Module Extensions Over Classical Lie Superalgebras

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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 is a complex classical simple Lie superalgebra and that E is an indecomposable injective 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 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 g, for the number of isomorphism classes of simple highest weight g-modules appearing as g-subfactors of E.