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## Abstract

Let L be an affine Kac-Moody algebra, with central element c, and let λ∈C. We study two-sided ideals in the central quotient Uλ(L):=U(L)/(c−λ) of the universal enveloping algebra of L, and prove:

Theorem 1. If λ≠0 then Uλ(L) is simple.

Theorem 2. The algebra U0(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.

Theorem 1. If λ≠0 then Uλ(L) is simple.

Theorem 2. The algebra U0(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.

Original language | English |
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Publisher | ArXiv |

Publication status | Published - Dec 2021 |

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