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Semi-infinite highest weight categories

Jonathan Brundan, Catharina Stroppel

Abstract

We develop axiomatics of highest weight categories and quasi-hereditary algebras in order to incorporate two semi-infinite situations which are in Ringel duality with each other; the underlying posets are either upper finite or lower finite. We also consider various more general sorts of stratified categories. In the upper finite cases, we give an alternative characterization of these categories in terms of based quasi-hereditary algebras and based stratified algebras, which are certain locally unital algebras possessing triangular bases.

Semi-infinite highest weight categories

Abstract

We develop axiomatics of highest weight categories and quasi-hereditary algebras in order to incorporate two semi-infinite situations which are in Ringel duality with each other; the underlying posets are either upper finite or lower finite. We also consider various more general sorts of stratified categories. In the upper finite cases, we give an alternative characterization of these categories in terms of based quasi-hereditary algebras and based stratified algebras, which are certain locally unital algebras possessing triangular bases.

Paper Structure

This paper contains 37 sections, 148 theorems, 266 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Some finiteness properties on Abelian categories
  3. Finite and locally finite Abelian categories
  4. Locally unital algebras
  5. Schurian categories
  6. Essentially finite Abelian categories
  7. Recollement
  8. Generalizations of highest weight categories
  9. Stratifications and the associated standard and costandard objects
  10. Finite and essentially finite $\varepsilon$-stratified categories
  11. Upper finite $\varepsilon$-stratified categories
  12. Shared lemmas for $\S\S$\ref{['ssn']}--\ref{['sst']}
  13. Lower finite $\varepsilon$-stratified categories
  14. Refining stratifications in fully stratified categories
  15. Tilting modules and semi-infinite Ringel duality
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $\mathcal{R}$ be a finite Abelian category equipped with a stratification $(\mathbf{B},L,\rho,\Lambda,\leq)$ and $\varepsilon:\Lambda\rightarrow\{\pm\}$ be a sign function. Then $\mathcal{R}$ is $\varepsilon$-stratified if and only if $\mathcal{R}^{\operatorname{op}}$ is $(-\varepsilon)$-stratif

Theorems & Definitions (323)

  • Theorem 1.1: Dlab,…
  • Theorem 1.2: Ringel, Happel, …
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 313 more