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Monoidal Ringel duality and monoidal highest weight envelopes

Johannes Flake, Jonathan Gruber

Abstract

We show that a large class of non-abelian monoidal categories can be realized as subcategories of tilting objects in abelian monoidal categories with a highest weight structure. The construction relies on a monoidal enhancement of Brundan-Stroppel's semi-infinite Ringel duality and applies to many of Sam-Snowden's triangular categories and Knop's tensor envelopes of regular categories. We also explain how monoidal Ringel duality gives rise to monoidal structures on categories of representations of affine Lie algebras at positive levels.

Monoidal Ringel duality and monoidal highest weight envelopes

Abstract

We show that a large class of non-abelian monoidal categories can be realized as subcategories of tilting objects in abelian monoidal categories with a highest weight structure. The construction relies on a monoidal enhancement of Brundan-Stroppel's semi-infinite Ringel duality and applies to many of Sam-Snowden's triangular categories and Knop's tensor envelopes of regular categories. We also explain how monoidal Ringel duality gives rise to monoidal structures on categories of representations of affine Lie algebras at positive levels.
Paper Structure (19 sections, 57 theorems, 192 equations)

This paper contains 19 sections, 57 theorems, 192 equations.

Table of Contents

  1. Monoidal categories
  2. Monoidal categories
  3. Karoubi envelopes
  4. Tensor product of complexes
  5. Presheaves and Day convolution
  6. Ind-completions
  7. Highest weight categories
  8. Preliminaries on highest weight categories
  9. Tilting objects and Ringel duality
  10. Lower to upper Ringel duality and functor categories
  11. Upper to lower Ringel duality and exceptional sequences
  12. Monoidal Ringel duality
  13. Lower to upper monoidal Ringel duality
  14. Upper to lower monoidal Ringel duality
  15. Monoidal abelian envelopes
  16. ...and 4 more sections

Key Result

Theorem A

Let $\mathcal{R}$ be a subobject-finite exact regular Mal'cev category with a degree function $\delta$, taking values in an algebraically closed field of characteristic zero. Then there is a lower finite highest weight category $\mathcal{C}$ with a rigid symmetric monoidal structure such that the fu as symmetric monoidal categories. If $\mathcal{T}(\mathcal{R},\delta)$ admits a monoidal abelian en

Theorems & Definitions (146)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Definition 1.1
  • Lemma 1.2
  • Definition 1.3
  • Lemma 1.4
  • proof
  • Lemma 1.5
  • ...and 136 more