Category O (or category
O
) is a mathematical object in representation theory of semisimple Lie algebras. It is a category whose objects are certain representations of a semisimple Lie algebra and morphisms are homomorphisms of representations.
Assume that
g
is a (usually complex) semisimple Lie algebra with a Cartan subalgebra
h
,
Φ
is a root system and
Φ
+
is a system of positive roots. Denote by
g
α
the root space corresponding to a root
α
∈
Φ
and
n
:=
⊕
α
∈
Φ
+
g
α
a nilpotent subalgebra.
If
M
is a
g
-module and
λ
∈
h
∗
, then
M
λ
is the weight space
M
λ
=
{
v
∈
M
;
∀
h
∈
h
h
⋅
v
=
λ
(
h
)
v
}
.
The objects of category O are
g
-modules
M
such that
-
M
is finitely generated
-
M
=
⊕
λ
∈
h
∗
M
λ
-
M
is locally
n
-finite, i.e. for each
v
∈
M
, the
n
-module generated by
v
is finite-dimensional.
Morphisms of this category are the
g
-homomorphisms of these modules.
Each module in a 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 injectives.
O is closed to submodules, quotients and finite direct sums
Objects in O are
Z
(
g
)
-finite, i.e. if
M
is an object and
v
∈
M
, then the subspace
Z
(
g
)
v
⊆
M
generated by
v
under the action of the center of the universal enveloping algebra, is finite-dimensional.
All finite-dimensional
g
-modules and their
g
-homomorphisms are in category O.
Verma modules and generalized Verma modules and their
g
-homomorphisms are in category O.
