# An inequality for weights of affine Lie algebras, level, and dual Coxeter number

Suppose $$\mathfrak{g}$$ is an (untwisted) affine Lie algebra with the normalized invariant form $$(\cdot | \cdot)$$. Let $$\lambda \in \mathfrak{h}^\ast$$ be a dominant integral weight such that $$\lambda(d)=0$$ for $$d$$ the derivation in the loop algebra construction; that is, $$\lambda = c_0\Lambda_0 +c_1\Lambda_1+\cdots+c_n\Lambda_n$$, where $$c_i \in \mathbb{Z}_{\geq0}$$, $$\Lambda_i$$ the fundamental weights. Let $$l$$ be the level of $$\lambda$$.

While doing some (albeit limited) explicit computations for various $$\mathfrak{g}$$ and $$\lambda$$, I am encountering an inequality of the form $$2l(\lambda|\rho)-h^\vee(\lambda|\lambda) \geq 0$$ where $$h^\vee$$ is the dual Coxeter number of $$\mathfrak{g}$$ and $$\rho$$ is as usual a choice of weight satisfying $$\rho(\alpha_i^\vee)=1$$ for all simple coroots $$\alpha_i^\vee$$.

Question: Should this inequality hold for all such $$\lambda$$?

I have little intuition for whether this should be expected; this very well could be an artifact of the specific $$\lambda$$ that appear in my computations. For example, I would be somewhat satisfied if this were to hold under the stronger condition that each $$c_i \in \{0,1\}$$.

• What is $d$?... Jul 2, 2020 at 15:05
• @LSpice $d$ is the derivation coming from the loop algebra construction of $\mathfrak{g}$; really, I was just emphasizing that $\lambda$ has no $\delta$ term in the summand. I will add that. Jul 2, 2020 at 15:09

## 1 Answer

The answer to this is found as theorem 13.11 in Kac, "Infinite dimensional Lie Algebras". To be specific, we have $$2k(\Lambda|\rho) \geq h^{\vee} (\Lambda| \Lambda)$$ for all $$\Lambda \in P^k_+$$, with equality if and only if $$\Lambda = k \Lambda_j$$ mod $$\mathbb{C} \delta$$. Here $$j \in J$$, where $$J$$ is a set depending on the Kac labeling of the Dynkin diagram in question (J should correspond to the set of simple roots with Kac label 1, roughly).

• This is actually fantastic! Thank you--I am still woefully unfamiliar with the content of the last two chapters of Kac's book, but I'm not completely surprised that this was "hidden" in there! Jul 15, 2020 at 2:25