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The irreducible representations of two intermediate Casimir elements associated to the recoupling of three identical irreducible representations of U_q(\mathfrak {sl}_2) are considered. It is shown that these intermediate Casimirs are related by a conjugation involving braid group representations. Consequently, the entries of the braid group matrices are explicitly given in terms of the q-Racah polynomials which appear as 6j-symbols in the Racah problem for U_q(\mathfrak {sl}_2). Formulas for these polynomials are derived from the algebraic relations satisfied by the braid group representations. [ABSTRACT FROM AUTHOR]

The biorthogonal rational functions of 3F2 type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties are described by three difference operators X,Y,Z which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space. The pairwise commutators of the operators X,Y,Z generate a quadratic algebra which is akin to the algebras of Askey-Wilson type attached to hypergeometric polynomials. [ABSTRACT FROM AUTHOR]

We introduce Heun algebras of Lie type. They are obtained from bispectral pairs associated to simple or solvable Lie algebras of dimension three or four. For su(2), this leads to the Heun-Krawtchouk algebra. The corresponding Heun-Krawtchouk operator is identified as the Hamiltonian of the quantum analogue of the Zhukovsky-Voltera gyrostat. For su(1,1), one obtains the Heun algebras attached to the Meixner, Meixner-Pollaczek, and Laguerre polynomials. These Heun algebras are shown to be isomorphic to the the Hahn algebra. Focusing on the harmonic oscillator algebra ho leads to the Heun-Charlier algebra. The connections to orthogonal polynomials are achieved through realizations of the underlying Lie algebras in terms of difference and differential operators. In the su(1,1) cases, it is observed that the Heun operator can be transformed into the Hahn, Continuous Hahn, and Confluent Heun operators, respectively. [ABSTRACT FROM AUTHOR]

The Bannai-Ito polynomials are shown to arise as Racah coefficients for sl-1(2). This Hopf algebra has four generators, including an involution, and is defined with both commutation and anticommutation relations. It is also equivalent to the parabosonic oscillator algebra. The coproduct is used to show that the Bannai-Ito algebra acts as the hidden symmetry algebra of the Racah problem for sl-1(2). The Racah coefficients are recovered from a related Leonard pair. [ABSTRACT FROM AUTHOR]