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In this paper we classify real trivectors in dimension 9. The correspondingclassification over the field C of complex numbers was done by Vinberg andElashvili in 1978. One of the main tools used for their classification was theconstruction of the representation of SL(9,C) on the space of complextrivectors of C^9 as a theta-representation corresponding to a Z/3Z-grading ofthe simple complex Lie algebra of type E8. This divides the trivectors intothree groups: nilpotent, semisimple, and mixed trivectors. Our classificationfollows the same pattern. We use Galois cohomology to obtain the classificationover R. For the nilpotent orbits this is in principle rather straightforwardand the main problem is to determine the first Galois cohomology sets of a longlist of centralizers (we compute the centralizers using computer). For thesemisimple and mixed orbits we develop new methods based on Galois cohomology,first and second. We also consider real theta-representations in general, andderive a number of results that are useful for the classification of theirorbits.

Real graded Lie algebras, Galois cohomology, and classification of trivectors in R^9