Quasi-exactly Solvable Lie Superalgebras of Differential Operators
Federico Finkel, Artemio González-López, Miguel A. Rodríguez
Abstract
In this paper, we study Lie superalgebras of
2×2
matrix-valued first-order differential operators on the complex line. We first completely classify all such superalgebras of finite dimension. Among the finite-dimensional superalgebras whose odd subspace is nontrivial, we find those admitting a finite-dimensional invariant module of smooth vector-valued functions, and classify all the resulting finite-dimensional modules. The latter Lie superalgebras and their modules are the building blocks in the construction of QES quantum mechanical models for spin 1/2 particles in one dimension.