geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
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).
Under a cup product the BGG sequence becomes a curved A-infinity algebra. (Calderbank-Diemer 00, section 6)
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:
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:
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
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)
Last revised on July 24, 2024 at 17:27:26. See the history of this page for a list of all contributions to it.