Algebraic structures and combinatorial models

Cluster algebras are a class of commutative rings defined by simple recursive rules, yet they encode surprisingly rich geometric and representation-theoretic data — a tension that has driven much of modern algebra and topology over the past two decades. Researchers study how these algebras interact with triangulated and derived categories, which are algebraic frameworks for tracking relationships between mathematical objects up to homotopy, and with quiver representations, which translate network-like diagrams into linear algebra problems that reveal hidden symmetry. This machinery connects to quantum groups and Calabi-Yau algebras, whose structures underlie conformal field theory and the mathematics of topological phases of matter, making the subject relevant far beyond pure combinatorics. Central open directions include understanding the full scope of categorical models that realize cluster phenomena, classifying Calabi-Yau structures in higher homological dimensions, and pinning down the precise role of modular tensor categories in bridging representation theory with low-dimensional topology.

Works

101,226

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989,238

Keywords

Cluster AlgebrasTriangulated CategoriesDerived CategoriesQuiver RepresentationsHomological DimensionsQuantum Groups