Irreducibly acting subgroups of $Gl(n,\rr)$

Details Irreducibly acting subgroups of $Gl(n,\rr)$ Irreducibly acting subgroups of $Gl(n,\rr)$

2005-07-04

arxiv.org/abs/math/0507047v1

Mathematics

Differential Geometry

21 pages

Scientific paper

In this note we prove the following three algebraic facts which have applications in the theory of holonomy groups and homogeneous spaces: Any irreducibly acting connected subgroup $G \subset Gl(n,\rr)$ is closed. Moreover, if $G$ admits an invariant bilinear form of Lorentzian signature, $G$ is maximal, i.e. it is conjugated to $SO(1,n-1)_0$. Finally we calculate the vector space of $G$-invariant symmetric bilinear forms, show that it is at most 3-dimensional, and determine the maximal stabilizers for each dimension.

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