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I have seen at many places the notions that Lie Algebras are infinitesimal objects and they look really close at a point. But I never understood this. They are abstract algebraic objects different from rings in that they are equipped with a weird sort of product and a weird Jacobi identity. Any hints on how to make this connection to infinitesimals?

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    Interesting question! I don't know the answer myself but Sophus Lie heavily uses infinitesimals in his paper on Lie groups (which I am currently reading).2010-08-09
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    In fact Abraham Robinson showed how to rigorize the classical intuitive arguments via nonstandard analysis - see my recent post for references http://math.stackexchange.com/questions/18282010-08-09
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    Are there classical intuitive arguments that were not easily expressible using standard approaches (e.g., calling an "infinitesimal" a sequence or function tending to zero, or calculating with nilpotents in Taylor series)?2010-08-09

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