Real Lie Algebras of Differential Operators and Quasi-Exactly Solvable Potentials

We first establish some general results connecting real and complex Lie algebras of first-order differential operators. These are applied to completely classify all finite-dimensional real Lie algebras of first-order differential operators in $R^2$. Furthermore, we find all algebras which are quasi-exactly solvable, along with the associated finite-dimensional modules of analytic functions. The resulting real Lie algebras are used to construct new quasi-exactly solvable Schroedinger operators on $R^2$.