Ideals in enveloping algebras of affine Kac–Moody algebras

Abstract

Let $L$ be an affine Kac–Moody algebra, with central element $c$, and let $\lambda \in ℂ$. We study two-sided ideals in the central quotient $U_{\lambda }(L):=U(L)∕(c-\lambda )$ of the universal enveloping algebra of $L$ and prove:

1. If $\lambda \ne 0$ then $U_{\lambda }(L)$ is simple.

2. The algebra $U_0(L)$ has just-infinite growth, in the sense that any proper quotient has polynomial growth.

As an immediate corollary, we show that the annihilator of any nontrivial integrable highest-weight representation of $L$ is centrally generated, extending a result of Chari for Verma modules.

We also show that universal enveloping algebras of loop algebras and current algebras of finite-dimensional simple Lie algebras have just-infinite growth, and prove similar results to the two results above for quotients of symmetric algebras of these Lie algebras by Poisson ideals.