Kartik Prabhu
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It will. But that is also not explicitly given anywhere in terms of some nice coordinates.
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Leo C. Stein 🦁
replied on Twitter with
I agree that they look like they know what they’re doing! My instinct was to construct a map from e->g, and use the differential of the map to translate the measure from T_e to T_g. And I thought that was what the author was doing, too. Except then I remembered that exp is not
Kartik Prabhu
replied on Twitter with
I have looked at that paper/notes before, but I am not sure I understand it. If you look at the last formula in section 1.3 on pg. 3, the k and k’ are in SU(2) and the r is a real radial coordinate. So that measure looks like it is 7-dimensional which seems wrong.
John Carlos Baez
replied on Twitter with
Ugh! They use the Cartan decomposition G = KAK where A is a maximal abelian subgroup of G. For G = SL(2,C) you can take A to be diagonal matrices with det = 1, and that’s a 2d Lie group, but their formula for it in Section 1.1 makes it look 1d. “r” should really be complex!!!