Module extensions over classical Lie superalgebras

Abstract

We study certain filtrations of indecomposable injective modules over classical Lie superalgebras, applying a general approach for noetherian rings developed by Brown, Jategaonkar, Lenagan, and Warfield. To indicate the consequences of our analysis, suppose that g \mathfrak {g} is a complex classical simple Lie superalgebra and that E E is an indecomposable injective g \mathfrak {g} -module with nonzero (and so necessarily simple) socle L L . (Recall that every essential extension of L L , and in particular every nonsplit extension of L L by a simple module, can be formed from g \mathfrak {g} -subfactors of E E .) A direct transposition of the Lie algebra theory to this setting is impossible. However, we are able to present a finite upper bound, easily calculated and dependent only on g \mathfrak {g} , for the number of isomorphism classes of simple highest weight g \mathfrak {g} -modules appearing as g \mathfrak {g} -subfactors of E E .