A classification of the finite dimensional, indecomposable 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 thesis examines the finite dimensional representations of e (2) having two generators. Given a representation with a fixed pair of generators we associate a graph: the graph is dependent on the choice of generators and for each isomorphism class of representations there are infinitely many possible graphs. In term of graphs, we give a criterion for the indecomposability of such representations and describe an invariant for the indecomposable representations.; Next, we classify the finite dimensional, indecomposable representations of e (2) having two generators. We describe a procedure that enables us to select a single graph for each isomorphism class. This graph uniquely identifies the class.