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Stream: community: events

Topic: Symmetric spaces and the tenfold way

John Baez (May 09 2023 at 19:33):

I'm giving a talk on Monday May 15, 2023, 17:00 UTC or 10:00 for me in California:

Symmetric spaces and the tenfold way

Abstract: The tenfold way has many manifestations. It began as a tenfold classification of states of matter based on their behavior under time reversal and charge conjugation. 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. 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.

Access via zoom link:
https://hu-berlin.zoom.us/j/62643922622?pwd=V0JtTHBoMnc0aFdoMXdKN2VDUDM2UT09
Password:
542525

John Baez (May 09 2023 at 19:35):

I may not have time to get too deep into the category theory in my talk, but you can see it near the end of my slides.

John Baez (May 09 2023 at 19:39):

The final upshot is this:

Let the [[Clifford algebra]] $\mathrm{Cliff}_{n}$ be the free real or complex algebra on $n$ anticommuting square roots of -1. Let

$F \colon \mathsf{Rep}(\mathrm{Cliff}_{n}) \to \mathsf{Rep}(\mathrm{Cliff}_{n-1})$

be the forgetful functor. Then the [[essential fibers]] of this functor are disjoint unions of compact [[symmetric spaces]]. Moreover we get all 10 infinite series of compact symmetric spaces this way.