Extended affine Lie algebras and other generalizations of affine Lie algebras – a survey

Motivation. The theory of affine (Kac-Moody) Lie algebras has been a tremendous success story. Not only has one been able to generalize essentially all of the well-developed theory of finite-dimensional simple Lie algebras and their associated groups to the setting of affine Lie algebras, but these algebras have found many striking applications in other parts of mathematics. It is natural to ask – why? What is so special about affine Lie algebras? What really makes them “tick”? One way to understand this, is to generalize affine Lie algebras and see where things go wrong or don’t. After all, it is conceivable that there is a whole theory of Lie algebras, ready to be discovered, for which affine Lie algebras are just one example.