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(Quasi-)admissible modules over symmetrizable Kac-Moody superalgebras

Maria Gorelik, Victor Kac

TL;DR

This work generalizes the theory of quasi-admissible and admissible modules from anisotropic settings to arbitrary symmetrizable Kac-Moody superalgebras, extending the foundational KW framework within Gorelik–Serganova's broader approach. It introduces and analyzes the Δ_L root subsystem, the L ↦ L′ correspondence, and the notion of π-quasi-admissibility, culminating in a criterion linking quasi-admissibility to partial integrability of the associated g′-modules. The paper proves an Arakawa-type theorem for vertex algebras attached to non-twisted affine Kac–Moody superalgebras and provides a comprehensive classification of admissible levels for many affine superalgebras (including A, D, G, F families and B(m|n) variants). It also treats boundary and subprincipal admissible levels, the Enright functor, and the structure of V_k(g)-modules, contributing key structural results and explicit level formulas with broad implications for representation theory and vertex-algebra constructions.

Abstract

The theory of admissible modules over symmetrizable anisotropic Kac-Moody superalgebras, introduced by Kac and Wakimoto in late 80's, is a well-developed subject with many applications, including representation theory of vertex algebras. Recently this theory was developed in a more general setup by Gorelik and Serganova. In the present paper we develop in this more general setup the theory of admissible modules over arbitrary symmetrizable Kac-Moody superalgebras.

(Quasi-)admissible modules over symmetrizable Kac-Moody superalgebras

TL;DR

This work generalizes the theory of quasi-admissible and admissible modules from anisotropic settings to arbitrary symmetrizable Kac-Moody superalgebras, extending the foundational KW framework within Gorelik–Serganova's broader approach. It introduces and analyzes the Δ_L root subsystem, the L ↦ L′ correspondence, and the notion of π-quasi-admissibility, culminating in a criterion linking quasi-admissibility to partial integrability of the associated g′-modules. The paper proves an Arakawa-type theorem for vertex algebras attached to non-twisted affine Kac–Moody superalgebras and provides a comprehensive classification of admissible levels for many affine superalgebras (including A, D, G, F families and B(m|n) variants). It also treats boundary and subprincipal admissible levels, the Enright functor, and the structure of V_k(g)-modules, contributing key structural results and explicit level formulas with broad implications for representation theory and vertex-algebra constructions.

Abstract

The theory of admissible modules over symmetrizable anisotropic Kac-Moody superalgebras, introduced by Kac and Wakimoto in late 80's, is a well-developed subject with many applications, including representation theory of vertex algebras. Recently this theory was developed in a more general setup by Gorelik and Serganova. In the present paper we develop in this more general setup the theory of admissible modules over arbitrary symmetrizable Kac-Moody superalgebras.

Paper Structure

This paper contains 135 sections, 294 equations.

Table of Contents

  1. Introduction
  2. Comment
  3. Index of definitions and notation
  4. Contragredient Lie superalgebras $\mathfrak g(A,\tau)$
  5. Definition of $\mathfrak g(A,\tau)$
  6. Triangular and root space decompositions
  7. Odd reflexions
  8. Remark
  9. Integrable $\mathfrak g(A,\tau)$
  10. Indecomposable $\mathfrak g(A,\tau)$
  11. Invariant bilinear form on symmetrizable $\mathfrak g(A,\tau)$
  12. Spine, principal roots, Weyl group, real and imaginary roots
  13. Remark
  14. Weyl vector $\rho=\rho_{\Sigma}$
  15. Categories $\mathcal{O}^{\operatorname{fin}}\subset \mathcal{O}_{\Sigma}\subset \mathcal{O}^{\inf}$
  16. ...and 120 more sections

Theorems & Definitions (50)

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  • ...and 40 more