I'm new to the subject of topological insulators. I've been trying to understand how they come about the Cartan classification for topological insulators. I've been following these slides kitp.ucsb.edu/online/freedmanfest/ludwig/pdf/Andreas_Ludwig.pdf. I do think I get how they arrive at $U(N)/O(N)$, but it is not clear to me how they arrive at the other classes. I understand that the constraints on the hamiltonians have something to do with it, but when I sit down to try to get $U(2N)/Sp(2N)$ for example, I am not able to get this. Can someone explain how they arrive at $U(2N)/Sp(2N)$? Is the line of reasoning to obtain it the same for all the other classes?
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