Rigid modules over preprojective algebras

Abstract

Let $\la$ be a preprojective algebra of simply laced Dynkin type $\Delta$. We study maximal rigid $\la$-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 $\Delta$. As an application we obtain that all cluster monomials of $\C[N]$ belong to the dual semicanonical basis.

34 pages. Final Version.To appear in Invent. Math. Minor changes

References in corpus (2)

• Semicanonical bases and preprojective algebras
• Extension-orthogonal components of nilpotent varieties

Cited by in corpus (32)

• Mutation in triangulated categories and rigid Cohen-Macaulay modules
• Cluster structures for 2-Calabi-Yau categories and unipotent groups
• Kac-Moody groups and cluster algebras
• Quiver varieties and cluster algebras
• Cluster algebras and quantum affine algebras
• Cluster structures on quantum coordinate rings
• Reduction of $\tau$-tilting modules and torsion pairs
• Wall and Chamber Structure for finite-dimensional Algebras
• On the Cluster Category of a Marked Surface
• Generic bases for cluster algebras and the Chamber Ansatz
• Categorification of skew-symmetrizable cluster algebras
• Geometric conditions for $\square$-irreducibility of certain representations of the general linear group over a non-archimedean local field
• Quotients of exact categories by cluster tilting subcategories as module categories
• Classifying tilting complexes over preprojective algebras of Dynkin type
• d-Representation-finite self-injective algebras
• Cluster algebras in algebraic Lie theory
• Mutation of frozen Jacobian algebras
• The Existence of Maximal $n$-Orthogonal Subcategories
• On the Finiteness of the Derived Equivalence Classes of some Stable Endomorphism Rings
• A quantum analogue of generic bases for affine cluster algebras
• Calabi-Yau properties of Postnikov diagrams
• A quantum cluster algebra of Kronecker type and the dual canonical basis
• Cluster algebras arising from cluster tubes
• On $n$-cubic Pyramid Algebras
• Cluster categories from Grassmannians and root combinatorics
• A non-simply laced version for cluster structures on 2-Calabi-Yau categories
• Cluster algebras and preprojective algebras : the non simply-laced case
• Maximal rigid objects without loops in connected 2-CY triangulated categories are cluster-tilting objects
• Existence of a 2-cluster tilting module does not imply finite complexity
• Morphisms and extensions between bricks over preprojective algebras of type A
• A geometric model for the non-$\tau$-rigid modules of type ${\tilde{D}}_n$
• Derived equivalences of self-injective 2-Calabi--Yau tilted algebras