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

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

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, \(G\)...

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Veröffentlicht in:arXiv.org 2005-07
Hauptverfasser: Di Scala, Antonio J, Leistner, Thomas, Neukirchner, Thomas
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Invariants
Set theory
Subgroups
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description 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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Subgroups
title Irreducibly acting subgroups of \(Gl(n,\rr)\)
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