Categories with projective functors

Abstract

We introduce a notion of a category with full projective functors. It encodes certain common properties of categories appearing in representation theory of Lie groups, Lie algebras and quantum groups. We describe the left or right exact functors which naturally commute with projective functors and provide a unified approach to the verification of relations between such functors. 2000 Mathematics Subject Classification 17B10.

Serre functors for Lie algebras and superalgebras

“…The paper [25] establishes the important property that the Serre functor naturally commutes with projective functors (in the sense of [20]). This silently uses the fact that the category O for Lie algebras is a category with full projective functors in the sense of [20].…”

“…In Section 3 we recall the theory of categories with full projective functors following [20]. In Section 4 and Section 5 we prove the main results for Lie algebras and superalgebras, respectively.…”

“…We assume that F is closed (up to isomorphism) under direct sums and composition. Following [20] we say that (A, M, F) is a category with full projective functors, or, simply, an fpf-category, provided that…”

“…Let F denote the family of all such functors. Then (O λ , ∆(µ), F) is an fpf-category, see [20,Proposition 16].…”

A Twisted Approach to Kostant's Problem

“…In [18] it was shown that Arkhipov's functors are adjoint to Joseph's completion functors (see [15]), which suggests a close connection to Kostant's problem. We base our arguments mostly on the results of [3] and also use some results from [17,18,22].…”