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Transactions of the American Mathematical Society

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$ {\germ F}$-categories and $ {\germ F}$-functors in the representation theory of Lie algebras

Author: Ben Cox
Journal: Trans. Amer. Math. Soc. 343 (1994), 433-453
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 $ \mathfrak{a}$ be a Lie subalgebra of a Lie algebra $ \mathfrak{g}$ and $ \Gamma $ be a functor on some category of $ \mathfrak{a}$-modules. We then consider the following general question: For a $ \mathfrak{g}$-module E what hypotheses on $ \Gamma $ and E are sufficient to insure that $ \Gamma (E)$ admits a canonical structure as a $ \mathfrak{g}$-module? The article offers an answer through the introduction of the notion of $ \mathfrak{F}$-categories and $ \mathfrak{F}$-functors. The last section of the article treats various examples of this theory.

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