§ Classification of lie algebras, dynkin diagrams

§ Classification of complex lie algebras

§ Theorem (Leri)

§ Remarks

§ Adjoint Map

§ Killing form

§ Calculation wrt basis: adad map.

§ Calculation wrt basis: Killing form.

§ adad is anti-symmetric with respect to the killing form.

§ Key Definition for classification: Cartan subalgebra

§ Existence of cartan subalgebra

§ Analysis of Cartan subalgebra.

§ Root set is closed under negation

§ Root set is not linearly independent

§ Fundamental roots

§ complex span of fundamental roots is the dual of the cartan subalgebra

§ Defn: HRH_{\mathbb R}^*

§ Defn: Killing form on HH^*

§ KK^* on HRH^*_{\mathbb R}

§ Recovering Φ\Phi from Π\Pi

§ Theorem: Weyl group is generated by fundamental roots

§ Theorem: Roots are prouced by action of Weyl group on fundamental roots

§ Showdown