# Lately in Lie Groups (week 8ish)

Over the past couple of weeks we covered:

1. Engel’s Thm
2. Lie’s Thm
3. Irred. rep of arb $\mathfrak{g}$ in via 1-dim reps and irreps of $\mathfrak{g}_{ss} = \mathfrak{g}/Rad(\mathfrak{g})$.
4. Jordan decomposition of a semisimple lie algebra
5. Complete reducibility of representations of semisimple lie algebras.
6. Unitary Trick
7. Killing Form $B( , )$
8. Cartan’s Criterion
9. $\mathfrak{g}$ solvable iff $B(\mathfrak{g}, [\mathfrak{g},\mathfrak{g}])=0$.
10. $\mathfrak{g}$ semisimple iff Killing form is nondegenerate.
11. $\mathfrak{g}$ semisimple implies its a direct sum of semisimple.

Slightly more details: lieweek8ish

references to results from Fulton and Harris.