𝔉-categories and 𝔉-functors in the representation theory of Lie algebras

Cox, Ben

doi:10.1090/S0002-9947-1994-1191610-7

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

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

DOI: https://doi.org/10.1090/S0002-9947-1994-1191610-7

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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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