Finite dimensional representations of the Euclidean algebra e(2) having two generators

The Euclidean algebra e(2) is the Lie algebra of the group E(2) of Euclidean transformations of the plane. This paper examines finite dimensional representations of e(2) having two generators. To each representation with two generators we associate a graph. In term of graphs, we give a criterion for the indecomposability of such representations and describe an invariant for indecomposable representations. We also classify the indecomposable representations of dimensions 5 and 6, regardless of the number of generators (dimensions less than 5 have been classified). In each case there are finitely many such representations. Next, we show that for each dimension ≥8 there are infinitely many nonequivalent indecomposable representations