128.32.252.15: /* Kost's method: t approximation */

2012-01-04T20:09:12Z

Kost's method: t approximation

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'''Category O''' (or '''category <math>\mathcal{O}</math>''') is a [[mathematics|mathematical]] object in [[representation theory]] of [[semisimple Lie algebra]]s. It is a [[category (mathematics)|category]] whose objects are
certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.

== Introduction ==
Assume that <math>\mathfrak{g}</math> is a (usually complex) semisimple Lie algebra with a [[Cartan subalgebra]]
<math>\mathfrak{h}</math>, <math>\Phi</math> is a [[Root system of a semi-simple Lie algebra|root system]] and <math>\Phi^+</math> is a system of [[Root system#Positive_roots_and_simple_roots|positive roots]]. Denote by <math>\mathfrak{g}_\alpha</math>
the [[root space]] corresponding to a root <math>\alpha\in\Phi</math> and <math>\mathfrak{n}:=\oplus_{\alpha\in\Phi^+} \mathfrak{g}_\alpha</math> a [[Nilpotent Lie algebra|nilpotent]] subalgebra.

If <math>M</math> is a <math>\mathfrak{g}</math>-module and <math>\lambda\in\mathfrak{h}^*</math>, then <math>M_\lambda</math> is the [[Weight_(representation_theory)#Weight_space_of_a_representation|weight space]]
:<math>M_\lambda=\{v\in M;\,\, \forall\,h\in\mathfrak{h}\,\,h\cdot v=\lambda(h)v\}.</math>

== Definition of category O ==
The objects of category O are <math>\mathfrak{g}</math>-modules <math>M</math> such that
# <math>M</math> is finitely generated
# <math>M=\oplus_{\lambda\in\mathfrak{h}^*} M_\lambda</math>
# <math>M</math> is locally <math>\mathfrak{n}</math>-finite, i.e. for each <math>v\in M</math>, the <math>\mathfrak{n}</math>-module generated by <math>v</math> is finite-dimensional.

Morphisms of this category are the <math>\mathfrak{g}</math>-homomorphisms of these modules.

== Basic properties ==
{{Expand section|date=September 2011}}
*Each module in a category O has finite-dimensional [[Weight_(representation_theory)#Weight_space_of_a_representation|weight spaces]].
*Each module in category O is a [[Noetherian module]].
*O is an [[abelian category]]
*O has [[Projective_object |enough projectives]] and [[Injective_object |injectives]].
*O is closed to [[submodule]]s, quotients and finite direct sums
*Objects in O are <math>Z(\mathfrak{g})</math>-finite, i.e. if <math>M</math> is an object and <math>v\in M</math>, then the subspace <math>Z(\mathfrak{g}) v\subseteq M</math> generated by <math>v</math> under the action of the [[Center (algebra)|center]] of the [[universal enveloping algebra]], is finite-dimensional.

== Examples ==
{{Expand section|date=September 2011}}
* All finite-dimensional <math>\mathfrak{g}</math>-modules and their <math>\mathfrak{g}</math>-homomorphisms are in category O.
* [[Verma module]]s and [[generalized Verma module]]s and their <math>\mathfrak{g}</math>-homomorphisms are in category O.

== See also ==
*[[Highest-weight module]]
*[[Universal enveloping algebra]]
*[[Highest-weight category]]

== References ==
*{{Citation | last1=Humphreys | first1=James E. | author1-link=James Humphreys (mathematician) | 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}}

[[Category:Representation theory of Lie algebras]]

Extensions of Fisher's method - Revision history

128.32.252.15: /* Kost's method: t approximation */