# Infinite dimensional representations of cubic and quintic algebras and special functions

## TL;DR

This paper develops a practical method to construct infinite-dimensional representations of polynomial algebras in quantum superintegrable systems, enabling the generation of complex special function states beyond traditional separation of variables.

## Contribution

It introduces a new approach for constructing infinite-dimensional representations of polynomial algebras in superintegrable systems, extending the ability to analyze spectral properties.

## Key findings

- Constructed many states using Airy, Bessel, and Whittaker functions.
- Provided a method similar to induced modules for polynomial algebras.
- Enabled analysis beyond separation of variables in 2D Darboux spaces.

## Abstract

Finite and Infinite-dimensional representations of symmetry algebras play a significant role in determining the spectral properties of physical Hamiltonians. In this paper, we introduce and apply a practical method to construct infinite dimensional representations of certain polynomial algebras which appear in the context of quantum superintegrable systems. Explicit construction of these representations is a non-trivial task due to the non-linearity of the polynomial algebras. Our method has similarities with the induced module construction approach in the context of Lie algebras and allows the construction of states of the superintegrable systems beyond the reach of separation of variables. Our main focus is the representations of the polynomial algebras underlying superintegrable systems in 2D Darboux spaces. We are able to construct a large number of states in terms of complicated expressions of Airy, Bessel and Whittaker functions which would be difficult to obtain in other ways.