Gentle Algebras Research Articles

There are no strictly shod algebras in hereditary gentle algebras

We prove that there are no strictly shod algebras in hereditary gentle algebras by geometric models. As an application, we give a classification of the silted algebras for Dynkin type A n and A ˜ n .

Spherical objects, transitivity and auto-equivalences of Kodaira cycles via gentle algebras

This paper studies the class of spherical objects over any Kodaira n -cycle of projective lines and provides a parametrization of their isomorphism classes in terms of closed curves on the n -punctured torus without self-intersections. Employing recent results on gentle algebras, we derive a topological model for the bounded derived category of any Kodaira cycle. The groups of triangle auto-equivalences of these categories are computed and are shown to act transitively on isomorphism classes of spherical objects. This answers a question by Polishchuk (2002) and extends earlier results by Burban–Kreußler (2012) and Lekili–Polishchuk (2019). The description of auto-equivalences is further used to establish faithfulness of a mapping class group action defined by Sibilla (2014). The final part describes the closed curves which correspond to vector bundles and simple vector bundles. This leads to an alternative proof of a result by Bodnarchuk–Drozd–Greuel (2012) which states that simple vector bundles on cycles of projective lines are uniquely determined by their multi-degree, rank and determinant. As a by-product, we obtain a closed formula for the cyclic sequence of any simple vector bundle on C_{n} as introduced by Burban–Drozd–Greuel (2001).

Triangulations of flow polytopes, ample framings, and gentle algebras

Triangulations of flow polytopes, ample framings, and gentle algebras

G-semisimple algebras

Let Λ be an Artin algebra and mod-(Gprj_-Λ) the category of finitely presented functors over the stable category Gprj_-Λ of finitely generated Gorenstein projective Λ-modules. This paper deals with those algebras Λ in which mod-(Gprj_-Λ) is a semisimple abelian category, and we call G-semisimple algebras. We study some basic properties of such algebras. In particular, it will be observed that the class of G-semisimple algebras contains important classes of algebras, including gentle algebras and more generally quadratic monomial algebras. Next, we construct an epivalence (called representation equivalence in the terminology of Auslander), i.e. a full and dense functor that reflects isomorphisms, from the stable category of Gorenstein projective representations Gprj_(Q,Λ) of a finite acyclic quiver Q to the category of representations rep(Q,Gprj_-Λ) over Gprj_-Λ, provided Λ is a G-semisimple algebra over an algebraic closed field. Using this, we will show that the path algebra ΛQ of the G-semisimple algebra Λ is CM-finite if and only if Q is Dynkin. In the last part, we provide a complete classification of indecomposable Gorenstein projective representations within Gprj(An,Λ) of the linear quiver An over a G-semisimple algebra Λ. We also determine almost split sequences in Gprj(An,Λ) with certain ending terms. We apply these results to obtain insights into the cardinality of the components of the stable Auslander-Reiten quiver Gprj(An,Λ).

Homological dimensions of gentle algebras via geometric models

Homological dimensions of gentle algebras via geometric models

Gentle Algebras Arising from Surfaces with Orbifold Points of Order 3, Part I: Scattering Diagrams

To each triangulation of any surface with marked points on the boundary and orbifold points of order three, we associate a quiver (with loops) with potential whose Jacobian algebra is finite dimensional and gentle. We study the stability scattering diagrams of such gentle algebras and use them to prove that the Caldero–Chapoton map defines a bijection between τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-rigid pairs and cluster monomials of the generalized cluster algebra associated to the surface by Chekhov and Shapiro.

Associahedra for finite‐type cluster algebras and minimal relations between g‐vectors

AbstractWe show that the mesh mutations are the minimal relations among the ‐vectors with respect to any initial seed in any finite‐type cluster algebra. We then use this algebraic result to derive geometric properties of the ‐vector fan: we show that the space of all its polytopal realizations is a simplicial cone, and we then observe that this property implies that all its realizations can be described as the intersection of a high‐dimensional positive orthant with well‐chosen affine spaces. This sheds a new light on and extends earlier results of Arkani‐Hamed, Bai, He, and Yan in type and of Bazier‐Matte, Chapelier‐Laget, Douville, Mousavand, Thomas, and Yıldırım for acyclic initial seeds. Moreover, we use a similar approach to study the space of polytopal realizations of the ‐vector fans of another generalization of the associahedron: nonkissing complexes (also known as support ‐tilting complexes) of gentle algebras. We show that the space of realizations of the nonkissing fan is simplicial when the gentle bound quiver is brick and 2‐acyclic, and we describe in this case its facet‐defining inequalities in terms of mesh mutations. Along the way, we prove algebraic results on 2‐Calabi–Yau triangulated categories, and on extriangulated categories that are of independent interest. In particular, we prove, in those two setups, an analogue of a result of Auslander on minimal relations for Grothendieck groups of module categories.

Jordan recoverability of some subcategories of modules over gentle algebras

Gentle algebras form a class of finite-dimensional algebras introduced by I. Assem and A. Skowroński in the 1980s. Modules over such an algebra can be described by string and band combinatorics in the associated gentle quiver from the work of M.C.R. Butler and C.M. Ringel. Any module can be naturally associated to a quiver representation. A nilpotent endomorphism of a quiver representation induces linear transformations over vector spaces at each vertex. Generically among all nilpotent endomorphisms, a well-defined Jordan form exists for these representations. We focus on subcategories additively generated by all the indecomposable representations of a gentle quiver, including a fixed vertex in their support. We show a characterization of the vertices such that the objects of this subcategory are determined up to isomorphism by their generic Jordan form.

Exceptional sequences in the derived category of a gentle algebra

In this paper, using the correspondence of gentle algebras and dissections of marked surfaces, we study full exceptional sequences in the perfect derived category {{,mathrm{mathsf {K^b(A)}},}} of a gentle algebra {{,mathrm{textsf{A}},}}. We show that full exceptional sequences in {{,mathrm{mathsf {K^b(A)}},}} exist if and only if the associated marked surface has no punctures and has at least two marked points on the boundary. Furthermore, by using induction on cuts of surfaces, we characterize when an exceptional sequence can be completed to a full exceptional sequence. If the genus of the associated surface is zero then we show that the action of the braid group together with the grading shift on full exceptional sequences in {{,mathrm{mathsf {K^b(A)}},}} is transitive. For the case of surfaces of higher genus, we reduce the problem of transitivity to the problem of the existence of certain sequences of pairs of exceptional objects. Finally, we interpret the duality of a full exceptional sequence induced by the longest element in the associated symmetric group using Koszul duality.

On support τ-tilting graphs of gentle algebras

Let A be a finite-dimensional gentle algebra over an algebraically closed field. We investigate the combinatorial properties of support τ-tilting graph of A. In particular, it is proved that the support τ-tilting graph of A is connected and has the so-called reachable-in-face property. This property was conjectured by Fomin and Zelevinsky for exchange graphs of cluster algebras which was recently confirmed by Cao and Li.

A complete derived invariant for gentle algebras via winding numbers and Arf invariants

Gentle algebras are in bijection with admissible dissections of marked oriented surfaces. In this paper, we further study the properties of admissible dissections and we show that silting objects for gentle algebras are given by admissible dissections of the associated surface. We associate to each gentle algebra a line field on the corresponding surface and prove that the derived equivalence class of the algebra is completely determined by the homotopy class of the line field up to homeomorphism of the surface. Then, based on winding numbers and the Arf invariant of a certain quadratic form over {mathbb {Z}}_2, we translate this to a numerical complete derived invariant for gentle algebras.

A geometric realization of silting theory for gentle algebras

A gentle algebra gives rise to a dissection of an oriented marked surface with boundary into polygons and the bounded derived category of the gentle algebra has a geometric interpretation in terms of this surface. In this paper we study silting theory in the bounded derived category of a gentle algebra in terms of its underlying surface. In particular, we show how silting mutation corresponds to the changing of graded arcs and that in some cases silting mutation results in the interpretation of the octahedral axioms in terms of the flipping of diagonals in a quadrilateral as in the work of Dyckerhoff and Kapranov (J Eur Math Soc 20(6):1473–1524, 2018) in the context of triangulated surfaces. We also show that the cutting of the underlying surface along a curve without self-intersections corresponds to silting reduction. This is analogous to the Calabi–Yau reduction of surface cluster categories as shown by Marsh and Palu (Proc Lond Math Soc (3) 108(2):411–440, 2014) and Qiu and Zhou (Compos Math 153(9):1779–1819, 2017).

Corrigendum: A geometric model for the module category of a gentle algebra

Abstract In this note we correct an oversight in [A geometric model for the module category of a gentle algebra, Int. Math. Res. Not. IMRN 2021, no. 15, 11357–11392] regarding morphisms between indecomposable modules of a gentle algebra. The oversight only occurs when bands are involved, which can be easily corrected by considering infinite periodic strings associated to bands and the universal cover of the surface associated to the gentle algebra.

The role of gentle algebras in higher homological algebra

Abstract We investigate the role of gentle algebras in higher homological algebra. In the first part of the paper, we show that if the module category of a gentle algebra Λ contains a d-cluster tilting subcategory for some d ≥ 2 {d\geq 2} , then Λ is a radical square zero Nakayama algebra. This gives a complete classification of weakly d-representation finite gentle algebras. In the second part, we use a geometric model of the derived category to prove a similar result in the triangulated setup. More precisely, we show that if 𝒟 b ⁡ ( Λ ) {\operatorname{\mathcal{D}}^{b}(\Lambda)} contains a d-cluster tilting subcategory that is closed under [ d ] {[d]} , then Λ is derived equivalent to an algebra of Dynkin type A. Furthermore, our approach gives a geometric characterization of all d-cluster tilting subcategories of 𝒟 b ⁡ ( Λ ) {\operatorname{\mathcal{D}}^{b}(\Lambda)} that are closed under [ d ] {[d]} .

Derived equivalence classification of Brauer graph algebras

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∞-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.

Derived categories of skew-gentle algebras and orbifolds

AbstractSkew-gentle algebras are a generalisation of the well-known class of gentle algebras with which they share many common properties. In this work, using non-commutative Gröbner basis theory, we show that these algebras are strong Koszul and that the Koszul dual is again skew-gentle. We give a geometric model of their bounded derived categories in terms of polygonal dissections of surfaces with orbifold points, establishing a correspondence between curves in the orbifold and indecomposable objects. Moreover, we show that the orbifold dissections encode homological properties of skew-gentle algebras such as their singularity categories, their Gorenstein dimensions and derived invariants such as the determinant of theirq-Cartan matrices.

Derived equivalences between skew-gentle algebras using orbifolds

Skew-gentle algebras are skew-group algebras of gentle algebras equipped with a certain $\Z_2$-action. Building on the bijective correspondence between gentle algebras and dissected surfaces, we obtain in this paper a bijection between skew-gentle algebras and certain dissected orbifolds that admit a double cover. We prove the compatibility of the $\Z_2$-action on the double cover with the skew-group algebra construction. This allows us to investigate the derived equivalence relation between skew-gentle algebras in geometric terms: We associate to each skew-gentle algebra a line field on the orbifold, and on its double cover, and interpret different kinds of derived equivalences of skew-gentle algebras in terms of diffeomorphisms respecting the homotopy class of the line fields associated to the algebras.

Finite gentle repetitions of gentle algebras and their Avella-Alaminos–Geiss invariants

Among finite dimensional algebras over a field K, the class of gentle algebras is known to be closed by derived equivalences. Although a classification up to derived equivalences is usually a difficult problem, Avella-Alaminos and Geiss have introduced derived invariants for gentle algebras A, which can be calculated combinatorially from their bound quivers, applicable to such classification. Ladkani has given a formula to describe the dimensions of the Hochschild cohomologies of A in terms of some values of its Avella-Alaminos–Geiss invariants. This in turn implies a cohomological meaning of these values. Since most of the other values do not appear in this formula, it will be a natural question to ask if there is a similar cohomological meaning for such values. In this article, we construct a sequence of gentle algebras indexed by positive integers by a procedure which we call finite gentle repetitions, in order to relate these values of Avella-Alaminos–Geiss invariants of A to the dimensions of Hochschild cohomologies of On the way we will see that the Avella-Alaminos–Geiss invariants of are completely determined by those of A. Therefore one may expect that the finite gentle repetitions would preserve derived equivalences in a nice situation. In the last part of this article, we will see how the generalized Auslander–Platzeck–Reiten reflection can be related to the finite gentle repetition.

Schemes of modules over gentle algebras and laminations of surfaces

We study the affine schemes of modules over gentle algebras. We describe the smooth points of these schemes, and we also analyze their irreducible components in detail. Several of our results generalize formerly known results, e.g. by dropping acyclicity, and by incorporating band modules. A special class of gentle algebras are Jacobian algebras arising from triangulations of unpunctured marked surfaces. For these we obtain a bijection between the set of generically tau -reduced decorated irreducible components and the set of laminations of the surface. As an application, we get that the set of bangle functions (defined by Musiker–Schiffler–Williams) in the upper cluster algebra associated with the surface coincides with the set of generic Caldero-Chapoton functions (defined by Geiß–Leclerc–Schröer).

Non-kissing complexes and tau-tilting for gentle algebras

We interpret the support τ \tau -tilting complex of any gentle bound quiver as the non-kissing complex of walks on its blossoming quiver. Particularly relevant examples were previously studied for quivers defined by a subset of the grid or by a dissection of a polygon. We then focus on the case when the non-kissing complex is finite. We show that the graph of increasing flips on its facets is the Hasse diagram of a congruence-uniform lattice. Finally, we study its g \mathbf {g} -vector fan and prove that it is the normal fan of a non-kissing associahedron.