Combinatorial derived invariants for gentle algebras

By Diana Avella-Alaminos, C. Geiss. 2006

94 citations recorded by the source corpus.

Abstract We define derived equivalent invariants for gentle algebras, constructed in an easy combinatorial way from the quiver with relations defining these algebras. Our invariants consist of pairs of natural numbers and contain important information about the algebra and the structure of the stable Auslander–Reiten quiver of its repetitive algebra. As a by-product we obtain that the number of arrows of the quiver of a gentle algebra is invariant under derived equivalence. Finally, our invariants separate the derived equivalence classes of gentle algebras with at most one cycle.

Subfields: Atomic and Molecular Physics, and Optics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/b1c216fc13db49ac3f3f4470046f79e243b73545

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In Atomic and Molecular Physics, and Optics

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

A complete derived invariant for gentle algebras via winding numbers and Arf invariants (2019, methodology): “We note that the curves in B are computed implicitly by the algorithm in [12].”
Morphisms between indecomposable complexes in the bounded derived category of a gentle algebra (2014, background): “The class presents us with a particularly good candidate to begin a systematic ‘hands on’ study of derived categories of non-hereditary algebras for a number of reasons: • Certain general aspects of their structure are already known: the Avella Alaminos– Geiß invariant describes certain fractionally”
The almost split triangles for perfect complexes over gentle algebras (2009, background): “An important role in these investigations is played by an invariant introduced by Avella-Alaminos and Geiss [6].”
Derived equivalences of gentle algebras via Fukaya categories (2018, methodology): “t we obtain a suﬃcient criterion for homologically smooth graded gentle algebras to be derived equivalent. Our criterion uses numerical invariants generalizing those given by Avella-Alaminos-Geiss in [7], as well as some other numerical invariants. As an application, we ﬁnd many new cases when the A”