Indecomposable modules of 2-step solvable Lie algebras in arbitrary characteristic

L. CAGLIERO,; F. SZCHETMAN

COMMUNICATIONS IN ALGEBRA

TAYLOR & FRANCIS INC

Lugar: Londres; Año: 2014

0092-7872

 Let $F$ be an algebraically closed field andconsider the Lie algebra $g=langle x angleltimes a$, where$ad, x$ acts diagonalizably on the abelian Lie algebra $a$.Refer to a $g$-module as admissible if $[g,g]$ acts vianilpotent operators on it, which is automatic if $chr(F)=0$. Inthis paper we classify all indecomposable $g$-modules $U$ which are admissible as well as uniserial,in the sense that $U$ has a unique composition series.When $chr(F)=0$ we recover results previously obtained elsewhere,although the present methods are drastically simpler. When $F$ hasprime characteristic $p$ the classification of admissibleuniserial $g$-modules of length $mleq p$ is essentially the sameas in characteristic 0, although when~$m>p$ the modular analoguesof the isomorphism of classes from characteristic~0 now split intoseveral distinct classes, according to the orbits of anintransitive group action, while, on the other hand, infinitely manynew isomorphism classes arise which only exist in prime characteristic.