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Let $\hat{\mathfrak{g}} = L\mathfrak{g}\oplus \mathbb{C}K$ be the affine Lie algebra corresponding to a simple, finite-dimensional Lie algebra $\mathfrak{g}$. Many texts note that one can define the Serre relations

$$(ad E_i)^{1-{a}_{ij}}E_j=\sum_{n=0}^{1-{a}_{ij}}(-1^n)\binom{1-{a}_{ij}}{n}(E_i)^{1-{a}_{ij}-n}E_j (E_i)^n=0$$

and similarly for the generators $F_i$, where ${a}_{ij}$ denote the entries of the Cartan matrix. However, I have not yet found a reference that goes through with the actual computation of these relations. Does anyone know where I could find such a reference?

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    $\begingroup$ What would it mean to "compute" the relations? They are imposed. And isn't this just the usual construction of Kac-Moody algebras, which include but are more general than affine Lie algebras. $\endgroup$ – Sam Hopkins Dec 12 '19 at 20:16
  • $\begingroup$ @SamHopkins I'm guessing my question is that I don't understand where these relations come from. $\endgroup$ – Joe Wolf Dec 12 '19 at 20:25
  • $\begingroup$ Are you comfortable with the construction of simple (finite-dimensional) Lie algebras from root system data? It's completely analogous. $\endgroup$ – Sam Hopkins Dec 12 '19 at 20:26
  • $\begingroup$ @SamHopkins Would you know of a good reference where this construction is done for the affine/Kac-Moody case? $\endgroup$ – Joe Wolf Dec 12 '19 at 20:36
  • $\begingroup$ I think the textbook "Infinite dimensional Lie algebras" by Kac is a canonical source. But doing a little googling, these notes (see in particular Prop. 1.2.16) also look nice: people.kth.se/~namini/PartIIIEssay.pdf , as do these notes (see Lemma 3.1.1): darkwing.uoregon.edu/~klesh/teaching/IDLALN3.pdf $\endgroup$ – Sam Hopkins Dec 12 '19 at 20:43

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