Vol. 10, No. 7, 2016

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Lifting preprojective algebras to orders and categorifying partial flag varieties

Laurent Demonet and Osamu Iyama

Vol. 10 (2016), No. 7, 1527–1579
Abstract

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

Keywords
orders, Cohen–Macaulay modules, finite-dimensional algebras, preprojective algebras, categorification, cluster algebras, partial flag varieties, exact categories
Mathematical Subject Classification 2010
Primary: 16G30
Secondary: 13F60, 16G10, 16G50, 18E10, 18E30
Milestones
Revised: 26 April 2016
Accepted: 13 June 2016
Published: 27 September 2016
Authors
 Laurent Demonet Graduate School of Mathematics Nagoya University Furocho Chikusaku Nagoya 464-8602 Japan Osamu Iyama Graduate School of Mathematics Nagoya University Furocho Chikusaku Nagoya 464-8602 Japan