Classification of the Irreducible Representations of «

Let g be a nonabelian Lie algebra over an algebraically closed field K of characteristic 0. One is interested in the (algebraically) irreducible representations of g acting on a vector space which is allowed to be infinite dimensional. The subject of enveloping algebras is largely concerned with these, but even in the simplest nonabelian case, with g = I) the 3-dimensional (nilpotent) Heisenberg algebra, as Dixmier remarks in discussing the situation when K = C in the preface to [2], "a deeper study reveals the existence of an enormous number of irreducible representations of f). . • It seems that these representations defy classification. A similar phenomenon exists for g = «1(2), and most certainly for all noncommutative Lie algebras." However, as we shall see, the situation for f) and for «1(2) turns out to be far nicer than hoped for. Indeed we announce here a determination and classification of all irreducible representations of f), of «[(2), and of the 2-dimensional nonabelian Lie algebra, and thus of the prototypes respectively of nilpotent, simple, and solvable Lie algebras. As a guide to the meaning of "classification" and because our results use the same invariants, consider a classical situation of an (associative) algebra for which the irreducible representations have long been classified, namely, the algebra B of formal linear differential operators with rational function coefficients, i.e., B = K(q)[p], the (noncommutative) polynomials in an indeterminate p where multiplication is determined by the relation pq qp = 1. Then B is a left principal ideal domain. Therefore [3] a ^-module M is simple if and only if M = B/Bb for some i G 5 which is irreducible (i.e., b = ac implies a or c is a unit); and B/Bb = B/Ba if and only if a and b are similar, i.e., there exists c EB such that (b, c) = 1 and a = [b, c]c _ 1 where {b, c) is a