteros-Herranz. For a "non-minimal" map the twist has an additional factor obtainable in terms of the similarity transformation relating the map in question to the minimal one. Our general prescription may be used to evaluate the series expansion in powers of h of the twist operator corresponding to an arbitrary "non-minimal" map. The classical and the Jordanian antipode maps may also be interrelated by suitable similarity transformations.