There is a correspondence between Lie algebras and groups on the level of their "algebra", in that many "purely algebraic" theorems in group theory correspond exactly to "purely algebraic" theorems in Lie algebras, and one can often see the exact form of the translation, pretend your group $G$ was a Lie group, and take the Lie functor.

Some examples of this are the definitions and properties of solvability/nilpotence, simplicity, semidirect products, etc.

This also manifests in the fact that the category of group representations for any group has way more structure than might be naively expected, coming from Hopf algebra structure of $k[G]$. We also have this for Lie algebras, coming from the Hopf algebra structure of $U_\mathfrak{g}$, and morally, this hopf algebra structure is the "linearised" version of the Hopf algebra structure on groups.

What is the correct formal way to view the similarity between these theories? Is there a precise formulation of this translation principle, in terms of the (first order?) theories of the axioms of groups/lie algebras?

  • $\begingroup$ Is there a first order theory whose models are Lie groups? $\endgroup$ – Alex Youcis May 5 at 6:18
  • $\begingroup$ @AlexYoucis No - in general first-order logic is not well-equipped for axiomatizing structures with any kind of topological component. $\endgroup$ – Alex Kruckman May 5 at 15:19
  • $\begingroup$ @AlexKruckman That's what I thought. Then I'm confused at what the question is even asking. $\endgroup$ – Alex Youcis May 5 at 20:45
  • $\begingroup$ Groups and Lie algebras are both just defined with equational axioms, and a lot of their “formal” theory seems the same. It seems unsatifying to have to invent Lie groups to get a dictionary between them, so the question is about making precise why the axiomatic systems feel so similar. $\endgroup$ – user277182 May 6 at 1:19
  • $\begingroup$ Lie groups were not "invented to get some dictionary". They came from the need of the theory of differential equations and now play role in many parts of mathematics. (Topology, differential geometry, mathematical physics, etc.) $\endgroup$ – Moishe Kohan May 6 at 13:03

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