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Representation Theory

ISSN 1088-4165



Module extensions over classical
Lie superalgebras

Author: Edward S. Letzter
Journal: Represent. Theory 3 (1999), 354-372
MSC (1991): Primary 16P40, 17A70; Secondary 17B35
Published electronically: October 5, 1999
MathSciNet review: 1711503
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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 $E$.

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Additional Information

Edward S. Letzter
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843

Received by editor(s): November 20, 1998
Received by editor(s) in revised form: July 14, 1999
Published electronically: October 5, 1999
Additional Notes: This research was partially supported by grants from the National Science Foundation.
Article copyright: © Copyright 1999 American Mathematical Society