Derived equivalence classification of Brauer graph algebras

Abstract

We classify Brauer graph algebras up to derived equivalence by showing that the set of derived invariants introduced by Antipov is complete. These algebras first appeared in representation theory of finite groups and can be defined for any suitably decorated graph on an oriented surface. Motivated by the connection between Brauer graph algebras and gentle algebras we consider $A_{\infty }$-trivial extensions of partially wrapped Fukaya categories associated to surfaces with boundary. This construction naturally enlarges the class of Brauer graph algebras and provides a way to construct derived equivalences between Brauer graph algebras with the same derived invariants. As part of the proof we provide an interpretation of derived invariants of Brauer graph algebras as orbit invariants of line fields under the action of the mapping class group.

arXiv comment42 pages, 3rd version: added Theorem B; closed gap in Proposition 7.6. and added Lemma 7.5.; expanded Remark 7.13

References in corpus (12)

• Silting mutation in triangulated categories
• Flat surfaces and stability structures
• Suspending Lefschetz fibrations, with an application to Local Mirror Symmetry
• A geometric model for the derived category of gentle algebras
• Classification of two-term tilting complexes over Brauer graph algebras
• Infinity inner products on A-infinity algebras
• On auto-equivalences and complete derived invariants of gentle algebras
• Sheaves on local Calabi-Yau varieties
• Tame block algebras of Hecke algebras of classical type
• A primer on A-infinity-algebras and their Hochschild homology
• From groups to clusters
• Self-injective cellular algebras of polynomial growth representation type

Cited by in corpus (3)

• Representation type of cyclotomic quiver Hecke algebras of type $A_{\ell }^{(1)}$
• Generalized Kauer moves and derived equivalences of Brauer graph algebras
• Representation type of higher level cyclotomic quiver Hecke algebras in affine type C