Classification of linked indecomposable modules of a family of solvable Lie algebras over an arbitrary field of characteristic 0

摘要

Let g be a finite-dimensional Lie algebra over a field of characteristic 0, with solvable radical r and nilpotent radical n = [g, r]. Given a finite-dimensional g-module U, its nilpotency series 0 subset of U-(1) subset of ... subset of U-(m) = U is defined so that U-(1) is the 0-weight space of n in U, U-(2)/U-(1) is the 0-weight space of n in U/U-(1), and so on. We say that U is linked if each factor of its nilpotency series is a uniserial g-module, i.e. its g-submodules form a chain. Every uniserial g-module is linked, every linked g-module is indecomposable with irreducible socle, and both converses fail. In this paper, we classify all linked g-modules when g = < x > (sci) a and ad x acts diagonalizably on the abelian Lie algebra a. Moreover, we identify and classify all uniserial g-modules amongst them.