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

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