Lifting preprojective algebras to orders and categorifying partial flag varieties

Abstract

We describe a categorification of the cluster algebra structure of multi-homogeneous coordinate rings of partial flag varieties of arbitrary Dynkin type using Cohen-Macaulay modules over orders. This completes the categorification of Geiss-Leclerc-Schröer by adding the missing coefficients. To achieve this, for an order $A$ and an idempotent $e\in A$, we introduce a subcategory ${\mathrm{CM}}_{e}A$ of $\mathrm{CM}A$ and study its properties. In particular, under some mild assumptions, we construct an equivalence of exact categories $({\mathrm{CM}}_{e}A)/[Ae]\cong \mathrm{Sub}Q$ for an injective $B$-module $Q$ where $B:=A/(e)$. These results generalize work by Jensen-King-Su concerning the cluster algebra structure of the Grassmannian ${\mathrm{Gr}}_m({\mathbb{C}}^n)$.

37 pages. Several important improvements, new results and examples added. A part of the previous version (explicit construction of orders at the end) has been removed to be put in a forthcoming article. Accepted for publication to Algebra and Number Theory

References in corpus (4)

• Dimer models and cluster categories of Grassmannians
• A categorification of Grassmannian cluster algebras
• Ice quivers with potentials associated with triangulations and Cohen-Macaulay modules over orders
• Ice quivers with potential arising from once-punctured polygons and Cohen-Macaulay modules

Cited by in corpus (4)

• Classifying exact categories via Wakamatsu tilting
• Internally Calabi-Yau algebras and cluster-tilting objects
• A categorification of acyclic principal coefficient cluster algebras
• Mutation via Hovey twin cotorsion pairs and model structures in extriangulated categories