/pdf/differential-systems-associated-with-simple-graded-lie-1y92cs5frv.pdf

On the equivalence problems associated with simple graded Lie algebras

Differential Systems Associated with Simple Graded Lie Algebras

The Bernstein-Gelfand-Gelfand sequences extend the complexes of
homogeneous vector bundles to curved Cartan geometries

background

https://www.jstor.org/stable/pdfplus/3062111.pdf

Bernstein-Gelfand-Gelfand sequences

Parabolic geometries may be considered as curved analogues of the homogeneous spaces G/P where G is a semisimple Lie group and P C G a parabolic subgroup. Conformal geometries and CR geometries are examples of such structures. We present a uniform description of a calculus, called tractor calculus, based on natural bundles with canonical linear connections for all parabolic geometries. It is shown that from these bundles and connections one can recover the Cartan bundle and the Cartan connection. In particular we characterize the normal Cartan connection from this induced bundle/connection perspective. We construct explicitly a family of fundamental first order differential operators, which are analogous to a covariant derivative, iterable and defined on all natural vector bundles on parabolic geometries. For an important subclass of parabolic geometries we explicitly and directly construct the tractor bundles, their canonical linear connections and the machinery for explicitly calculating via the tractor calculus.

/pdf/tractor-calculi-for-parabolic-geometries-3p1ohi504k.pdf

Tractor calculi for parabolic geometries

This is the first part of a series of papers. The whole series aims to develop the tools for the study of all almost Hermitian symmetric structures in a unified way. In particular, methods for the construction of invariant operators, their classification and the study of their properties will be worked out. In this paper we present the invariant differentiation with respect to a Cartan connection and we expand the differentials in the terms of the underlying linear connections belonging to the structures in question. Then we discuss the holonomic and non-holonomic jet extensions and we suggest methods for the construction of invariant operators.

Invariant operators on manifolds with almost Hermitian symmetric structures, I. Invariant differentiation

This paper demonstrates the power of the calculus developed in the two previous parts of the series for all real forms of the almost Hermitian symmetric structures on smooth manifolds, including, e.g., conformal Riemannian and almost quaternionic geometries. Exploiting some finite-dimensional representation theory of simple Lie algebras, we give explicit formulae for distinguished invariant curved analogues of the standard operators in terms of the linear connections belonging to the structures in question, so in particular we prove their existence. Moreover, we prove that these formulae for kth order standard operators, kD 1; 2;:::, are universal for all geometries in question.

Invariant operators on manifolds with almost Hermitian symmetric structures

/pdf/on-the-equivalence-problems-associated-with-simple-graded-1fakwhcwh3.pdf

On the equivalence problems associated with simple graded Lie algebras

Citations

Differential Systems Associated with Simple Graded Lie Algebras

Bernstein-Gelfand-Gelfand sequences

Tractor calculi for parabolic geometries

Invariant operators on manifolds with almost Hermitian symmetric structures, I. Invariant differentiation

Invariant operators on manifolds with almost Hermitian symmetric structures

Related Papers (5)

Differential Systems Associated with Simple Graded Lie Algebras

On differential systems, graded Lie algebras and pseudo-groups

Lie Algebra Cohomology and the Generalized Borel-Weil Theorem

Real hypersurfaces in complex manifolds

Parabolic Geometries I