Ph.D Thesis
Ph.D Student Subag Eyal Contraction of Representations of the Three Dimentional Lie Algebras as Direct Limit Department of Mathematics Professor Ehud Moshe Baruch Professor Ady Mann

Abstract

It quite often happens that one physical theory approximates another under some limiting procedure. One of the prominent examples for this phenomenon is the interpretation of classical Newtonian mechanics as a limit of relativistic mechanics when the relative velocity tends to zero. The symmetry group of classical Newtonian mechanics is the Galilei group while that of relativistic mechanics is the Poincaré group. More than sixty years ago Segal, and Inönü and Wigner showed that the Lie algebra of the Galilei Group is a limit of the Lie algebra of the Poincaré group. These kinds of limits of Lie  algebras are now known as contractions. They have been studied in various contexts and found useful in many applications.

One aspect of the theory that drew considerable attention along the years is contraction of Lie algebra representations. Contraction of Lie algebra representations is a limiting procedure of obtaining representations of the contracted Lie algebra from representations of the Lie algebra that is being contracted. The theory of contraction of representations is far from being complete and even for low dimensional Lie algebras further investigation is needed. In this research thesis for any Inönü-Wigner contraction of a real three dimensional Lie algebra we construct the corresponding contractions of representations. Most of these contractions are new and in other examples we give a different proof from the known one. Our method is quite canonical in the sense that in all cases we deal with realizations of the representations on some spaces of functions; we contract the differential operators on those spaces along with the representation spaces themselves by taking certain pointwise limit of functions. We call such contractions strong contractions. We show that this pointwise limit gives rise to a direct limit space in which we also obtain an L2 convergence.

We also give a general definition in terms of direct limit for contraction of Lie algebra representations that generalizes the definition of strong contraction of representations. All the known examples for contractions of Lie algebra representations fit within this formalism and as a consequence it follows that any contraction of Lie algebra representations is naturally a direct limit construction. We investigate the relation between the two definitions and contraction of representations by means of convergence of matrix elements. In addition, contractions of the skew-Hermitian irreducible representations of so(n) to those of iso(n-1) are given in the appendix.