Quasi-Exact Solvability and Deformations of Sl(2) Algebra

TL;DR

This paper explores how certain differential equations, including Heun, relate to deformations of sl(2) algebra, revealing new quasi-exactly solvable systems beyond traditional sl(2) symmetry.

Contribution

It demonstrates the connection between algebraic deformations and quasi-exact solvability, including explicit representations of cubic algebra and answers Turbiner's question.

Findings

Finite-dimensional cubic algebra representations describe new quasi-exactly solvable systems.

Known sl(2) representations are recovered under specific conditions.

Deformed symmetries underlie systems not associated with standard sl(2) symmetry.

Abstract

Algebraic structure of a class of differential equations including Heun is shown to be related with the deformations of sl(2) algebra. These include both quadratic and cubic ones. The finite dimensional representation of cubic algebra is explicitly shown to describe a quasi-exactly solvable system, not connected with sl(2) symmetry. Known finite dimensional representations of sl(2) emerge under special conditions. We answer affirmatively the question raised by Turbiner: Are there quasi- exactly solvable problems which can not be represented in terms of sl(2) generators? and give the explicit deformed symmetry underlying this system.