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Ideals in enveloping algebras of affine Kac-Moody algebras

Rekha Biswal, Susan J. Sierra

Abstract

Let $L$ be an affine Kac-Moody algebra, with central element $c$, and let $λ\in \mathbb C$. 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: Theorem 1. If $λ\neq 0$ then $U_λ(L)$ 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.

Ideals in enveloping algebras of affine Kac-Moody algebras

Abstract

Let be an affine Kac-Moody algebra, with central element , and let . We study two-sided ideals in the central quotient of the universal enveloping algebra of , and prove: Theorem 1. If then is simple. Theorem 2. The algebra 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 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.
Paper Structure (14 sections, 33 theorems, 105 equations)

This paper contains 14 sections, 33 theorems, 105 equations.

Table of Contents

  1. Introduction
  2. Bases and basic facts
  3. Basic facts about $\mathfrak g$
  4. Two monomial orders on $S(\mathfrak g[t,t^{-1}]^\sigma)$
  5. Growth
  6. A reduction result for symmetric algebras of current algebras
  7. Preparatory results
  8. The main reduction result
  9. Useful technical results
  10. Just-infinite growth
  11. Simplicity of nontrivial central quotients
  12. The $\mathfrak{sl}_2$ case
  13. The general case
  14. Other applications

Key Result

Theorem 1.1

Let $\lambda \in \Bbbk$.

Theorems & Definitions (70)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof
  • Proposition 1.4
  • proof
  • Proposition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.4
  • ...and 60 more