The superalgebra spl(p, q) and differential operators

Series of finite-dimensional representations of the superalgebras spl(p, q) can be formulated in terms of linear differential operators acting on a su itable space of polynomials. We sketch the general ingredients necessary to construct these representations and present examples related to spl(2, 1) and spl(2, 2). By revisiting the products of projectivised representations of sl(2), we are able to construct new sets of differential operators prese rving some space of polynomials in two or more variables. In particular, th is allows us to express the representation of spl(2, 1) in terms of matrix differential operators in two variables. The corresponding operators provid e the building blocks for the construction of quasi-exactly solvable system s of two and four equations in two variables. We also present a quommutator deformation of spl(2, 1) which, by construction provides an appropriate ba sis for analyzing the quasi exactly solvable systems of finite difference e quations.