Preprojective algebras and cluster algebras

Abstract

We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.

31 pages, one figure. This is a survey of our recent work. It will appear in the proceedings of the ICRA 12 conference held in Torun in 2007

References in corpus (2)

• Acyclic Calabi-Yau categories
• Cluster algebra structures and semicanonical bases for unipotent groups

Cited by in corpus (24)

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• Cluster algebras via cluster categories with infinite-dimensional morphism spaces
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• Quantum Unipotent Subgroup and dual canonical basis
• Purity for graded potentials and quantum cluster positivity
• Cluster characters II: A multiplication formula
• On Generalized Cluster Categories
• Tilting theory and cluster algebras
• A comparison of Newton-Okounkov polytopes of Schubert varieties
• Q-system Cluster Algebras, Paths and Total Positivity
• Denominators in cluster algebras of affine type
• Cluster automorphism groups of cluster algebras of finite type
• Fock-Goncharov conjecture and polyhedral cones for $U\subset S{L}_n$ and base affine space $S{L}_n/U$
• On cluster algebras with coefficients and 2-Calabi-Yau categories
• The affine Grassmannian and the Springer resolution in positive characteristic
• Cluster algebras arising from cluster tubes
• The canonical basis and the quantum Frobenius morphism
• Demazure modules, Fusion products and Q--systems
• Rigid Indecomposable Modules in Grassmannian Cluster Categories
• Newton-Okounkov convex bodies of Schubert varieties and polyhedral realizations of crystal bases
• The valuation pairing on an upper cluster algebra
• Polyhedral and Tropical Geometry of Flag Positroids
• Folding procedure for Newton-Okounkov polytopes of Schubert varieties
• Koszul calculus of preprojective algebras