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Commuting linear operators and algebraic decompositions

Rod A. GoverJosef Šilhan — 2007

Archivum Mathematicum

For commuting linear operators P 0 , P 1 , , P we describe a range of conditions which are weaker than invertibility. When any of these conditions hold we may study the composition P = P 0 P 1 P in terms of the component operators or combinations thereof. In particular the general inhomogeneous problem P u = 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...

Algorithmic computations of Lie algebras cohomologies

Šilhan, Josef — 2003

Proceedings of the 22nd Winter School "Geometry and Physics"

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 L i E offers the data structures and corresponding procedures for computing...

On a new normalization for tractor covariant derivatives

Matthias HammerlPetr SombergVladimír SoučekJosef Šilhan — 2012

Journal of the European Mathematical Society

A regular normal parabolic geometry of type G / P on a manifold M gives rise to sequences D i of invariant differential operators, known as the curved version of the BGG resolution. These sequences are constructed from the normal covariant derivative ω on the corresponding tractor bundle V , where ω is the normal Cartan connection. The first operator D 0 in the sequence is overdetermined and it is well known that ω yields the prolongation of this operator in the homogeneous case M = G / P . Our first main result...

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