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 $L^\ell_k$ 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 $L^\ell_k$ 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 ${\rm SL}(n+1)$ into ${\rm Spin}(n+1,n+1)$. The construction associates a split-signature $(n,n)$-conformal spin structure to a projective structure of dimension $n$. 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 $k$-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 $LiE$ offers the data structures and corresponding procedures for computing with semisimple Lie algebras. Thus, using $LiE$ 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/$\sim$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 $P_0,P_1,\dots ,P_\ell$ we describe a range of conditions which are weaker than invertibility. When any of these conditions hold we may study the composition $P=P_0P_1\cdots P_\ell$ in terms of the component operators or combinations thereof. In particular the general inhomogeneous problem $Pu=f$ 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 $P$ may be derived from generalised symmmetries of the component operators $P_i$ in the system