Positive Structures in Lie Theory

0.1. In late 19th century and early 20th century, a new branch of mathematics was born: Lie theory or the study of Lie groups and Lie algebras (Lie, Killing, E.Cartan, H.Weyl). It has become a central part of mathematics with applications everywhere. More recent developments in Lie theory are as follows. -Analogues of simple Lie groups over any field (including finite fields where they explain most of the finite simple groups): Chevalley 1955; -infinite dimensional versions of the simple Lie algebras/simple Lie groups: Kac and Moody 1967, Moody and Teo 1972; -theory of quantum groups: Drinfeld and Jimbo 1985.