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Why were Lie algebras called infinitesimal groups in the past? And why did mathematicians begin to avoid calling them infinitesimal groups and switch to calling them Lie algebras?

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    $\begingroup$ 1. Because that's what they are, 2. historical backlash against the use of infinitesimals which is no longer justified (see en.wikipedia.org/wiki/Smooth_infinitesimal_analysis). $\endgroup$ – Qiaochu Yuan Oct 11 '12 at 6:23
  • $\begingroup$ Also, the term infinitesimal group is now used (at least in some contexts) to refer to an algebraic group scheme that has only one point over any field (such as the Frobenius kernels). $\endgroup$ – Tobias Kildetoft Oct 11 '12 at 6:33
  • $\begingroup$ «which is no longer justified» in a general sense, but a Lie algebra is not an «infinitessimal group» in any technical sense this term might have, no? $\endgroup$ – Mariano Suárez-Álvarez Oct 11 '12 at 6:43
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    $\begingroup$ I think @QiaochuYuan is right about the historical reason. However, I would maintain that it is still good to have different names. The intuition which the original theorists were capturing with the term "infinitesimal group" is very close to what we now call a formal group: A formal power series which acts like a group multiplication. It is true in characteristic zero that the categories of formal groups and Lie algebras are equivalent. (continued) $\endgroup$ – David E Speyer Oct 11 '12 at 13:05
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    $\begingroup$ However, formal groups and Lie algebras look like very different objects, and it is nontrivial that they are equivalent. I think it is good to call them by different names, so that we can then state the theorem "the categories of formal groups and Lie algebras are equivalent". In addition, in characteristic $p$, they are not equivalent, so that is another good reason to have both concepts. $\endgroup$ – David E Speyer Oct 11 '12 at 13:07
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I don't know anything about the actual history, but you might want to look at this blog post by Terry Tao. Terry defines a "local group" to be something that looks like a neighborhood of the identity in a topological group and a "local Lie group" to be something that looks like a neighborhood of the identity in a Lie group.

He then defines a "group germ" to be an equivalence class of local groups, where the equivalence relation should be thought of as "become the same under passing to a sufficiently small open neighborhood of the identity". Thus, the group germ remembers all the information which can be seen on an arbitrarily small neighborhood of the identity. I think this is a very good modern rigorous analogue of the notion of an infinitesimal group. Terry then proves Lie's Third Theorem (Theorem 2 in his notes): Germs of local lie groups are in bijection with Lie algebras.

If you prefer algebra to analysis, the corresponding idea is a formal group. Again, it is a true but nontrivial theorem that formal groups in characteristic zero are equivalent to Lie algebra.

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Thomas Hawkins's Emergence of the Theory of Lie Groups: An Essay in the History of Mathematics 1869–1926 (2000) answers your question.

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    $\begingroup$ It may be helpful to editors here if you could summarize Hawkins' argument. $\endgroup$ – Mikhail Katz Dec 28 '17 at 8:43

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