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 $\mathfrak {g}$ is a complex classical simple Lie superalgebra and that $E$ is an indecomposable injective $\mathfrak {g}$-module with nonzero (and so necessarily simple) socle $L$. (Recall that every essential extension of $L$, and in particular every nonsplit extension of $L$ by a simple module, can be formed from $\mathfrak {g}$-subfactors of $E$.) 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 $\mathfrak {g}$, for the number of isomorphism classes of simple highest weight $\mathfrak {g}$-modules appearing as $\mathfrak {g}$-subfactors of $E$.