Structural data and invariants of nine dimensional real Lie algebras with nontrivial Levi decomposition

Abstract

In this work the authors analyze various structural properties of indecomposable nine dimensional real Lie algebras having a non-trivial Levi decomposition. These algebras, classified in the early 90s by Turkowski, actually show patterns that do not occur in Lie algebras of lower dimension. The main objective is to determine a fundamental system of invariants for the coadjoint representation. Due to the large number of essential parameters, a direct analytical approach is not useful, due to the quantity of special cases to be analyzed. In order to integrate the corresponding systems of PDEs, the authors consider the theory of semi-invariants developed by Trofimov and a reduction of the problem to certain subalgebras, the invariants of which can be computed easily or are already known. The procedure is based in the labelling of representations using subgroup chains (the so called missing label operator problem), which allows to express the invariants of the total algebra as functions of invariant of subalgebras. The method also provides those Lie algebras that arise as an extension of some subalgebra, therefore providing information on the trivial cohomology.