Question

There are a lot of results in textbooks concerned with canonical forms of matrices under certain complex groups of transformations, e.g. GL(n|C), O(n|C),... Could anybody give me references where the canonical forms of real matrices under the action of SO(p,q|R) were found. Of most interest is the canonical form of antisymmetric matrices, i.e. that of the adjoint representation. Other related results, e.g. on canonical forms under SU(p,q), SP(p,q) are also appreciated. Thanks!

Answer 1

In Chapter 2 of the PhD thesis of Charles Boubel (in French, though), you can find the normal forms of a symmetric or antisymmetric bilinear form under the automorphism group of a nondegenerate bilinear form, again either symmetric or skewsymmetric. In particular you can find there the canonical forms for elements of the Lie algebra of SO(p,q|R) under the adjoint action.

A summary of this work can be found (in German, though) in the Diplom thesis of Thomas Neukirchner Normalform eines schief- oder selbstadjungierten Endomorphismus auf einem pseudo-euklidischen oder symplektischen Vektorraum über R oder C — eine Arbeit von Charles Boubel, which is a Humboldt University preprint, and also for some special signatures in a paper of Felipe Leitner's Imaginary Killing spinors in lorentzian geometry (math.DG/0302024) and also in a paper of mine and Joan Simón's Supersymmetric Kaluza-Klein reductions of AdS backgrounds (hep-th/0401206). As in Leitner's paper, we were mostly concerned with the case q=2, though.

I hope this helps. I have PDFs of Boubel's thesis, which I can make available if you want.

Answer 2

Another reference is: D. Djokovic, J. Patera, P. Winternitz, H. Zassenhaus, Normal forms of elements of classical real and complex Lie and Jordan algebras, J. Math. Phys. 24 (1983), N6, 1363-1374 http://dx.doi.org/10.1063/1.525868 .