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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.

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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."

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I think you might like Affine Lie Algebras and Quantum Groups by Jurgen Fuchs. Also, Lie Algebras of Finite and Affine Type by Roger Carter is pretty good.

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A quick and dirty (not quite mathematical) introduction is in Francesco, Mathieu, Senechal's "Conformal Field theory" (chapter 14, affine lie algebras). For a more mathematical but still not very detailed treatment, Terry Gannon's book "Moonshine beyond the monster" is a good option.

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Here is a comment which isn't a direct answer but may be helpful. Kac "Infinite dimensional Lie algebras" is extremely detailed and thorough and is a sort of bible to me, but many of the constructions are somewhat unmotivated. If you find yourself thinking about affine algebras in the context of Virasoro algebras, Heisenberg algebras etc, a very good companion to Kac "Infinite Dimensional Lie algebras" is Kac, Raina and Rozhkovskaya, "Highest weight reps of infinite dimensional Lie algebras". It is very readable and the development is very motivated, and helps me through the (difficult) chapters 12, 13, 14 of "Big Kac".

I am not sure how often these topics appear in quantum groups, however.

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