But Cartan also showed that there are ten infinite families of compact symmetric spaces! After explaining symmetric spaces, we show how they arise naturally from forgetful functors between categories of representations of Clifford algebras. Mathematically, it relies on the fact that there are ten super division algebras and ten kinds of Clifford algebras, where two Clifford algebras are of the same kind if they have equivalent super-categories of super-representations. It began as a tenfold classification of states of matter based on their behavior under time reversal and charge conjugation. The slides have material on category theory that I didn’t get around to talking about.Ībstract: The tenfold way has many manifestations. You can see a video of my new talk here and see my slides here. This talk is a followup to an earlier talk, also about the tenfold way. I gave my talk in Nicohl Furey’s series Algebra, Particles and Quantum Theory on Monday May 15, 2023.
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