BGG resolution
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
![]() |
![]() |
![]() |
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where ,
and
means that
has been omitted. Let
be the constant term of
. Then
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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
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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
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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
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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
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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].
References
[a1] | I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "Structure of representations generated by vectors of highest weight" Funkts. Anal. Prilozh. , 5 : 1 (1971) pp. 1–9 |
[a2] | I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "A certain category of ![]() |
[a3] | I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "Differential operators on the base affine space and a study of ![]() |
[a4] | A. Rocha-Caridi, "Splitting criteria for ![]() ![]() |
[a5] | A. Rocha-Caridi, N.R. Wallach, "Projective modules over graded Lie algebras" Math. Z. , 180 (1982) pp. 151–177 |
[a6] | A. Rocha-Caridi, N.R. Wallach, "Highest weight modules over graded Lie algebras: Resolutions, filtrations and character formulas" Trans. Amer. Math. Soc. , 277 : 1 (1983) pp. 133–162 |
BGG resolution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=BGG_resolution&oldid=14028