Strong contraction of the representations of the three dimensional Lie algebras

TL;DR

This paper introduces a canonical method for contracting representations of three-dimensional Lie algebras, using pointwise limits of functions to produce strong contractions and direct limit spaces, with many new results and alternative proofs.

Contribution

It develops a canonical approach to strong contraction of Lie algebra representations via pointwise limits, expanding the understanding of representation contractions.

Findings

Constructed strong contractions for all Inonu-Wigner contractions of 3D Lie algebras.

Produced many new contraction examples and provided alternative proofs for existing ones.

Established a framework linking differential operators, function spaces, and contraction limits.

Abstract

For any Inonu-Wigner contraction of a three dimensional Lie algebra we construct the corresponding contractions of representations. 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. Many of these contractions are new and in other examples we give a different proof.