Abstract
The Bannai-Ito algebra is presented together with some of its applications. Its relations with the Bannai-Ito polynomials, the Racah problem for the sl−1(2) algebra, a superintegrable model with reflections and a Dirac-Dunkl equation on the 2-sphere are surveyed.
Highlights
Exploration through the exact solution of models has a secular tradition in mathematical physics
Exact solvability is possible in the presence of symmetries, which come in various guises and which are described by a variety of mathematical structures
Exact solutions are expressed in terms of special functions, whose properties encode the symmetries of the systems in which they arise
Summary
Exploration through the exact solution of models has a secular tradition in mathematical physics. Exact solutions are expressed in terms of special functions, whose properties encode the symmetries of the systems in which they arise. This can be represented by the following virtuous circle: Exact solvability k x Symmetries f ' Special functions. The classical path is the following: start with a model, find its symmetries, determine how these symmetries are mathematically described, work out the representations of that mathematical structure and obtain its relation to special functions to arrive at the solution of the model. The following path will be taken: Algebra −→ Orthogonal polynomials −→ Symmetries −→ Exact solutions. A list of open questions is provided in lieu of a conclusion
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