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Smooth representations of affine Kac-Moody algebras

Vyacheslav Futorny, Xiangqian Guo, Yaohui Xue, Kaiming Zhao

TL;DR

The paper addresses the problem of constructing and classifying new smooth modules for untwisted affine Kac-Moody algebras, extending beyond positive-energy representations. It develops a framework of induced smooth modules from carefully chosen subalgebras, links them to generalized Takiff algebras, and analyzes irreducibility for the fundamental case of affine sl2. A complete irreducibility and isomorphism picture is given for both the standard affine algebra and its extended version, including critical level phenomena and families of irreducible quotients parameterized by spectral data. The results provide a substantial expansion of smooth modules applicable to vertex algebra contexts and offer concrete criteria and quotients that capture a wide range of infinite-dimensional representations with rich algebraic structure.

Abstract

Smooth modules for affine Kac-Moody algebras have a prime importance for the quantum field theory as they correspond to the representations of the universal affine vertex algebras. But, very little is known about such modules beyond the category of positive energy representations. We construct a new class of smooth modules over affine Kac-Moody algebras. In a particular case, these modules are isomorphic to those induced from generalized Whittaker modules for Takiff Lie algebras. We establish the irreducibility criterion for constructed modules in the case of the affine sl(2) Lie algebra.

Smooth representations of affine Kac-Moody algebras

TL;DR

The paper addresses the problem of constructing and classifying new smooth modules for untwisted affine Kac-Moody algebras, extending beyond positive-energy representations. It develops a framework of induced smooth modules from carefully chosen subalgebras, links them to generalized Takiff algebras, and analyzes irreducibility for the fundamental case of affine sl2. A complete irreducibility and isomorphism picture is given for both the standard affine algebra and its extended version, including critical level phenomena and families of irreducible quotients parameterized by spectral data. The results provide a substantial expansion of smooth modules applicable to vertex algebra contexts and offer concrete criteria and quotients that capture a wide range of infinite-dimensional representations with rich algebraic structure.

Abstract

Smooth modules for affine Kac-Moody algebras have a prime importance for the quantum field theory as they correspond to the representations of the universal affine vertex algebras. But, very little is known about such modules beyond the category of positive energy representations. We construct a new class of smooth modules over affine Kac-Moody algebras. In a particular case, these modules are isomorphic to those induced from generalized Whittaker modules for Takiff Lie algebras. We establish the irreducibility criterion for constructed modules in the case of the affine sl(2) Lie algebra.
Paper Structure (18 sections, 24 theorems, 131 equations)

This paper contains 18 sections, 24 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Affine Lie algebras and smooth modules
  3. Affine Lie algebras
  4. Affine vertex algebras and Segal-Sugawara construction.
  5. Construction of smooth modules
  6. Connection to Takiff modules
  7. Affine Lie algebra $A^{(1)}_1$
  8. Irreducibility and isomorphism
  9. Reduction to the case $N_1=-1$.
  10. Singular vectors in $\widehat{M}(\varphi_{-1,N})$
  11. Irreducibility in the noncritical case
  12. Irreducibility in the critical case
  13. Isomorphisms
  14. Smooth modules over $\widetilde{\mathfrak{g}}$
  15. Irreducibility of smooth $\widetilde{\mathfrak{g}}$-modules
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $\widehat{\mathfrak{g}}=\widehat{\mathfrak{sl}}_2$. Suppose $n, m\in\mathbb{Z}$ with $n+m\geq0$ and $\varphi$ is a Lie algebra homomorphism from $\mathcal{S}_{n, m}$ to $\mathbb C$. The $\widehat{\mathfrak{g}}$-module $\widehat{M}(\varphi)$ is irreducible if and only if $\varphi(h(n+m+1))\neq0$

Theorems & Definitions (45)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Example 3.2
  • Example 3.3
  • ...and 35 more