Ideals in enveloping algebras of affine Kac-Moody algebras

Abstract

Let $L$ be an affine Kac-Moody algebra, with central element $c$, and let $\lambda \in \mathbb{C}$. 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: Theorem 1. If $\lambda \ne 0$ then $U_{\lambda }(L)$ is simple. Theorem 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 Theorems 1 and 2 for quotients of symmetric algebras of these Lie algebras by Poisson ideals.

34 pages, comments welcome; v2 added author affiliations and emails; v3 substantial revision with Theorem 1 added

References in corpus (1)

• The generalized Weyl Poisson algebras and their Poisson simplicity criterion