ABSTRACT
This chapter reviews important elementary concepts from linear algebra, Lie theory, and differential geometry. Every element of a semisimple Lie algebra can be expressed as the sum of a nilpotent element and a semisimple element. In turn, this leads to the notions of nilpotent and semisimple orbits in a Lie algebra (or its dual). The chapter explores the previously mentioned orbits that are symplectic manifolds and the basic implications of having such a geometric structure. It devises some sort of classification scheme for the nilpotent and semisimple elements in a complex semisimple Lie algebra g. Certain arguments can be simplified if we exploit the differential geometry of a coadjoint orbit. The chapter also reviews the relevant concepts, ultimately describing a complex symplectic structure on any coadjoint orbit. Among other useful properties, the adjoint and coadjoint orbits in the sequel always have even (complex) dimension.
