# Reference request: Serre relations for affine Lie algebras

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?

• 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. – Sam Hopkins Dec 12 '19 at 20:16
• @SamHopkins I'm guessing my question is that I don't understand where these relations come from. – Joe Wolf Dec 12 '19 at 20:25
• Are you comfortable with the construction of simple (finite-dimensional) Lie algebras from root system data? It's completely analogous. – Sam Hopkins Dec 12 '19 at 20:26
• @SamHopkins Would you know of a good reference where this construction is done for the affine/Kac-Moody case? – Joe Wolf Dec 12 '19 at 20:36
• 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 – Sam Hopkins Dec 12 '19 at 20:43