THE LIE ALGEBRAIC STRUCTURE OF DIFFERENTIAL-OPERATORS ADMITTING INVARIANT SPACES OF POLYNOMIALS

We prove that the scalar and 2 x 2 matrix differential operators which preserve the simplest scalar and vector-valued polynomial modules in two variables have a fundamental Lie algebraic structure. Our approach is based on a general graphical method which does not require the mod ules to be irreducible under the action of the corresponding Lie (supe r)algebra. This method fan be generalized to modules of polynomials in an arbitrary number of variables. We given generic examples of partia lly solvable differential operators which are not Lie algebraic. We sh ow that certain vector-valued modules give rise to new realizations of finite-dimensional Lie superalgebras by first-order differential oper ators. (C) 1998 Academic Press.