In the

representation theory Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...

semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of modules, direct sum of Simple Lie algebra, simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper Lie algebra#Subalgebras.2C ideals ...

s, Category O (or category

\mathcal

) is a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

whose

objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...

are certain

representations ''Representations'' is an interdisciplinary journal in the humanities published quarterly by the University of California Press. The journal was established in 1983 and is the founding publication of the New Historicism movement of the 1980s. It ...

of a

semisimple In mathematics, semi-simplicity is a widespread concept in disciplines such as linear algebra, abstract algebra, representation theory, category theory, and algebraic geometry. A semi-simple object is one that can be decomposed into a sum of ''sim ...

Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...

and

morphism In mathematics, a morphism is a concept of category theory that generalizes structure-preserving maps such as homomorphism between algebraic structures, functions from a set to another set, and continuous functions between topological spaces. Al ...

s are

homomorphisms In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...

of representations.

Introduction

Assume that

\mathfrak

is a (usually

complex Complex commonly refers to: * Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe ** Complex system, a system composed of many components which may interact with each ...

) semisimple Lie algebra with a

Cartan subalgebra In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if ,Y\in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak). They were introduced by ...

\mathfrak

\Phi

is a

root system In mathematics, a root system is a configuration of vector space, vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and ...

and

\Phi^+

is a system of

positive roots In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation ...

. Denote by

\mathfrak_\alpha

the root space corresponding to a root

\alpha\in\Phi

and

\mathfrak:=\bigoplus_ \mathfrak_\alpha

nilpotent In mathematics, an element x of a ring (mathematics), ring R is called nilpotent if there exists some positive integer n, called the index (or sometimes the degree), such that x^n=0. The term, along with its sister Idempotent (ring theory), idem ...

subalgebra. If

M

is a

\mathfrak

-module and

\lambda\in\mathfrak^*

, then

M_\lambda

is the weight space :

M_\lambda=\.

Definition of category O

The objects of category

\mathcal O

are

\mathfrak

-modules

M

such that #

M

is finitely generated #

M=\bigoplus_ M_\lambda

M

is locally

\mathfrak

-finite. That is, for each

v \in M

, the

\mathfrak

-module generated by

v

is finite-dimensional. Morphisms of this category are the

\mathfrak

-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 In abstract algebra, a Noetherian module is a module that satisfies the ascending chain condition on its submodules, where the submodules are partially ordered by inclusion. Historically, Hilbert was the first mathematician to work with the pr ...

. *O is an

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 prototypical example of an abelian category is the category o ...

*O has

enough projectives In category theory, the notion of a projective object generalizes the notion of a projective module. Projective objects in abelian categories are used in homological algebra. The dual notion of a projective object is that of an injective object. ...

and injectives. *O is closed under taking

submodule In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since t ...

s, quotients and finite direct sums. *Objects in O are

Z(\mathfrak)

-finite, i.e. if

M

is an object and

v\in M

, then the subspace

Z(\mathfrak) v\subseteq M

generated by

v

under the action of the

center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentrici ...

of the

universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representa ...

, is finite-dimensional.

Examples

* All finite-dimensional

\mathfrak

-modules and their

\mathfrak

-homomorphisms are in category O. *

Verma module Verma modules, named after Daya-Nand Verma, are objects in the representation theory of Lie algebras, a branch of mathematics. Verma modules can be used in the classification of irreducible representations of a complex semisimple Lie algebra. Sp ...

s and

generalized Verma module In mathematics, generalized Verma modules are a generalization of a (true) Verma module, and are objects in the representation theory of Lie algebras. They were studied originally by James Lepowsky in the 1970s. The motivation for their study is t ...

s and their

\mathfrak

-homomorphisms are in category O.

References

*{{Citation , last1=Humphreys , first1=James E. , author1-link=James E. Humphreys , title=Representations of semisimple Lie algebras in the BGG category O , publisher=AMS , year=2008 , isbn=978-0-8218-4678-0 , url=http://www.math.umass.edu/~jeh/bgg/main.pdf , url-status=dead , archiveurl=https://web.archive.org/web/20120321142849/http://www.math.umass.edu/~jeh/bgg/main.pdf , archivedate=2012-03-21 Representation theory of Lie algebras

Introduction

Definition of category O

Basic properties

Examples

See also

References