Lie-algebras and linear operators with invariant subspaces

TL;DR

This paper classifies linear differential and difference operators with polynomial invariant subspaces, linking their structure to Lie algebras and exploring their role in quasi-exactly-solvable spectral problems.

Contribution

It provides a comprehensive classification of such operators using Lie algebra representations and extends understanding of their algebraic structure.

Findings

Operators are represented as polynomial elements of universal enveloping algebras.

Low-dimensional classifications involve specific Lie algebras like sl_2, sl_3, and osp(2,2).

Connection established between these operators and quasi-exactly-solvable spectral problems.

Abstract

A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of some algebra of differential (difference) operators in finite-dimensional representation plus an operator annihilating the finite-dimensional invariant subspace. In low dimensions a classification is given by algebras $sl_2({\bold R})$ (for differential operators in ${\bold R}$) and $sl_2({\bold R})_q$ (for finite-difference operators in ${\bold R}$), $osp(2,2)$ (operators in one real and one Grassmann variable, or equivalently, $2 \times 2$ matrix operators in ${\bold R}$), $sl_3({\bold R})$, $sl_2({\bold R}) \oplus sl_2({\bold R})$ and $gl_2 ({\bold R}) \ltimes {\bold R}^{r+1}\ , r$ a natural number (operators in ${\bold R^2}$). A classification of linear operators possessing infinitely many finite-dimensional invariant subspaces with a basis in polynomials is presented. A connection to the recently-discovered quasi-exactly-solvable spectral problems is discussed.