What is a good reference for a fast approach to construct affine Kac-Moody algebras from finite-dimensional simple Lie algebras? 


I know that Kac's book and many others do a very detailed and progressive construction, but I mean a understandable and direct realization as in the Hong and Kang's book about Quantum groups.



affine Kac-Moody algebras
I am a big fan of Carter's book. It's very nicely laid out and I found it quite easy to read.


Here's an older reference: Kass, Moody, Patera, and Slansky's "Affine Lie Algebras, Weight Multiplicities, and Branching Rules"


This text is focused only on affine algebras. It is kind of light on proofs but provides a lot of nice details. Also, it's co-written by Physicists so there is an extra sprinkling of Physics flavor throughout.


Also, if you do decide to wade through Kac, you may want to pick up a copy of Minoru Wakimoto's "Infinite-Dimensional Lie Algebras" ISBN: 0821826549 (be careful Wakimoto has two books with almost the same exact title published at nearly the same time). Wakimoto's book makes a nice companion to Kac's book and is filled with great quotes such as: "sl2 representation theory is like Mt. Fuji reflected in a beautiful lake."


