Gordana Todorov

Dynkin diagrams and higher Auslander algebras, Cluster Tilting Objects and Preprojective Algebras

January 9, 2025

Source: JMM 2025

Preprojective modules

There are three ways to think about preprojective modules:

  1. Preprojective Algebra $\Pi(Q)$ provides an algebraic approach where preprojective modules are part of the structure of the preprojective algebra associated with a quiver.

  2. $T(\text{Ext}_H^1)$ provides a homological perspective where preprojective modules can be understood in terms of Ext groups in the derived category, capturing extensions and relationships between modules.

  3. $\bigoplus_{i \geq 0} \tau^{-i} H$ gives a more categorical perspective, describing preprojective modules in terms of shifts under the Auslander-Reiten translation, which are crucial for the study of mutation and cluster-tilting objects in the cluster category.

Auslander Bijection

There is a 1-1 bijection between algebras of finite representation type and Auslander algebras $T$ satisfying $\dim T \le 2 \le \text{domdim }T$.

Why Care About Derived Categories?

The derived category of an abelian category $\mathcal{A}$ is important because it is the setting for homological algebra in $\mathcal{A}$.