Constructing Lie algebras of first order differential operators

We extend Guillemin and Sternberg’s Realization Theorem for transitive Lie algebras of formal vector fields to certain Lie algebras of formal first order differential operators, and show that Blattner’s proof of the Realization Theorem allows for a computer implementation that automatically reproduces many realizations derived in the existing literature, and that can also be used to compute new realizations. Applications include the explicit construction of quasi-exactly solvable Hamiltonians, and of finite-dimensional irreducible modules over semisimple Lie algebras.