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Preservation of Loewy Diagrams Under Exact Functors

Matthew Rupert

Abstract

We derive sufficient conditions for exact functors on locally finite abelian categories to preserve Loewy diagrams of objects. We apply our results to determine sufficient conditions for induction functors associated to simple current extensions of vertex algebras to preserve Loewy diagrams.

Preservation of Loewy Diagrams Under Exact Functors

Abstract

We derive sufficient conditions for exact functors on locally finite abelian categories to preserve Loewy diagrams of objects. We apply our results to determine sufficient conditions for induction functors associated to simple current extensions of vertex algebras to preserve Loewy diagrams.
Paper Structure (7 sections, 9 theorems, 38 equations)

This paper contains 7 sections, 9 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Loewy Diagrams
  4. Algebra Objects
  5. Invertible Objects
  6. Preservation of Loewy Diagrams
  7. Induction Functors

Key Result

Theorem 1.1

CKLCKMHKL Let $\mathcal{C}$ be a category of modules over some vertex operator algebra $V$ with a natural vertex tensor category structure. Then the following are equivalent: Further, the category of $V^e$-modules which lie in $\mathcal{C}$ as $V$-modules is braided equivalent to $\mathrm{Rep}^0V^e$, now viewing $V^e \in \mathcal{C}$ as a commutative algebra object.

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • Definition 3.1
  • ...and 12 more