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Partial flag varieties and preprojective algebras. (English) Zbl 1151.16009

The authors study preprojective algebras of type \(A\), \(D\) and \(E\) (i.e. simply-laced Dynkin type). In particular, they consider full subcategories whose objects are given by the submodules of a fixed injective module over such a preprojective algebra. A module in such a category is said to be rigid if it has no self-extensions (of degree 1), and it is said to be maximal rigid if it has a maximal number of non-isomorphic indecomposable direct summands.
It is shown that, via a certain mutation operation, the collection of maximal rigid modules gives rise to a cluster structure on the coordinate ring of a corresponding partial flag variety associated to the semisimple simply-connected complex algebraic group associated to the Dynkin diagram of the preprojective algebra. This implies that all the cluster monomials in this cluster structure lie in the dual semicanonical basis. (See also the related article [A. B. Buan, O. Iyama, I. Reiten and J. Scott, Cluster structures for 2-Calabi-Yau categories and unipotent groups, preprint, arXiv:math/0701557 (2007)].)

MSC:

16G20 Representations of quivers and partially ordered sets
14M15 Grassmannians, Schubert varieties, flag manifolds
16D90 Module categories in associative algebras
16G70 Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers
17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
20G05 Representation theory for linear algebraic groups
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