Classification of linked indecomposable modules of a family of solvable Lie algebras

L. CAGLIERO,; F. SZCHETMAN

JOURNAL OF ALGEBRA AND ITS APPLICATIONS

WORLD SCIENTIFIC PUBL CO PTE LTD

Lugar: London, UK; Año: 2015

0219-4988

Let $\g$ be a finite dimensional Liealgebra over a field of characteristic 0, with solvable radical$\r$ and nilpotent radical $\n=[\g,\r]$. Given a finitedimensional $\g$-module $U$, its nilpotency series$0\subset U^{(1)}\subset\cdots\subset U^{(m)}=U$ 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 saythat $U$ is linked if each factor of its nilpotency series is auniserial $\g/\n$-module, i.e., its $\g/\n$-submodules form achain. Every uniserial $\g$-module is linked, every linked$\g$-module is indecomposable with irreducible socle, and bothconverses fail.In this paper we classify all linked $\g$-modules when $\g=\langlex\rangle\ltimes \a$ and $\ad\, x$ actsdiagonalizably on the abelian Lie algebra $\a$. Moreover, weidentify and classify all uniserial $\g$-modules amongst them.