Cayley form, comass, and triality isomorphisms
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Following an idea of Dadok, Harvey and Morgan, we apply the triality property of Spin(8) to calculate the comass of self-dual 4-forms on ℝ8. In particular, we prove that the Cayley form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio (equation (1.5)) is SO(8)-conjugate to the Cayley form. We also use triality to prove that the stabilizer in SO(8) of the Cayley form is Spin(7). The results have applications in systolic geometry, calibrated geometry, and Spin(7) manifolds.
KeywordsWeight Space Maximal Torus Dynkin Diagram Cartan Subalgebra Coadjoint Representation
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