What is the generator of translation in the Beltrami-Klein model of the hyperbolic plane?
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$\begingroup$ I've voted to close. math.stackexchange.com is a more appropriate forum for your question. $\endgroup$– Ryan BudneyJun 2, 2011 at 7:05
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$\begingroup$ I've also voted to close. The notion of "the generator of translation" also makes no sense, as r0b0t's answer below suggests. $\endgroup$– José Figueroa-O'FarrillJun 2, 2011 at 13:29
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1 Answer
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Hyperbolic plane is a homogeneous space $G/H$ where $G$ acts by isometries and so any reasonable defined `translation' is given by the natural left action of $G$. Decide what elements of $G$ will you call translations and then for any abstract generator $X\in\mathfrak{g}$ such that $\exp{tX}$ is translation, compose the natural left action with diffeomorphism of $G/H$ to your favorite model. Differentiate the resulting map at zero and you are done.
If I am not mistaken, this question is more suitable for math.SE.