On polynomial symmetry algebras underlying superintegrable systems in Darboux spaces

TL;DR

This paper reviews three approaches to understanding polynomial symmetry algebras in superintegrable systems within Darboux spaces, highlighting methods for spectrum analysis, state construction, and discovering new models.

Contribution

It introduces and compares three novel methods for analyzing and constructing polynomial symmetry algebras in superintegrable systems in Darboux spaces.

Findings

Finite-dimensional representations via deformed oscillator algebra.

Construction of infinite-dimensional representations using induced modules.

Discovery of new superintegrable models through commutant analysis.

Abstract

We review three different approaches to polynomial symmetry algebras underlying superintegrable systems in Darboux spaces. The first method consists of using deformed oscillator algebra to obtain finite-dimensional representations of quadratic algebras. This allow one to gain information on the spectrum of the superintegrable systems. The second method has similarities with the induced module construction approach in the context of Lie algebras and can be used to construct infinite dimensional representations of the symmetry algebras. Explicit construction of these representations is a non-trivial task due to the non-linearity of the polynomial algebras. This method allows the construction of states of the superintegrable systems beyond the reach of separation of variables. As a result, we are able to construct a large number of states in terms of Airy, Bessel and Whittaker functions which would be difficult to obtain in other ways. We also discuss the third approach which is based on the notion of commutants of subalgebras in the enveloping algebra of a Poisson algebra or a Lie algebra. This allows us to discover new superintegrable models in the Darboux spaces and to construct their integrals and symmetry algebras via polynomials in the enveloping algebras.