Category O

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title: "Category O" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["representation-theory-of-lie-algebras"] topic_path: "general/representation-theory-of-lie-algebras" source: "https://en.wikipedia.org/wiki/Category_O" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

In the representation theory of semisimple Lie algebras, Category O (or category \mathcal{O}) is a category whose objects are certain representations of a semisimple Lie algebra, and whose morphisms are homomorphisms of representations.

Introduction

Assume that \mathfrak{g} is a (usually complex) semisimple Lie algebra with a Cartan subalgebra \mathfrak{h}. Let \Phi be its root system and let \Phi^+ be a choice of positive roots. Denote by \mathfrak{g}\alpha the root space corresponding to a root \alpha\in\Phi, and set \mathfrak{n}:=\bigoplus{\alpha\in\Phi^+}\mathfrak{g}_\alpha, a nilpotent subalgebra.

If M is a \mathfrak{g}-module and \lambda\in\mathfrak{h}^*, then the \lambda-weight space of M is :M_\lambda={v\in M:\forall h\in\mathfrak{h},; h\cdot v=\lambda(h)v}.

Definition of category O

The objects of category \mathcal O are \mathfrak{g}-modules M such that:

M is finitely generated;
M=\bigoplus_{\lambda\in\mathfrak{h}^*} M_\lambda;
M is locally \mathfrak{n}-finite, i.e. for each v\in M, the \mathfrak{n}-submodule generated by v is finite-dimensional.

Morphisms in this category are the \mathfrak{g}-module homomorphisms.

Basic properties

Each module in category \mathcal O has finite-dimensional weight spaces.
Each module in category \mathcal O is a Noetherian module.
\mathcal O is an abelian category.
\mathcal O has enough projectives and enough injectives.
\mathcal O is closed under taking submodules, quotients, and finite direct sums.
Objects in \mathcal O are Z(\mathfrak{g})-finite: if M is an object and v\in M, then the subspace Z(\mathfrak{g})v\subseteq M generated by v under the action of the center of the universal enveloping algebra is finite-dimensional.

Koszul duality

A homological feature of category \mathcal O is that, after choosing graded lifts of blocks, certain blocks can be described by Koszul algebras. In particular, Beilinson, Ginzburg, and Soergel showed that (for suitable graded realizations) the endomorphism algebra of a projective generator of a block (notably the principal block) is a Koszul algebra A.{{Cite journal | last1 = Beilinson | first1 = Alexander | last2 = Ginzburg | first2 = Victor | last3 = Soergel | first3 = Wolfgang | title = Koszul duality patterns in representation theory | journal = Journal of the American Mathematical Society | volume = 9 | year = 1996 | pages = 473–527 Equivalently, the corresponding graded block of \mathcal O is (via a projective generator) equivalent to the category of finite-dimensional graded modules over A.

In this setting, Koszul duality relates two graded blocks: one block is equivalent to \mathrm{gr}\text{-}A, while a second (dual) graded block is equivalent to \mathrm{gr}\text{-}A^{!}, where A^{!} is the Koszul dual algebra of A. 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 :D^{b}(\mathcal O_{\mathrm{block}}^{\mathrm{gr}});\simeq;D^{b}(\mathcal O_{\mathrm{block}^\vee}^{\mathrm{gr}}), where \mathcal O_{\mathrm{block}}^{\mathrm{gr}}\simeq \mathrm{gr}\text{-}A and \mathcal O_{\mathrm{block}^\vee}^{\mathrm{gr}}\simeq \mathrm{gr}\text{-}A^{!}.

Koszul duality for category \mathcal O is closely connected with geometric and combinatorial structures such as the geometry of the flag variety, perverse sheaves, and Kazhdan–Lusztig theory.{{Cite journal | last = Soergel | first = Wolfgang | title = Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe | journal = Journal of the American Mathematical Society | volume = 3 | year = 1990 | pages = 421–445

Examples

All finite-dimensional \mathfrak{g}-modules and their \mathfrak{g}-homomorphisms are in category \mathcal O.
Verma modules and generalized Verma modules and their \mathfrak{g}-homomorphisms are in category \mathcal 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

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