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

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- Trans. Amer. Math. Soc.
**343**(1994), 433-453 Request permission

## 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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## Additional Information

- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**343**(1994), 433-453 - MSC: Primary 17B67; Secondary 17B55
- DOI: https://doi.org/10.1090/S0002-9947-1994-1191610-7
- MathSciNet review: 1191610