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.

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