Indecomposable Finite-dimensional Representations of a Class of Lie Algebras and Lie Superalgebras

The topic of indecomposable finite-dimensional representations of the Poincaré group was first studied in a systematic way by S. Paneitz ([5][6]). In these investigations only representations with one source were considered, though by duality, one representation with 2 sources was implicitly present. The idea of nilpotency was mentioned indirectly in Paneitz’s articles, but a more down-to-earth method was chosen there. The results form a part of a major investigation by S. Paneitz and I. E. Segal into physics based on the conformal group. Induction from indecomposable representations plays an important part in this theory. See ([7]) and references cited therein. The defining representation of the Poincaré group, when given as a subgroup of SU(2,2) (see below), is indecomposable. This representation was studied by the present author prior to the articles by Paneitz in connection with a study of special aspects of Dirac operators and positive energy representations of the conformal group ([4]). Indecomposable representations in theoretical physics have also been used in a major way in a study by G. Cassinelli, G. Olivieri, P. Truini, and V. S. Varadarajan ([1]). The main object is the Poincaré group. In an appendix to the article, the indecomposable representations of the 2-dimensional Euclidean group are considered, and many results are obtained. This group can also be studied by our method, but we will not pursue this here. One small complication is that the circle group is abelian. In the article at hand, we wish to sketch how, by utilizing nilpotency to its fullest extent while using methods from the theory of universal enveloping algebras, a complete description of the indecomposable representations may be reached. In practice, the combinatorics is still formidable, though. It turns out that the method applies to both a class of ordinary Lie algebras and to a similar class of Lie superalgebras. Besides some examples, due to the level of complexity we will only describe a few precise results. One of these is a complete classification of which ideals