# Lie Groups lecture 5

This lecture ( lie5 ) was given on 9/10/09

### Excerpts

One of the main things we did was prove the

Lie Product Formula: For $X,Y \in M_n(\mathbb{C}$

$\exp(X+Y) = \lim_{m \to \infty} (\exp X/m \cdot \exp Y/m)^m$

sketch of proof: prove prop. 2.8 in Brian Hall’s book that says if $||B||<1/2$ then $\log (I + B) = B + O(||B||^2)$.  Worrying only about terms of order $1/m$, we get $\alpha := \exp X/m \cdot \exp Y/m \approx I + X/m + Y/m$ so for large m apply the proposition

$\log \alpha \approx X/m + Y/m$

$\alpha = \exp \log \alpha \approx \exp(X/m + Y/m)$

$\alpha^m \approx \exp(X+Y)$

as m goes to infinity, the error term goes to zero.

Next we proved a big result that says every 1-parameter subgroup ($m \colon \mathbb{R} \to G$ that is matrix lie group homo.) is of the form $\exp tX$ for some $X \in M_n(\mathbb{C})$.

Using this, and the fact that $\det e^X = e^{tr X}$ (which can be proved by reducing $X$ to the Jordan Normal form (since conjugation doesn’t change det) and that $X = X_D + X_N$ splits X into commuting parts so effectively $\det e^X = \det e^{X_D}$.)  We classified various lie algebras.