Monoidal Ringel duality and monoidal highest weight envelopes

TL;DR

This paper introduces a framework for realizing non-abelian monoidal categories as subcategories of tilting objects within abelian monoidal categories that have a highest weight structure, extending the concept of Ringel duality.

Contribution

It develops a monoidal enhancement of semi-infinite Ringel duality, applicable to various categories including those studied by Sam-Snowden and Knop, and explores implications for affine Lie algebra representations.

Findings

Realization of non-abelian monoidal categories as subcategories of tilting objects.

Extension of Ringel duality to a monoidal setting.

Application to categories of affine Lie algebra representations.

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.