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.


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