nLab BGG resolution

Idea

A BGG resolution is a certain projective resolution of certain representations.

More in detail, give a finite dimensional semisimple complex Lie algebra $\mathfrak{g}$ with Cartan subalgebra $\mathfrak{h}$, and a positive root system, and let $P^+\subset \mathfrak{h}^*$ be the set of dominant integral weights. Then for every $\lambda\in P^+$ consider the corresponding finite dimensional left $U(\mathfrak{g})$-module $L(\lambda)$. Certain resolutions of $L(\lambda)$ are defined in a series of papers of Bernstein, Gel’fand and Gel’fand (BBG 7x) and are now called BGG resolutions. There are also generalizations, e.g. for Kac-Moody algebras.

These resolutions have a natural incarnation in terms of complexes of sections of tractor bundles over flag varieties or more generally over homogeneous parabolic Klein geometries. As such, there are generalizations of the construction to more general parabolic Cartan geometries, called curved BGG sequences. See for instance (Calderbank-Diemer 00, theorem 3.6).

BGG resolutions may be used to construct resolutions of sheaves of constant functions on Klein geometries/coset space $G/H$ that are more efficient (smaller) that the general resolution given by the de Rham complex (the Poincare lemma). In this way BGG resolutions are used notably for computation in Leray spectral sequences as they appear in Penrose transforms (Baston-Eastwood 89, chapter 8).

Properties

Curved $A_\infty$-Structure

Under a cup product the BGG sequence becomes a curved A-infinity algebra. (Calderbank-Diemer 00, section 6)

Related concepts

• Verma module

• parabolic geometry

References

Original case over flag variety

• eom: Alvany Rocha, BGG resolution; wikipedia, category O

• Joseph N. Bernstein, Israel M. Gelfand, Sergei I. Gelfand: Differential operators on the base affine space and a study of $\mathfrak{g}$-modules, in: I. M. Gelfand (ed.), Lie groups and their representations, Adam Hilger (1975) 21-64 [pdf, pdf]

• Joseph N. Bernstein, Israel M. Gelfand, Sergei I. Gelfand: Structure of representations generated by vectors of highest weight, Funkts. Anal. Prilozh. 5: 1 (1971) pp. 1–9; A certain category of -modules, Funkts. Anal. Prilozh. 10: 2 (1976) pp. 1–8;

• James E. Humphreys, Representations of semisimple Lie algebras in the BGG category $\mathcal{O}$, 2008, pdf

Via configuration spaces of points:

• M. Falk, Vladimir Schechtman?, Alexander Varchenko, BGG resolutions via configuration spaces [arxiv/1309.7811]

We study the blow-ups of configuration spaces. These spaces have a structure of what we call an Orlik-Solomon manifold; it allows us to compute the intersection cohomology of certain flat connections with logarithmic singularities using some Aomoto type complexes of logarithmic forms. Using this construction we realize geometrically the $sl_2$ Bernstein - Gelfand - Gelfand resolution as an Aomoto complex.

See also:

• Sergey Arkhipov, A new construction of the semi-infinite BGG resolution, q-alg/9605043

Discussion in the context of the Penrose transform includes

• Michael Eastwood, Variations on the de Rham complex, Notices Amer. Math. Soc. 46 (1999), no. 11, 1368–1376 pdf

• R. J. Baston, M. G. Eastwood, The Penrose transform, Oxford Univ. Press, New York, 1989; MR92j:32112

Curved generalization to parabolic Cartan geometries

The generalization from coset spaces to parabolic Cartan geometries (curved BGG sequences) is discussed in

• R. J. Baston, Verma modules and differential conformal invariants, J. Diff. Geom. 32 (1990) 851–898.

• Andreas Čap, Jan Slovák, Vladimír Souček, Bernstein-Gelfand-Gelfand sequences, ESI preprint 722 (1999).

• Andreas Čap, Jan Slovák, Vladimír Souček, Bernstein-Gelfand-Gelfand sequences, Ann. Math. (II) 154:1 (2001) pp. 97-113 (jstor)

• David Calderbank, Tammo Diemer, Differential invariants and curved Bernstein-Gelfand-Gelfand sequences, J.Reine Angew.Math. 537 (2001) 67-103 (arXiv:math/0001158)

• Andreas Čap, Vladimír Souček, Curved Casimir operators and the BGG machinery, SIGMA 3, 2007, 111 (arxiv:0708.3180 doi)

category: representation theory