Classification of real trivectors in dimension nine

TL;DR

This paper classifies real trivectors in nine-dimensional space by extending complex classification methods using Galois cohomology, providing a comprehensive understanding of their structure over the real numbers.

Contribution

It extends the complex classification of trivectors to the real case using Galois cohomology, offering a detailed real classification framework.

Findings

Complete classification of real trivectors in dimension nine.

Identification of the structure of trivectors over R.

Extension of complex classification techniques to real case.

Abstract

We classify real trivectors in dimension 9. The corresponding classification over the field C of complex numbers was obtained by Vinberg and Elashvili in 1978. One of the main tools used for their classification was the construction of the representation of SL(9,C) on the space of complex trivectors of C^9 as a theta-representation corresponding to a Z/3Z-grading of the simple complex Lie algebra of type E_8. This divides the trivectors into three groups: nilpotent, semisimple, and mixed trivectors. Our classification follows the same pattern. We use Galois cohomology, first and second, to obtain the classification over R.