13 research outputs found
Conformal operators on weighted forms; their decomposition and null space on Einstein manifolds
There is a class of Laplacian like conformally invariant differential
operators on differential forms which may be considered the
generalisation to differential forms of the conformally invariant powers of the
Laplacian known as the Paneitz and GJMS operators. On conformally Einstein
manifolds we give explicit formulae for these as explicit factored polynomials
in second order differential operators. In the case the manifold is not Ricci
flat we use this to provide a direct sum decomposition of the null space of the
in terms of the null spaces of mutually commuting second order
factors.Comment: minor changes; we correct typos, add further explanation and clarify
the treatment of the higher order operators in the case of even dimensions;
results unchange
Inverting covariant exterior derivative
The algorithm for inverting covariant exterior derivative understood as a sum
of the exterior derivative and an exterior multiplication by a one-form is
provided. It works for a sufficiently small star-shaped region of a fibered set
- a local version of a fiber manifold. The relation to operational calculus and
operator theory is outlined. The upshot of this paper is to show that, using
the linear homotopy operator, the solution of the covariancy constant and
related equations is an easy task.Comment: 26 pages, 1 figur
A Projective-to-Conformal Fefferman-Type Construction
We study a Fefferman-type construction based on the inclusion of Lie groups
into . The construction associates a
split-signature -conformal spin structure to a projective structure of
dimension . We prove the existence of a canonical pure twistor spinor and a
light-like conformal Killing field on the constructed conformal space. We
obtain a complete characterisation of the constructed conformal spaces in terms
of these solutions to overdetermined equations and an integrability condition
on the Weyl curvature. The Fefferman-type construction presented here can be
understood as an alternative approach to study a conformal version of classical
Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by
the authors. The present work therefore gives a complete exposition of
conformal Patterson-Walker metrics from the viewpoint of parabolic geometry
The conformal Killing equation on forms -- prolongations and applications
We construct a conformally invariant vector bundle connection such that its
equation of parallel transport is a first order system that gives a
prolongation of the conformal Killing equation on differential forms. Parallel
sections of this connection are related bijectively to solutions of the
conformal Killing equation. We construct other conformally invariant
connections, also giving prolongations of the conformal Killing equation, that
bijectively relate solutions of the conformal Killing equation on -forms to
a twisting of the conformal Killing equation on (k - l)-forms for various
integers l. These tools are used to develop a helicity raising and lowering
construction in the general setting and on conformally Einstein manifolds.Comment: 37 page
Algorithmic computations of Lie algebras cohomologies
summary:From the text: The aim of this work is to advertise an algorithmic treatment of the computation of the cohomologies of semisimple Lie algebras. The base is Kostant's result which describes the representation of the proper reductive subalgebra on the cohomologies space. We show how to (algorithmically) compute the highest weights of irreducible components of this representation using the Dynkin diagrams. The software package offers the data structures and corresponding procedures for computing with semisimple Lie algebras. Thus, using it has been easy to implement the (theoretical) algorithm.\par The web implementation of the resulting algorithm is available online at the following address www.math.muni.cz/silhan/lac. (These pages compute moreover cohomologies of real semisimple Lie algebras. These cohomologies will be described elsewhere)
Commuting linear operators and algebraic decompositions
summary:For commuting linear operators we describe a range of conditions which are weaker than invertibility. When any of these conditions hold we may study the composition in terms of the component operators or combinations thereof. In particular the general inhomogeneous problem reduces to a system of simpler problems. These problems capture the structure of the solution and range spaces and, if the operators involved are differential, then this gives an effective way of lowering the differential order of the problem to be studied. Suitable systems of operators may be treated analogously. For a class of decompositions the higher symmetries of a composition may be derived from generalised symmmetries of the component operators in the system