ABSTRACT
This chapter derives the classical Dynkin-Kostant classification of complex nilpotent orbits. First, it establishes a one-to-one correspondence between nilpotent orbits and conjugacy classes of certain small semisimple subalgebras of g. The proof that this correspondence is surjective depends on the famous Jacobson-Morozov Theorem and a conjugacy theorem of Kostant proves it is injective. Second, the chapter shows that conjugacy classes of these small subalgebras are in one-to-one correspondence with certain distinguished semisimple orbits; this uses a second conjugacy theorem, this time due to Mal'cev. The chapter describes how the nilpotent orbits stand in one-to-one correspondence with partitions of the integer n. It proves two conjugacy theorems that lead to a bijective correspondence between nilpotent orbits and distinguished semisimple orbits. Finally, the chapter establishes a one-to-one correspondence between nilpotent orbits and a certain finite collection of semisimple orbits.
