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A Lie group is a group that is also a smooth manifold (in a way compatible with the group structure). A Lie algebra is the tangent space of a Lie group (at the identity).

In this paper, the author defines an infinite continuous group as a set of transformation satisfying a system of partial differential equations (disregarding all groups which cannot be defined by differential equations), and then goes on to prove that every infinite continuous group has infinitesimal transformations (an infinitesimal transformation being an infinitely small transformation that --somehow!-- allows us to drop all higher order "infinitely small" terms and only keep the first order ones, and an infinitely small transformation being a transformation that differs from the identity by "infinitely little").

For Wikipedia, Lie algebras formalize infinitesimal transformations, like groups formalize symmetry.


It's my understanding that Weierstrass/et al. set out to "banish infinitesimals from mathematics", finally arriving, some centuries after the original breakthroughs of Leibniz/Newton, at limits.

Now, with limits it's possible (but cumbersome?) to prove important results such as

$$dy = {dy \over dx}dx$$

or

$${f(x) \over g(y)} = {dy \over dx} \hspace8pt \Longrightarrow \hspace8pt f(x)dx = g(y)dy,$$

or

$${dy \over dx} = {1 \over {dx \over dy}}$$

or

$$\int df = f,$$

all of which have intuitive (and rigorous!) proofs using an infinitesimal-enriched continuum.


Why do we find so many references to "infinitesimal" stuff in the Lie group literature and not in the calculus literature? Do we really need infinitesimal transformations to study Lie groups?

My guess is that all of Lie theory can be formulated using limits and neighborhoods, like calculus (doesn't Lie theory begin with smooth manifolds and tangent spaces, after all?), and that the difference in vocabulary is historical (Lie liked the word "infinitesimal", Weierstrass didn't).

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Your speculation is essentially correct... although a notion of "limit" was part of earlier (e.g., pre-Weierstrass, etc) calculus, as well. It's just that it was sometimes tangled up with infinitesimals, whereas Weierstrass (among others) succeeded in "defining limits" without infinitesimals (for better or for worse).

About Lie theory: even nowadays, it is common to speak of the "infinitesimal action" of a Lie group, by which one means the action of the Lie algebra obtained by differentiating the group action (when the limit exists, e.g., on smooth vectors in a representation): on the real line, for example, the infinitesimal action of translation $T_xf(y)=f(x+y)$ is ${d\over dx}|_{x=0}f(x+y) = f'(y)$.

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