Category O

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

Introduction

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

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

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

Definition of category O

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

1. ${\displaystyle M}$ is finitely generated;
2. ${\displaystyle M=\bigoplus _{\lambda \in {\mathfrak {h}}^{*}}M_{\lambda }}$;
3. ${\displaystyle M}$ is locally ${\displaystyle {\mathfrak {n}}}$-finite, i.e. for each ${\displaystyle v\in M}$, the ${\displaystyle {\mathfrak {n}}}$-submodule generated by ${\displaystyle v}$ is finite-dimensional.

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

Basic properties

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

Koszul duality

A homological feature of category ${\displaystyle {\mathcal {O}}}$ 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 ${\displaystyle A}$. ^[1] Equivalently, the corresponding graded block of ${\displaystyle {\mathcal {O}}}$ is (via a projective generator) equivalent to the category of finite-dimensional graded modules over ${\displaystyle A}$.

In this setting, Koszul duality relates two graded blocks: one block is equivalent to ${\displaystyle \mathrm {gr} {\text{-}}A}$, while a second (dual) graded block is equivalent to ${\displaystyle \mathrm {gr} {\text{-}}A^{!}}$, where ${\displaystyle A^{!}}$ is the Koszul dual algebra of ${\displaystyle A}$. ^[1] 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

${\displaystyle D^{b}({\mathcal {O}}_{\mathrm {block} }^{\mathrm {gr} })\;\simeq \;D^{b}({\mathcal {O}}_{\mathrm {block} ^{\vee }}^{\mathrm {gr} }),}$

where ${\displaystyle {\mathcal {O}}_{\mathrm {block} }^{\mathrm {gr} }\simeq \mathrm {gr} {\text{-}}A}$ and ${\displaystyle {\mathcal {O}}_{\mathrm {block} ^{\vee }}^{\mathrm {gr} }\simeq \mathrm {gr} {\text{-}}A^{!}}$. ^[1]

Koszul duality for category ${\displaystyle {\mathcal {O}}}$ is closely connected with geometric and combinatorial structures such as the geometry of the flag variety, perverse sheaves, and Kazhdan–Lusztig theory. ^[2]

Examples

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

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 (PDF), AMS, ISBN   978-0-8218-4678-0, archived from the original (PDF) on 2012-03-21

1. Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang (1996). "Koszul duality patterns in representation theory". Journal of the American Mathematical Society. 9: 473–527.
2. Soergel, Wolfgang (1990). "Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe". Journal of the American Mathematical Society. 3: 421–445.