A New Deformed Supersymmetric Oscillator
We construct and discuss the Fock-space representation for a deformed oscillator with "peculiar" statistics. We show that corresponding algebra represents deformed supersymmetric oscillator.
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Multivariable q-Racah polynomials
The Koornwinder-Macdonald multivariable generalization of the Askey-Wilson polynomials is studied for parameters satisfying a truncation condition such that the orthogonality measure becomes discrete with support on a finite grid. For this parameter regime the polynomials may be seen as a multivariable counterpart of the (one-variable) $q$-Racah polynomials. We present the discrete orthogonality measure, expressions for the normalization constants converting the polynomials into an orthonormal system (in terms of the normalization constant for the unit polynomial), and we discuss the limit $q\to 1$ leading to multivariable Racah type polynomials. Of special interest is the situation that $q$ lies on the unit circle; in that case it is found that there exists a natural parameter domain for which the discrete orthogonality measure (which is complex in general) becomes real-valued and positive. We investigate the properties of a finite-dimensional discrete integral transform for functions over the grid, whose kernel is determined by the multivariable $q$-Racah polynomials with parameters in this positivity domain.
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Integrality of two variable Kostka functions
The two variable Kostka functions are the scalar products of the Macdonald polynomials with the Schur polynomials with respect to the scalar product which makes the Hall-Littlewood polynomials pairwise orthogonal. A conjecture of Macdonald states that they are polynomials in q and t with non-negative integral coefficients. We prove that the Kostka functions are at least polynomials with integral coefficients. The main idea is to prove an analogous statement for the non-symmetric Macdonald polynomials by establishing recursion relations via the affine Hecke algebra.
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Quantization of Poisson Groups
Let $ G^τ$ be a connected simply connected semisimple algebraic group, endowed with generalized Sklyanin-Drinfel'd structure of Poisson group, let $ H^τ$ be its dual Poisson group. By means of quantum double construction and dualization via formal Hopf algebras, we construct new quantum groups $ U_{q,φ}^M(\frak{h}) $ --- dual of $ U_{q,φ}^{M'}(\frak{g}) $ --- which yield infinitesimal quantization of $ H^τ$ and $ G^τ\, $, we study their specializations at roots of 1 (in particular, their classical limits), thus discovering new quantum Frobenius morphisms. The whole description dualize for $ H^τ$ what was known for $ G^τ$, completing the quantization of the pair $ (G^τ,H^τ) $.
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Multiparameter Deformations of the Algebra $gl_n$ in Terms of Anyonic Oscillators
Generators of multiparameter deformations $U_{q;s_1,s_2,...,s_{n-1}}(gl_n)$ of the universal enveloping algebra $U(gl_n)$ are realized bilinearly by means of appropriately generalized form of anyonic oscillators (AOs). This modification takes into account the parameters $s_1,...,s_{n-1}$ and yields usual AOs when all the $s_i$ are set equal to unity.
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Symmetric and non-symmetric quantum Capelli polynomials
We define a family of symmetric and a family of non-symmetric polynomials in terms of vanishing conditions. These families depend on two paramters, q and t. Their main feature is that they consist of non-homogeneous polynomials. The symmetric polynomials form the quantized version of polynomials occuring in the context of generalized Capelli identities. We show that these quantum Capelli polynomials are also characterized by q-difference equations. More precisely, they are eigenfunctions of Cherednik type operators and transform under an affine Hecke algebra. Thus, we are able to identify their top homogeneous component as a Macdonald polynomial (symmetric or non-symmetric, respectively).
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On the parametrization of solutions of the Yang--Baxter equations
We study all five-, six-, and one eight-vertex type two-state solutions of the Yang-Baxter equations in the form $A_{12} B_{13} C_{23} = C_{23} B_{13} A_{12}$, and analyze the interplay of the `gauge' and `inversion' symmetries of these solution. Starting with algebraic solutions, whose parameters have no specific interpretation, and then using these symmetries we can construct a parametrization where we can identify global, color and spectral parameters. We show in particular how the distribution of these parameters may be changed by a change of gauge.
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