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Admissible modules over affine Lie superalgebras: The final step in the characterization

Malihe Yousofzadeh

TL;DR

The paper addresses the problem of classifying simple admissible modules over affine Lie superalgebras, with particular focus on zero-level representations. It extends the theory of twisted localization to the super setting and couples it with parabolic induction and coherent families to complete the classification. The authors prove that zero-level admissible modules contain simple submodules, develop a super version of twisted localization, and show that simple admissible modules with nonempty $R_{f-in}(M)\cap -R_{f-in}(M)$ are iterated twisted localizations of zero-level modules, expressible as twists of induced modules ${\rm Ind}_P(\Omega)$ from a cuspidal or structured subalgebra. They also determine which affine Lie superalgebras admit finite weight modules with $R_{re}=R^{in}(M)$. Overall, the work completes the final step in the characterization program for simple admissible modules over affine Lie superalgebras, providing a concrete, constructive description and a finite-type classification in many cases.

Abstract

Over the past three decades, there have been several attempts to characterize modules over affine Lie superalgebras. One of the main issues in this regard is dealing with zero-level modules. In this paper, we study these modules and {complete the characterization} of simple admissible modules over affine Lie superalgebras.

Admissible modules over affine Lie superalgebras: The final step in the characterization

TL;DR

The paper addresses the problem of classifying simple admissible modules over affine Lie superalgebras, with particular focus on zero-level representations. It extends the theory of twisted localization to the super setting and couples it with parabolic induction and coherent families to complete the classification. The authors prove that zero-level admissible modules contain simple submodules, develop a super version of twisted localization, and show that simple admissible modules with nonempty are iterated twisted localizations of zero-level modules, expressible as twists of induced modules from a cuspidal or structured subalgebra. They also determine which affine Lie superalgebras admit finite weight modules with . Overall, the work completes the final step in the characterization program for simple admissible modules over affine Lie superalgebras, providing a concrete, constructive description and a finite-type classification in many cases.

Abstract

Over the past three decades, there have been several attempts to characterize modules over affine Lie superalgebras. One of the main issues in this regard is dealing with zero-level modules. In this paper, we study these modules and {complete the characterization} of simple admissible modules over affine Lie superalgebras.
Paper Structure (5 sections, 13 theorems, 95 equations)

This paper contains 5 sections, 13 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. twisted localization
  3. Finite weight modules over affine Lie superelagebras
  4. Admissible modules
  5. Twisted Localization and end of the characterization

Key Result

Lemma 3.2

Suppose that $M$ is a simple finite weight $\mathfrak{L}$-module such that $T:=R_{\rm f-in}(M)\cap -R_{\rm f-in}(M)\neq \emptyset.$ Then, the level of $M$ is zero.

Theorems & Definitions (21)

  • Definition 2.1
  • Remark 3.1
  • Lemma 3.2
  • proof
  • Definition 3.3
  • Example 3.4
  • Lemma 3.5
  • proof
  • Lemma 3.6
  • Lemma 3.7
  • ...and 11 more