A huge literature has grown up on Kac-Moody algebras, partly because they are intrinsically interesting and partly because they have vast numbers of applications within both mathematics and physics. In particular, much of the language of theoretical physics is dominated by Kac-Moody algebras. Among Kac-Moody algebras, the ones most studied are the affine algebras. These, it turns out, are very closely related to the finite-dimensional simple Lie algebras, and a great deal of the finite theory can be lifted up to the affine setting. The best overviews of Kac-Moody algebras are the books by Kac , Moody and Pianzola and Wan .
Lie algebra is an incredibly exciting and interesting place to be.
There are large numbers of important unsolved questions, there is a rich theory in place, and there are hordes of applications. Lie algebras are also very beautiful. Recently, Lie algebras have spawned numerous variants, including Borcherds algebras, color algebras and the enormously popular, if somewhat misnamed, quantum groups. Humphreys, J.E., Introduction to Lie algebras and representation theory, Springer, 1972. Kac, V.G., Infinite dimensional Lie algebras, 3d edn, Cambridge University Press, 1990.