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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the determination of irreducible modules by restriction to a subalgebra

Authors: J. Lepowsky and G. W. McCollum
Journal: Trans. Amer. Math. Soc. 176 (1973), 45-57
MSC: Primary 17B10
MathSciNet review: 0323846
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Abstract: Let $ \mathcal{B}$ be an algebra over a field, $ \mathcal{A}$ a subalgebra of $ \mathcal{B}$, and $ \alpha $ an equivalence class of finite dimensional irreducible $ \mathcal{A}$-modules. Under certain restrictions, bijections are established between the set of equivalence classes of irreducible $ \mathcal{B}$-modules containing a nonzero $ \alpha $-primary $ \mathcal{A}$-submodule, and the sets of equivalence classes of all irreducible modules of certain canonically constructed algebras. Related results had been obtained by Harish-Chandra and R. Godement in special cases. The general methods and results appear to be useful in the representation theory of semisimple Lie groups.

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Keywords: Irreducible module, irreducible representation, finitely semisimple module, absolutely irreducible module, primary submodule, extension of submodules, Lie algebra, universal enveloping algebra, Poincaré-Birkhoff-Witt theorem, simple ring, full matrix algebra
Article copyright: © Copyright 1973 American Mathematical Society