Category O

In the representation theory of semisimple Lie algebras, Category O (or 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 $\mathfrak{𝔤}$ is a (usually complex) semisimple Lie algebra with a Cartan subalgebra $\mathfrak{𝔥}$. Let $\mathrm{\Phi }$ be its root system and let ${\mathrm{\Phi }}^{+}$ be a choice of positive roots. Denote by ${\mathfrak{𝔤}}_{\alpha }$ the root space corresponding to a root $\alpha \in \mathrm{\Phi }$, and set $\mathfrak{𝔫}:=\underset{\alpha \in {\mathrm{\Phi }}^{+}}{⨁}{\mathfrak{𝔤}}_{\alpha }$, a nilpotent subalgebra.

If $M$ is a $\mathfrak{𝔤}$-module and $\lambda \in {\mathfrak{𝔥}}^{*}$, then the $\lambda$-weight space of $M$ is

$M_{\lambda }=\{v\in M:\mathrm{\forall }h\in \mathfrak{𝔥},h\cdot v=\lambda (h)v\}.$

Definition of category O

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

1. $M$ is finitely generated;
2. $M=\underset{\lambda \in {\mathfrak{𝔥}}^{*}}{⨁}{M}_{\lambda }$;
3. $M$ 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 ${𝒪}$ has finite-dimensional weight spaces.
• Each module in category ${𝒪}$ is a Noetherian module.
• ${𝒪}$ is an abelian category.
• ${𝒪}$ has enough projectives and enough injectives.
• ${𝒪}$ is closed under taking submodules, quotients, and finite direct sums.
• Objects in ${𝒪}$ 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 ${𝒪}$ 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 $A$. Equivalently, the corresponding graded block of ${𝒪}$ 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 $\mathrm{g}\mathrm{r}\text{-}A$, while a second (dual) graded block is equivalent to $\mathrm{g}\mathrm{r}\text{-}{A}^{!}$, 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

$D^b({{𝒪}}_{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}^{\mathrm{g}\mathrm{r}})\simeq D^b({{𝒪}}_{{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}^{\vee }}^{\mathrm{g}\mathrm{r}}),$

where ${{𝒪}}_{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}^{\mathrm{g}\mathrm{r}}\simeq \mathrm{g}\mathrm{r}\text{-}A$ and ${{𝒪}}_{{\mathrm{b}\mathrm{l}\mathrm{o}\mathrm{c}\mathrm{k}}^{\vee }}^{\mathrm{g}\mathrm{r}}\simeq \mathrm{g}\mathrm{r}\text{-}{A}^{!}$.

Koszul duality for category ${𝒪}$ 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 $\mathfrak{𝔤}$-modules and their $\mathfrak{𝔤}$-homomorphisms are in category ${𝒪}$.
• Verma modules and generalized Verma modules and their $\mathfrak{𝔤}$-homomorphisms are in category ${𝒪}$.

See also

• Highest-weight module
• Universal enveloping algebra
• Highest-weight category
• Koszul algebra

References

• Humphreys, James E. (2008), Representations of semisimple Lie algebras in the BGG category O, AMS, ISBN 978-0-8218-4678-0, archived from the original on 2012-03-21, https://web.archive.org/web/20120321142849/http://www.math.umass.edu/~jeh/bgg/main.pdf