From groups to clusters

Abstract

We construct a new class of symmetric algebras of tame representation type that are also the endomorphism algebras of cluster tilting objects in 2-Calabi-Yau triangulated categories, hence all their non-projective indecomposable modules are $\Omega$-periodic of period dividing 4. Our construction is based on the combinatorial notion of triangulation quivers, which arise naturally from triangulations of oriented surfaces with marked points. This class of algebras contains the algebras of quaternion type introduced and studied by Erdmann with relation to certain blocks of group algebras. On the other hand, it contains also the Jacobian algebras of the quivers with potentials associated by Fomin-Shapiro-Thurston and Labardini-Fragoso to triangulations of closed surfaces with punctures, hence our construction may serve as a bridge between the modular representation theory of finite groups and the theory of cluster algebras.

60 pages; an invited contribution to appear in the proceedings of the DFG priority program "Representation Theory" conference, Bad Honnef, 2015; this is a survey which is a (hugely) expanded version of arXiv:1404.6834

References in corpus (10)

• Stability structures, motivic Donaldson-Thomas invariants and cluster transformations
• On triangulated orbit categories
• Calabi-Yau algebras
• Perverse equivalences, BB-tilting, mutations and applications
• On Jacobian algebras from closed surfaces
• 2-CY-tilted algebras that are not Jacobian
• Mutation classes of certain quivers with potentials as derived equivalence classes
• ${\tau }^2$-stable tilting complexes over weighted projective lines
• Algebras of quasi-quaternion type
• Which mutation classes of quivers have constant number of arrows?

Cited by in corpus (1)

• Derived equivalence classification of Brauer graph algebras