Lie theory, history

Question

I am trying to understand how Sophus Lie's work relates to linear algebraic group's correspondence with a Lie algebra, anyone could help? I'm having some trouble going from differential geometry to algebraic geometry.

Answer 1

Your question leaves room for many answers; any real algebraic group is also a real Lie group. Passing to its Lie algebra is perhaps going from "differential geometry to algebra". For more details on Lie groups, algebraic groups and Lie algebras see this MSE-question. Concerning "correspondence", I would think of Lie's three fundamental theorems. For example, consider

Lie's Third Theorem: Every finite-dimensional real Lie algebra $L$ is integrable, that is, there exists a Lie group $G$ with $Lie(G)\cong L$.

For a proof see, for example, the note by J. Ebert Van Est's exposition of Cartan's proof of Lie's third theorem, and the references.

Answer 2

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