Combinatorial derived invariants for gentle algebras

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.

22 pages, slightly reorganized and an example added, final version. To appear in J. Pure Appl. Algebra

Cited by in corpus (22)

• Singularity categories of gentle algebras
• Mutation classes of \tilde{A}_n-quivers and derived equivalence classification of cluster tilted algebras of type \tilde{A}_n
• Mutation classes of certain quivers with potentials as derived equivalence classes
• Gerstenhaber algebra structure on the Hochschild cohomology of quadratic string algebras
• On the derived categories of gentle and skew-gentle algebras: homological algebra and matrix problems
• On auto-equivalences and complete derived invariants of gentle algebras
• The derived category of surface algebras: the case of the torus with one boundary component
• Which mutation classes of quivers have constant number of arrows?
• Hochschild cohomology of gentle algebras
• Complete gentle and special biserial algebras are $g$-tame
• Derived equivalence classification of m-cluster tilted algebras of type A
• Derived classification of gentle algebras with two cycles
• Hochschild cohomology of Jacobian algebras from unpunctured surfaces: A geometric computation
• Some examples of $t$-structures for finite-dimensional algebras
• The AG-invariant for (m+2)-angulations
• The almost split triangles for perfect complexes over gentle algebras
• Morphisms between indecomposable complexes in the bounded derived category of a gentle algebra
• Maximal tori in $H{H}^1$ and the fundamental group
• Exceptional cycles for perfect complexes over gentle algebras
• The graded centers of derived discrete algebras
• On derived equivalence classification of gentle two-cycle algebras
• Hochschild cohomology of $m$-Cluster tilted algebras of type $\tilde{\mathbb{A}}$