Category O is a mathematical

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

 object in representation theory

Representation theory

Representation theory is a branch of mathematics that studies abstract algebraic structures by representing their elements as linear transformations of vector spaces, and studiesmodules over these abstract algebraic structures...

 of semisimple Lie algebras. It is a category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

 whose objects are
certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.

Introduction

Assume that is a (usually complex) semisimple Lie algebra with a Cartan subalgebra
, is a root system

Root system of a semi-simple Lie algebra

In mathematics, there is a one-to-one correspondence between reduced crystallographic root systems and semi-simple Lie algebras. We show the construction of a root system from a semi-simple Lie algebra and conversely, the construction of a semi-simple Lie algebra from a reduced crystallographic...

 and is a system of positive roots. Denote by
the root space corresponding to a root and a nilpotent subalgebra.

If is a -module and , then is the weight space

Definition of category O

The objects of category O are -modules such that

1. is finitely generated
2. 
3. is locally -finite, i.e. for each , the -module generated by is finite-dimensional.

Morphisms of this category are the -homomorphisms of these modules.

Basic properties

• Each module in a category O has finite-dimensional weight spaces.
• Each module in category O is a Noetherian module
  Noetherian module
  In abstract algebra, an Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion....
  
  .
• O is an abelian category
  Abelian category
  In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototype example of an abelian category is the category of abelian groups, Ab. The theory originated in a tentative...
  
  
• O is closed to submodules, quotients and finite direct sums
• Objects in O are -finite, i.e. if is an object and , then the subspace generated by under the action of the center
  Center (algebra)
  The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. It is often denoted Z, from German Zentrum, meaning "center". More specifically:...
  
   of the universal enveloping algebra
  Universal enveloping algebra
  In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
  
  , is finite-dimensional.

Examples

• All finite-dimensional -modules and their -homomorphisms are in category O.
• Verma module
  Verma module
  Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics.The definition of a Verma module looks complicated, but Verma modules are very natural objects, with useful properties...
  
  s and generalized Verma module
  Generalized Verma module
  In mathematics, generalized Verma modules are a generalization of a Verma module, and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky in the nineteen seventies. The motivation for their study is that their homomorphisms correspond to...
  
  s and their -homomorphisms) are in category O.

See also

• Highest-weight module
• Universal enveloping algebra
  Universal enveloping algebra
  In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
  
  
• Highest-weight category