Indecomposable representations of the Poincare algebra

Abstract

The authors consider indecomposable representations of the Poincare algebra iso(3,1) on the space Omega = Omega + Omega -H of its universal enveloping algebra. A master representation is obtained on Omega which induces representations on K, the invariant subalgebra of translations and on Omega - and Omega +. These representations are discussed, in particular in view of finite dimensional indecomposable representations of iso(3,1). The approach taken is analogous to the approach chosen by the authors in their analysis of indecomposable representations of the Lorentz algebra so(3,1) (1983). Thus, under restriction of iso(3,1) to so(3,1) the earlier results are recovered. The interpretation of the finite dimensional indecomposable representations of iso(3,1) then follows easily as a coupling of a finite number of irreducible so(3,1) representations to an indecomposable iso(3,1) representation, with the dimension of the irreducible representations strictly increasing or strictly decreasing. The bases for the finite dimensional indecomposable iso(3,1) representations are explicitly determined, and thus also their matrix elements via the inducing representations. A formula for their dimensionalities is obtained. The methods employed are purely algebraic and follow the line of work of Jacobson and Dixmier. (1962, 1978).