Remarks on Infinite-Dimensional Representations of the Heisenberg Algebra

Abstract

Infinite-dimensional representations of Lie algebras necessarily invoke the theory of unbounded operator algebras. Starting with the familiar example of the Heisenberg Lie algebra, we sketch the essential features of this interaction, distinguishing in particular the cases of integrable and nonintegrable representations.While integrable representations are well understood, nonintegrable representations are quite mysterious objects.We present here a short and didacticalminded overview of the subject.