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Jun 8, 2015 at 17:32 comment added Ben Wieland For the $\mathbb Q$ version to work it is important that the group to be nilpotent, so that the exponential preserve rational matrices. . . There is a smaller, though perhaps not simpler, non-commutative Lie group, the affine group $ax+b$. The exponential is still a bijection, so it gives an $\mathbb R$-group, but the exponential does not preserve rational elements. It contains $\mathbb Q$-monoids, but they are more complicated.
Jun 4, 2015 at 15:11 history edited André Henriques CC BY-SA 3.0
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Jun 4, 2015 at 11:31 history answered André Henriques CC BY-SA 3.0