# Why were Lie algebras called infinitesimal groups?

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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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). – Qiaochu Yuan Oct 11 '12 at 6:23
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). – Tobias Kildetoft Oct 11 '12 at 6:33
«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? – Mariano Suárez-Alvarez Oct 11 '12 at 6:43
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) – David Speyer Oct 11 '12 at 13:05
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. – David Speyer Oct 11 '12 at 13:07