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

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 A, we introduce a subcategory CMeA of CMA and study its properties. In particular, under some mild assumptions, we construct an equivalence of exact categories (CMeA)[Ae]SubQ 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 Grm(n).

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
Received: 25 September 2015
Revised: 26 April 2016
Accepted: 13 June 2016
Published: 27 September 2016
Laurent Demonet
Graduate School of Mathematics
Nagoya University
Nagoya 464-8602
Osamu Iyama
Graduate School of Mathematics
Nagoya University
Nagoya 464-8602