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Quasi-integrable modules over affine Lie superalgebras (Critical level)

Asghar Daneshvar, Hajar Kiamehr, Malihe Yousofzadeh

TL;DR

This work tackles the problem of classifying simple finite weight modules over twisted affine Lie superalgebras at zero (critical) level by developing a quasi-integrable framework. The authors introduce a multi-stage parabolic-induction strategy that passes through a sequence of subalgebras, ultimately reducing to finite-dimensional modules and leveraging center-triviality to apply known results. They prove a complete characterization of zero-level quasi-integrable modules, clarifying how real roots split into full-locally nilpotent or hybrid categories and showing how a three-step induction yields the full classification. The results advance the understanding of representation theory at the critical level and have potential implications for mathematical physics and related algebraic structures.

Abstract

Representation theory of Lie (super)algebras has attracted significant research interest for many years, especially due to its applications in theoretical physics; in this regard, the representation theory of affine Lie (super)algebras is of central importance. To characterize simple modules over affine Lie (super)algebras, it is necessary to study the cases of nonzero and critical levels separately. Although a vast amount of research has been done on the representation theory of affine Lie (super)algebras $\LL$, investigations concerning general modules at the critical level remain limited. In all existing studies, the characterization of the modules under consideration is reduced to the characterization of modules over some subalgebras of $\LL$. Depending on the structure of the original modules, these subalgebras -- and the corresponding modules -- have different natures some of which are already known, while others need to be studied separately. In this paper, we give a complete characterization of the modules over specific subalgebras $\gG$ of a twisted affine Lie superalgebra $\LL$ that arise in the study of general zero level simple finite weight $\LL$-modules. In particular, in the special case that $\gG=\LL,$ we obtain a complete characterization of quasi-integrable $\LL$-modules of level zero.

Quasi-integrable modules over affine Lie superalgebras (Critical level)

TL;DR

This work tackles the problem of classifying simple finite weight modules over twisted affine Lie superalgebras at zero (critical) level by developing a quasi-integrable framework. The authors introduce a multi-stage parabolic-induction strategy that passes through a sequence of subalgebras, ultimately reducing to finite-dimensional modules and leveraging center-triviality to apply known results. They prove a complete characterization of zero-level quasi-integrable modules, clarifying how real roots split into full-locally nilpotent or hybrid categories and showing how a three-step induction yields the full classification. The results advance the understanding of representation theory at the critical level and have potential implications for mathematical physics and related algebraic structures.

Abstract

Representation theory of Lie (super)algebras has attracted significant research interest for many years, especially due to its applications in theoretical physics; in this regard, the representation theory of affine Lie (super)algebras is of central importance. To characterize simple modules over affine Lie (super)algebras, it is necessary to study the cases of nonzero and critical levels separately. Although a vast amount of research has been done on the representation theory of affine Lie (super)algebras , investigations concerning general modules at the critical level remain limited. In all existing studies, the characterization of the modules under consideration is reduced to the characterization of modules over some subalgebras of . Depending on the structure of the original modules, these subalgebras -- and the corresponding modules -- have different natures some of which are already known, while others need to be studied separately. In this paper, we give a complete characterization of the modules over specific subalgebras of a twisted affine Lie superalgebra that arise in the study of general zero level simple finite weight -modules. In particular, in the special case that we obtain a complete characterization of quasi-integrable -modules of level zero.
Paper Structure (8 sections, 10 theorems, 129 equations, 2 tables)

This paper contains 8 sections, 10 theorems, 129 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Generic Information
  4. Parabolically Induced Modules.
  5. On split central extensions of affine Lie algebras
  6. Twisted affine Lie superalgebras
  7. Well-behaved interactions
  8. Main results

Key Result

Proposition 2.1

Suppose that $\lambda$ and $\mu$ are as above and set $P:=P_{\lambda,\mu}$. If $V$ is a simple finite weight $\mathfrak{k}$-module with then $V^{\mathfrak{k}_{_P}^+}$ is a simple finite weight $\mathfrak{k}_{_P}^\circ$-module and $V\simeq {\rm Ind}_{\lambda,\mu}^\mathfrak{k}(V^{\mathfrak{k}_{_P}^+}).$

Theorems & Definitions (19)

  • Proposition 2.1
  • Theorem 2.2
  • proof
  • Remark 2.3
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • Theorem 2.6: you8
  • Definition 2.7: you9
  • ...and 9 more