Category O

In the representation theory of semisimple Lie algebras, Category O is a category whose objects are certain representations of a semisimple Lie algebra, and whose morphisms are homomorphisms of representations.

Introduction

Assume that is a semisimple Lie algebra with a Cartan subalgebra. Let be its root system and let be a choice of positive roots. Denote by the root space corresponding to a root, and set
a nilpotent subalgebra.
If is a -module and, then the -weight space of is

Definition of category O

The objects of category are -modules such that:

is finitely generated;
;
is locally -finite, i.e. for each, the -submodule generated by is finite-dimensional.

Morphisms in this category are the -module homomorphisms.

Basic properties

Each module in category has finite-dimensional weight spaces.
Each module in category is a Noetherian module.
is an abelian category.
has enough projectives and enough injectives.
is closed under taking submodules, quotients, and finite direct sums.
Objects in are -finite: if is an object and, then the subspace generated by under the action of the center of the universal enveloping algebra is finite-dimensional.
Koszul duality

A homological feature of category is that, after choosing graded lifts of blocks, certain blocks can be described by Koszul algebras. In particular, Beilinson, Ginzburg, and Soergel showed that the endomorphism algebra of a projective generator of a block is a Koszul algebra.
Equivalently, the corresponding graded block of is equivalent to the category of finite-dimensional graded modules over.
In this setting, Koszul duality relates two graded blocks: one block is equivalent to, while a second graded block is equivalent to, where is the Koszul dual algebra of.
The associated Koszul duality functors induce a triangulated equivalence between the bounded derived categories of these graded realizations, i.e. an equivalence of the form
where and.
Koszul duality for category is closely connected with geometric and combinatorial structures such as the geometry of the flag variety, perverse sheaves, and Kazhdan–Lusztig theory.

Examples

All finite-dimensional -modules and their -homomorphisms are in category.
Verma modules and generalized Verma modules and their -homomorphisms are in category.