Indecomposable representations of the Euclidean algebra (3) from irreducible representations of the symplectic algebra (4, ℂ)

Abstract

The Euclidean group E(3) is the noncompact, semidirect product group E(3) ≊ SO(3) ℝ3. It is the Lie group of orientation-preserving isometries of 3-dimensional Euclidean space. The Euclidean algebra (3) is the complexification of the Lie algebra of E(3). We embed (3) into the 10-dimensional symplectic algebra (4, ℂ), the simple Lie algebra of type C2. We show that, up to conjugation by an element of Sp(4, ℂ), there is only one embedding of (3) into (4, ℂ), and then prove that the irreducible representations of (4, ℂ) remain indecomposable upon restriction to (3), thus creating a new class of indecomposable (3)-representations.