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The idea of nonstandard analysis is to combine finite quantities with infinitesimals. And, back in the day, Lie algebras were roughly considered the "infinitesimal elements" of Lie groups. Say we want to employ the idea of nonstandard analysis in Lie theory. Take a Lie group $G$ with Lie algebra $\mathfrak{g}$, suppose that $\rho:G\to GL(\mathfrak{g})$ is the adjoint representation, and define $\mathcal{U}(G)$ to be the free product $G\ast(\mathfrak{g},+)$ modulo the relations $g\cdot V\cdot g^{-1}=\rho(g)V$ for all $g\in G,V\in\mathfrak{g}$. Unfortunately this doesn't use the Lie bracket in $\mathfrak{g}$, but if we "differentiate" $a(t)\cdot V\cdot a(t)^{-1}$ with respect to $t$ and evaluate at $t=0$ we get the commutator $AV-VA$ and we also get the bracket $[A,V]\in\mathfrak{g}$. As another example, if we "differentiate" $a(t)Vb(t)Wc(t)$ at $t=0$ we get $AVW+VBW+VWC$.

So it would seem the "Lie algebra" of $\mathcal{U}(G)$ is the universal enveloping algebra $\mathcal{U}(\mathfrak{g})$.

Is this a thing? Is there a more precise sense in which this is true and meaningful? According to Wikipedia, the universal enveloping algebra corresponds to the algebra of left-invariant differential operators on $G$, and nonstandard analysis is a way to formulate differentiation algebraically, so that suggests there is something "real" going on here.

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    $\begingroup$ For Lie algebras you don't need hyperreal infinitesimals and dual numbers (with a nilsquare infinitesimal) suffice. In a "flabbier" context such as that of differential geometry, hyperreal infinitesimals become helpful. $\endgroup$ – Mikhail Katz Dec 9 '15 at 16:46

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