Semiholonomic Verma Modules

Eastwood, Michael; Slovák, Jan

doi:10.1006/jabr.1997.7136

Cited by 48 publications

(71 citation statements)

References 14 publications

(6 reference statements)

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“…This is a minor variation of the construction of a semiholonomic jet operator given in [17,20,23], except that we have presentedĴ k 0 E as a complicated subspace of an easily defined p-module, whereas in these references,Ĵ k 0 E is given as a complicated p-module structure on an easy vector space, namely k j=0 ⊗ j (g/p) * ⊗ E. The equations definingĴ k 0 E have a natural algebraic interpretation in the dual language of semiholonomic Verma modules.…”

Section: Appendix: Semiholonomic Jets and Verma Modulesmentioning

confidence: 99%

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Calderbank¹,

Diemer²

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

102

260

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Abstract. We give a simple construction of the Bernstein-Gelfand-Gelfand sequences of natural differential operators on a manifold equipped with a parabolic geometry. This method permits us to define the additional structure of a bilinear differential "cup product" on this sequence, satisfying a Leibniz rule up to curvature terms. It is not associative, but is part of an A∞-algebra of multilinear differential operators, which we also obtain explicitly. We illustrate the construction in the case of conformal differential geometry, where the cup product provides a wide-reaching generalization of helicity raising and lowering for conformally invariant field equations.

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Section: Appendix: Semiholonomic Jets and Verma Modulesmentioning

confidence: 99%

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Calderbank¹,

Diemer²

2001

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

102

260

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“…It is readily verified that this is invariant under the transformations (22) and that it is preserved by the spin-tractor connection (23). We subsequently calculate in a metric scale g without further comment.…”

Section: 2mentioning

confidence: 92%

“…of conformally invariant differential operators on the sphere [7,22]. This classification is based on the structure of generalised Verma modules and from this it follows that often the analogue, or replacement, for a conformal elliptic operator on the sphere is an elliptic complex of conformally invariant differential operators.…”

Section: Introductionmentioning

confidence: 99%

Yang-Mills Detour Complexes and Conformal Geometry

Gover

Somberg

Souček

2008

Commun. Math. Phys.

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Abstract. Working over a pseudo-Riemannian manifold, for each vector bundle with connection we construct a sequence of three differential operators which is a complex (termed a Yang-Mills detour complex) if and only if the connection satisfies the full Yang-Mills equations. A special case is a complex controlling the deformation theory of Yang-Mills connections. In the case of Riemannian signature the complex is elliptic. If the connection respects a metric on the bundle then the complex is formally self-adjoint. In dimension 4 the complex is conformally invariant and generalises, to the full Yang-Mills setting, the composition of (two operator) Yang-Mills complexes for (anti-)self-dual Yang-Mills connections. Via a prolonged system and tractor connection a diagram of differential operators is constructed which, when commutative, generates differential complexes of natural operators from the Yang-Mills detour complex. In dimension 4 this construction is conformally invariant and is used to yield two new sequences of conformal operators which are complexes if and only if the Bach tensor vanishes everywhere. In Riemannian signature these complexes are elliptic. In one case the first operator is the twistor operator and in the other sequence it is the operator for Einstein scales. The sequences are detour sequences associated to certain Bernstein-Gelfand-Gelfand sequences.

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“…The constants and differential operators that appear in the sharp inequalities that estimate quantities like those in (22) and (25), and in turn the determinant, also appear in the study of the complementary series of SO 0 (n + 1, 1).…”

Section: The Complementary Seriesmentioning

confidence: 99%

“…These lead, by endpoint differentiation, to the exponential class inequality (22). They are also used in deriving geometric inequalities like the estimate on F 2 (ω) in (25), based on [9,Corollary 3.6].…”

Section: −ν Hmentioning

confidence: 99%

Q-Curvature, Spectral Invariants, and Representation Theory

Branson¹

2007

SIGMA

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Abstract. We give an introductory account of functional determinants of elliptic operators on manifolds and Polyakov-type formulas for their infinitesimal and finite conformal variations. We relate this to extremal problems and to the Q-curvature on even-dimensional conformal manifolds. The exposition is self-contained, in the sense of giving references sufficient to allow the reader to work through all details. Originally we had planned to publish the lecture notes of the instructional lecturers. This paper was submitted by Professor Branson for that purpose. In fact this was the only paper we had received by the deadline. Hence we decided to cancel the plan for a proceedings volume. Just after that period, Thomas Branson unexpectedly passed away. We held the paper without knowing what we could do with it. When the editors of this proceedings volume invited me to submit an article, I realised that this would be an ideal place for Professor Thomas Branson's paper. I immediately submitted the paper to editors of the current proceedings. I would like to take this opportunity to express my sincere appreciation to the editors for their help.Xingwang Xu (National University of Singapore) E-mail: matxuxw@nus.edu.sg The functional determinantIn order to get a feel for the spectral theory of natural differential operators on compact manifolds, recall the idea of Fourier series, where one attempts to expand complex functions on the unit circle S 1 in C in the formThe trigonometric series is an expansion in real eigenfunctions of the Laplacian ∆ = −d 2 /dθ 2 (the eigenvalue being j 2 ). The exponential series is an expansion in eigenfunctions of the operator −id/dθ, which is a square root of the Laplacian; the eigenvalue is k.

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Semiholonomic Verma Modules

Cited by 48 publications

References 14 publications

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Differential invariants and curved Bernstein-Gelfand-Gelfand sequences

Yang-Mills Detour Complexes and Conformal Geometry

Q-Curvature, Spectral Invariants, and Representation Theory

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