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Finite induction functor for vertex operator algebras

Jianqi Liu

Abstract

In this paper, we introduce a new induction functor $\mathrm{Ind}^V_U$ between module categories corresponding to an embedding of vertex operator algebras (VOAs) $U \hookrightarrow V$. This induction functor is essentially defined at the level of the finite (Zhu) algebras, which we call the \emph{finite induction functor}. Under suitable conditions on $U$ and $V$, we prove that this functor satisfies the usual properties of induction functors, such as Frobenius reciprocity, functorial property for compositions, and an analogue of Artin's induction theorem for certain associated characters. To better understand the effect of this functor, we explicitly determine the finite induction of irreducible modules for standard subVOAs of the rank-one lattice/affine VOA $V_{A_1}$, as well as the finite induction of irreducible modules over a parabolic-type subVOA $V_P$ of the rank-two lattice/affine VOA $V_{A_2}$.

Finite induction functor for vertex operator algebras

Abstract

In this paper, we introduce a new induction functor between module categories corresponding to an embedding of vertex operator algebras (VOAs) . This induction functor is essentially defined at the level of the finite (Zhu) algebras, which we call the \emph{finite induction functor}. Under suitable conditions on and , we prove that this functor satisfies the usual properties of induction functors, such as Frobenius reciprocity, functorial property for compositions, and an analogue of Artin's induction theorem for certain associated characters. To better understand the effect of this functor, we explicitly determine the finite induction of irreducible modules for standard subVOAs of the rank-one lattice/affine VOA , as well as the finite induction of irreducible modules over a parabolic-type subVOA of the rank-two lattice/affine VOA .

Paper Structure

This paper contains 47 sections, 35 theorems, 277 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Finite induction functor for VOAs
  3. Definition of finite induction functor
  4. Properties and examples of the finite induction functor
  5. Parabolic-type subVOA $V_P$ of $V_{A_2}$
  6. Structure of the Zhu algebra $A(V_P)$
  7. Classification of irreducible $V_P$-modules
  8. Finite induction of irreducible $V_P$-modules under $V_P\hookrightarrow V_{A_2}$
  9. Finite induction and restriction functors for the VOA embedding $U\hookrightarrow V$
  10. Preliminaries on VOAs
  11. Module categories of VOAs
  12. Universal enveloping algebra of a VOA
  13. Zhu algebra of a VOA
  14. Generalized Verma module
  15. Mode transition algebras
  16. ...and 32 more sections

Key Result

Lemma 2.5

DGK2GGKL25 If the VOA $V$ satisfies the strongly unital property, then the adjoint pair $(\Phi^\mathsf{L} \dashv \Omega):\mathsf{Mod}(\mathsf{A})\rightleftarrows\mathsf{Adm}(V)$eq:adjointfunctor is an adjoint equivalence between categories. In this case, any admissible $V$-module $W$ is a generalize

Figures (3)

  • Figure 1:
  • Figure 2:
  • Figure 3:

Theorems & Definitions (82)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Example 2.4
  • Lemma 2.5
  • Proposition 2.6
  • proof
  • Corollary 2.7
  • Proposition 2.8
  • proof
  • ...and 72 more