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I have an explicit description of a Lie group in terms of matrices with entries in a quaternion algebra, and I want to determine if there exist isogenous covers. How can I do this?

Watson Ladd
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  • One needs to know many more details. At the very least: Can you compute the Lie algebra of your group? – Moishe Kohan Feb 09 '18 at 03:46
  • Yes, but the Lie algebra is the same for all groups in the same isogeny class, no? The group is U_2(D)= 2x2 matrices with entries in a quaternion algebra such that transpose-conjugate(B)SB=S where $S=(0, 1, 1,0)$. I'm working over a p-adic field. – Watson Ladd Feb 09 '18 at 04:18
  • Yes, the Lie algebra is the same, but once you know the Lie algebra, you can start to eliminate things by some basic representation theory. As a simple example, if you have the Lie algebra $sl(2)$ and you have an irreducible representation of even dimension then your group will be $SL(2)$ rather than $PSL(2)$. Aside: you should not call it a Lie group if you are working over p-adics. – Moishe Kohan Feb 09 '18 at 04:24

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