Indecomposable representations of the nonlinear Lie algebras

Abstract

On the two-state Heisenberg–Weyl basis, the indecomposable representations of the nonlinear Lie algebras of vector type, Rνλ (ν,λ=0,±1), one of which is generated by two angular momentum operators J0, J2 and any component of an irreducible tensor operator of rank 1, T(1νλ) (ν,λ=0,±1), of an SO(3) group, are studied in detail. We give the explicit expressions for the infinite-dimensional indecomposable representations of Rνλ, which further subduce the infinite-dimensional indecomposable representations on the invariant subspaces and induce the finite-dimensional or infinite-dimensional (indecomposable, irreducible) representations on the quotient spaces. As special cases, the standard angular momentum representations of Rνλ are given also.