categories and functors in the representation theory of Lie algebras
Author:
Ben Cox
Journal:
Trans. Amer. Math. Soc. 343 (1994), 433453
MSC:
Primary 17B67; Secondary 17B55
MathSciNet review:
1191610
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Abstract: The fields of algebra and representation theory contain abundant examples of functors on categories of modules over a ring. These include of course Horn, Ext, and Tor as well as the more specialized examples of completion and localization used in the setting of representation theory of a semisimple Lie algebra. In this article we let be a Lie subalgebra of a Lie algebra and be a functor on some category of modules. We then consider the following general question: For a module E what hypotheses on and E are sufficient to insure that admits a canonical structure as a module? The article offers an answer through the introduction of the notion of categories and functors. The last section of the article treats various examples of this theory.
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 A. Joseph, The Enright functor in the BernsteinGelfandGelfand category , Invent. Math. 57 (1980), 101118.
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 , Completion functors in the category , Lecture Notes in Math., vol. 1020, SpringerVerlag, Berlin and New York, 1982, pp. 80106.
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 , On the Demazure character formula, Ann. Sci. Ecole Norm. Sup. 18 (1985), 389419. MR 826100 (87g:17006a)
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 W. Kim, On the concept of completions and explicit determination of invariant pairing for general KacMoody Lie algebras, Comm. Algebra 16 (1988), 185222. MR 921949 (89d:17025)
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 A. Knapp, Lie groups, Lie algebras, and cohomology, Math. Notes, vol. 34, Princeton Univ. Press, Princeton, NJ, 1988. MR 938524 (89j:22034)
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 A. RochaCaridi and N. Wallach, Projective modules over graded Lie algebras. I, Invent. Math. 180 (1982), 151177. MR 661694 (83h:17018)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947199411916107
PII:
S 00029947(1994)11916107
Article copyright:
© Copyright 1994
American Mathematical Society
