Lifting preprojective algebras to orders and categorifying partial flag varieties

Demonet, Laurent; Iyama, Osamu

doi:10.2140/ant.2016.10.1527

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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

DOI: 10.2140/ant.2016.10.1527

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 ${CM}_{e} A$ of $CM A$ and study its properties. In particular, under some mild assumptions, we construct an equivalence of exact categories $({CM}_{e} A) ∕ [A e] ≅ Sub Q$ for an injective $B$ -module $Q$ , where $B : = A ∕ (e)$ . These results generalize work by Jensen, King and Su concerning the cluster algebra structure of the Grassmannian ${Gr}_{m} (ℂ^{n})$ .

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

Received: 25 September 2015

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

Vol. 10, No. 7, 2016