# Similarity Between Lie Algebra and Group Theory

In this semester I'm taking a lecture about Lie algebra. I'm quite surprised because theory of Lie algebra has very similar structure as theory of groups. Here is some of my observations:

1. Notion of nilpotent and solvable is almost same and their property is similar.

2. Theory of representation is also similar. For example, Schur's Lemma holds for representation of groups and Lie algebras.

3. Classification of Lie algebras and Finite groups has similar parts, like exceptional groups of Lie type and some matrix groups.

They are just some observations so it can be meaningless set of phenomenons. But, do you know any systematic reason of this phenomenon, then please tell me something. I have just naive explanation: commutator of groups and Lie brakets has very similar properties. Maybe Lie algebra - Lie group correspondence has something to do with, but in my intuition finite dimensional Lie algebra is similar with finite groups and Lie groups are infinite, so it may not be ultimate answer. Is there any systematic reason?

• To any group you can associate a Lie ring, namely, take associated factors of lower central series (which are abelian groups) with bracket being commutator: it's well defined modulo higher order terms, and makes $\bigoplus \gamma_n(G) / \gamma_{n+1}(G)$ a graded Lie ring. Universal envelopes of those objects are similar to graded factors of group ring, which is a Hopf algebra. If you tensor everything with rationals and look at "infinitesimal level", this connection becomes much more clear: math.univ-lille1.fr/~fresse/OperadHomotopyBook/…
– xsnl
Apr 5, 2017 at 21:44

There are several "systematic reasons". For example, we have the Lazard correspondence between $p$-groups and $p$-Lie rings (algebras). For Lie groups and Lie algebras we have the Lie group - Lie algebra correspondence. Nevertheless one should also see the differences between groups and Lie algebras. The classification of finite simple groups, and the classification of finite-dimensional simple Lie algebras is not so "similar", after all.