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

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].

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 -modules" Funkts. Anal. Prilozh. , 10 : 2 (1976) pp. 1–8
[a3]	I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, "Differential operators on the base affine space and a study of -modules" I.M. Gelfand (ed.) , Lie groups and their representations, Proc. Summer School on Group Representations , Janos Bolyai Math. Soc.&Wiley (1975) pp. 39–64
[a4]	A. Rocha-Caridi, "Splitting criteria for -modules induced from a parabolic and the Bernstein–Gelfand–Gelfand resolution of a finite dimensional, irreducible -module" Trans. Amer. Math. Soc. , 262 : 2 (1980) pp. 335–366
[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