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Rigid modules over preprojective algebras

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Abstract

Let Λ be a preprojective algebra of simply laced Dynkin type Δ. We study maximal rigid Λ-modules, their endomorphism algebras and a mutation operation on these modules. This leads to a representation-theoretic construction of the cluster algebra structure on the ring ℂ[N] of polynomial functions on a maximal unipotent subgroup N of a complex Lie group of type Δ. As an application we obtain that all cluster monomials of ℂ[N] belong to the dual semicanonical basis.

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Authors and Affiliations

  1. Instituto de Matemáticas, UNAM, Ciudad Universitaria, 04510, Mexico D.F., Mexico

    Christof Geiß

  2. Laboratoire LMNO, Université de Caen, F-14032, Caen Cedex, France

    Bernard Leclerc

  3. Mathematisches Institut, Universität Bonn, Beringstr. 1, D-53115, Bonn, Germany

    Jan Schröer

Authors
  1. Christof Geiß
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  2. Bernard Leclerc
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  3. Jan Schröer
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Correspondence to Christof Geiß, Bernard Leclerc or Jan Schröer.

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Mathematics Subject Classification (2000)

14M99, 16D70, 16E20, 16G20, 16G70, 17B37, 20G42

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Geiß, C., Leclerc, B. & Schröer, J. Rigid modules over preprojective algebras. Invent. math. 165, 589–632 (2006). https://doi.org/10.1007/s00222-006-0507-y

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