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Ringel's contributions to quasi-hereditary algebras

Changchang Xi

TL;DR

The paper surveys Claus Michael Ringel's pivotal contributions to quasi-hereditary algebras, spanning ring-theoretic and module-theoretic perspectives, dualities, and construction methods. It highlights how heredity ideals, tilting theory, and Ringel duality organize highest weight phenomena within finite-dimensional algebras, and it connects these ideas to standardly stratified frameworks, bocses, and exact Borel decompositions. It also outlines systematic construction schemes (Dlab–Ringel and Dlab–Heath–Marko) that generate broad families of quasi-hereditary algebras and discusses their homological consequences, such as recollements and derived-category stratifications. Overall, the survey showcases how Ringel’s methods unify representation theory, homological algebra, and categorical approaches to understanding quasi-hereditary phenomena and their applications.

Abstract

Quasi-hereditary algebras were introduced by Cline, Parshall and Scott to describe the highest weight categories of representations of semisimple Lie algebras and algebraic groups by the module categories of finite-dimensional algebras. Since then a lot of homological, structural and categorical properties of quasi-hereditary algebras have been discovered. This class of algebras seems quite common and occurs in many branches of mathematics. There are lots of important works on the subject. In this note we mainly survey some of Claus Michael Ringel's works or works jointly with his collaborators on quasi-hereditary algebras. Also, some of related works and recent developments on quasi-hereditary algebras are mentioned.

Ringel's contributions to quasi-hereditary algebras

TL;DR

The paper surveys Claus Michael Ringel's pivotal contributions to quasi-hereditary algebras, spanning ring-theoretic and module-theoretic perspectives, dualities, and construction methods. It highlights how heredity ideals, tilting theory, and Ringel duality organize highest weight phenomena within finite-dimensional algebras, and it connects these ideas to standardly stratified frameworks, bocses, and exact Borel decompositions. It also outlines systematic construction schemes (Dlab–Ringel and Dlab–Heath–Marko) that generate broad families of quasi-hereditary algebras and discusses their homological consequences, such as recollements and derived-category stratifications. Overall, the survey showcases how Ringel’s methods unify representation theory, homological algebra, and categorical approaches to understanding quasi-hereditary phenomena and their applications.

Abstract

Quasi-hereditary algebras were introduced by Cline, Parshall and Scott to describe the highest weight categories of representations of semisimple Lie algebras and algebraic groups by the module categories of finite-dimensional algebras. Since then a lot of homological, structural and categorical properties of quasi-hereditary algebras have been discovered. This class of algebras seems quite common and occurs in many branches of mathematics. There are lots of important works on the subject. In this note we mainly survey some of Claus Michael Ringel's works or works jointly with his collaborators on quasi-hereditary algebras. Also, some of related works and recent developments on quasi-hereditary algebras are mentioned.
Paper Structure (20 sections, 29 theorems, 37 equations)

This paper contains 20 sections, 29 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Ring theoretical definition of quasi-hereditary algebras
  3. Definitions and examples
  4. Examples
  5. Basic facts on quasi-hereditary algebras
  6. Hochschild homology and cohomology
  7. Some generalizations of quasi-hereditary algebras
  8. Module theoretical approach to quasi-hereditary algebras
  9. Module theoretical definition of quasi-hereditary algebras
  10. $\Delta$-good modules, characteristic tilting modules, and Ringel dualities
  11. Standardizations
  12. Standard modules of small projective dimension
  13. Finiteness of the category of $\Delta$-good modules
  14. Relations to bocs
  15. Decompositions of quasi-hereditary algebras
  16. ...and 5 more sections

Key Result

Lemma 2.2

dr1989 Let $e$ be an idempotent of a ring $R$. If the left module $_R(ReR)$ or the right module $(ReR)_R$ is projective, then the multiplication map $\mu: Re\otimes_{eRe}eR\rightarrow ReR, \; xe\otimes ey\mapsto xey,$ is bijective. Conversely, assume that $R$ is semiprimary with the Jacobson radical

Theorems & Definitions (42)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Proposition 2.5
  • Proposition 2.6
  • Theorem 2.7
  • Example 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 32 more