Ideals in enveloping algebras of affine Kac–Moody algebras

Biswal, Rekha; Sierra, Susan J.

doi:10.2140/ant.2025.19.1199

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Ideals in enveloping algebras of affine Kac–Moody algebras

Rekha Biswal and Susan J. Sierra

Vol. 19 (2025), No. 6, 1199–1230

DOI: 10.2140/ant.2025.19.1199

Abstract

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

If $λ \neq 0$ then $U_{λ} (L)$ is simple.
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.

Keywords

Kac–Moody algebra, affine algebra, highest-weight representation, Gelfand–Kirillov dimension, simple ring

Mathematical Subject Classification

Primary: 16P90, 16S30, 17B10, 17B67

Secondary: 16D30, 17B65

Milestones

Received: 21 October 2022

Revised: 27 May 2024

Accepted: 15 July 2024

Published: 14 May 2025

Authors

Rekha Biswal
School of Mathematical Sciences National Institute of Science Education and Research Bhubaneswar India

Susan J. Sierra
School of Mathematics University of Edinburgh Edinburgh United Kingdom

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 19, No. 6, 2025