The structure of a real
Lie group
can be
studied by considering representations of the complexification
of its
Lie algebra
(cf. also
Representation of a Lie algebra).
These are viewed as left modules over the
universal enveloping algebra
of
,
or
-modules.
The Lie algebras
considered here are the complexifications of real
semi-simple Lie algebras corresponding to real, connected, semi-simple
Lie
groups. A
Cartan subalgebra
,
that is, a maximal Abelian
subalgebra with the property that its adjoint representation on
is semi-simple, is chosen (cf. also
Cartan subalgebra).
A
root system
,
corresponding to the resulting decomposition of
,
is obtained. A further choice of a
positive root system
determines subalgebras
and
corresponding to the
positive and negative root spaces, respectively. The building blocks
in
the study of
are the finite-dimensional irreducible
-modules
.
They are indexed by the set
of
dominant integral weights
relative to
.
For any ring
with unity, a
resolution
of a left
-module
is an
exact chain complex of
-modules:
For example, let
be a complex Lie algebra, and let
,
where
is the
th
exterior power of
,
.
Let
where
,
and
means that
has been omitted. Let
be the constant
term of
.
Then
is the
standard resolution
of the trivial
-module
.
If
is a subalgebra, one considers the relative version
of
by setting
.
One observes that the
obvious modification of the
produces mappings
,
,
and that the resulting complex is similarly exact.
In
[a3]
two constructions of a resolution of
,
,
were obtained. They are described below.
Weak BGG resolution.
Let
and let
be the category of finitely-generated
-diagonalizable
-finite
-modules
([a2]).
Let
denote the centre of
.
If M is a
-module,
let
denote the set of eigenvalues of
.
For
,
let
denote the eigenspace associated to
.
The set
consists of only one element, denoted by
.
For
,
defines an
exact functor
in
.
If
,
let
be the image of
under the functor
.
is known as the
weak BGG resolution.
Its importance lies in the property of the
explained below. For
,
denotes the trivially extended action of
from
to
.
The
-module
is the
Verma module
associated to
.
Let
denote the set of
simple
(i.e. indecomposable in
,
positive roots. Let
be the group of automorphisms of
generated by the reflections
relative to
(cf. also
Weyl group).
Let
be the set of elements
that are minimally expressed as a product of
reflections
,
.
One writes
.
Each
has a
filtration (cf. also
Filtered algebra)
of
-modules
such that
and
,
where
and
.
If
is a Lie algebra and
is a resolution of the
-module
by projective
-modules,
and
is the image of
under the functor
,
then
.
The
cohomology groups
are defined as
.
If
,
and
,
the weak BGG resolution implies that
.
Strong BGG resolution.
For
one writes
if there exists a
such that
and
.
This relation induces a partial ordering
on
,
by setting
whenever there are
in
such that
.
It was shown in
[a1]
that
if and only if
.
Furthermore, every such homomorphism is
zero or injective. One fixes, for each pair
,
one such injection
.
Let
.
Therefore, a
-homomorphism
is determined by a complex matrix
with
and
.
It is shown in
[a3]
that there exist
,
,
,
for
,
such that
where
is the canonical surjection, is exact. This
strong BGG resolution
refines the weak BGG resolution
and, in particular, calculates the cohomology groups
.
In
[a4]
it was proved that the weak and the strong BGG
resolutions are isomorphic. The results of
[a4]
apply to the more general situation of parabolic subalgebras
.
They imply the existence of a complex in terms of the degenerate
principal series representations of
that has the same cohomology as
the de Rham
complex
[a4].
The BGG resolution has been extended to
Kac–Moody algebras (see
[a5]
and also
Kac–Moody algebra)
and to the
Lie algebra of vector fields on the circle
[a6].