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:
Mis finitely generated;M=\bigoplus_{\lambda\in\mathfrak{h}^*} M_\lambda;Mis locally\mathfrak{n}-finite, i.e. for eachv\in M, the\mathfrak{n}-submodule generated byvis finite-dimensional.
Morphisms in this category are the \mathfrak{g}-module homomorphisms.
Basic properties
- Each module in category
\mathcal Ohas finite-dimensional weight spaces. - Each module in category
\mathcal Ois a Noetherian module. \mathcal Ois an abelian category.\mathcal Ohas enough projectives and enough injectives.\mathcal Ois closed under taking submodules, quotients, and finite direct sums.- Objects in
\mathcal OareZ(\mathfrak{g})-finite: ifMis an object andv\in M, then the subspaceZ(\mathfrak{g})v\subseteq Mgenerated byvunder 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.[1]
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.[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
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^{!}.[1]
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.[2]
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.
See also
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
- ^ Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang (1996). "Koszul duality patterns in representation theory". Journal of the American Mathematical Society. 9: 473–527.
- ^ Soergel, Wolfgang (1990). "Kategorie O, perverse Garben und Moduln über den Koinvarianten zur Weylgruppe". Journal of the American Mathematical Society. 3: 421–445.