Skew-Gentle Algebra: Structure & Applications

• Skew-gentle algebras are finite-dimensional associative algebras defined via a skew-gentle triple that extends gentle algebras with special idempotent loops.
• They connect representation theory with topology by translating algebraic data into orbifold surface models and marked dissection diagrams.
• Their refined module combinatorics and homological properties enable clear classifications in derived categories and support cluster-tilting applications.

A skew-gentle algebra is a finite-dimensional associative algebra over an algebraically closed field, arising as a generalization of the well-studied class of gentle algebras. Skew-gentle algebras occupy a central position in the modern representation theory of finite-dimensional algebras due to three intertwined features: their close relationship to surface and orbifold topology, their tame representation type, and their centrality in the categorification of cluster structures and stability conditions.

1. Definition and Basic Structure

A skew-gentle algebra is defined via a "skew-gentle triple" $(Q, Sp, I)$, where:

• $Q=(Q_0, Q_1, s, t)$ is a finite quiver;
• $I\subset KQ$ is an admissible ideal generated by paths of length two;
• $Sp\subset Q_0$ is a set of special vertices.

The construction proceeds by first ensuring that $(Q, I)$ is a gentle pair, imposing at each vertex conditions:

• At most two arrows in and two arrows out;
• For each arrow, at most one zero-relation to the left/right and at most one non-relation.

To $Q$, we adjoin for each $i\in Sp$ a loop $\epsilon_i$, extending $Q$ to $Q^{sp}$ and $I$ to $I^{sp}=I\cup\{\epsilon_i^2 \mid i\in Sp\}$. The skew-gentle algebra is then defined as: $$A^{sg} = KQ^{sp}/\langle I \cup \{\epsilon_i^2-\epsilon_i \mid i \in Sp\}\rangle,$$ with each $\epsilon_i$ enforcing a nontrivial idempotent relation. The resulting algebras are tame but typically not gentle. The characteristic property is that, upon replacing each special loop with a nilpotent relation, one recovers a gentle algebra structure.

Equivalent characterizations include matrix algebra presentations as well as interpretations as skew-group algebras of gentle algebras equipped with a specific $\mathbb{Z}_2$-action (Amiot et al., 2019, Chen, 2022).

2. Surface and Orbifold Models

A fundamental insight is the tight correspondence between combinatorics of skew-gentle algebras and geometry of orbifold surfaces. Given a skew-gentle triple, one constructs a marked surface $(S, M, P)$ (with boundary marked points $M$, interior punctures $P$ indexed by $Sp$) and a partial triangulation $T$ (skew-tiling). Special vertices correspond to orbifold points of order 2 in the surface model (Labardini-Fragoso et al., 2020, Kucer et al., 2021). The polygons arising in this combinatorial dissection encode the quiver and relations of the algebra.

Every skew-gentle algebra is derived Morita equivalent to the surface model defined by its associated orbifold dissection. Derived equivalence classes correspond to homeomorphism classes of dissected orbifolds with order-2 orbifold points, modulo flips of dissections preserving the line field data (Amiot et al., 2019, Schroll, 2020).

This topological realization enables:

• Geometric classification of indecomposable objects in derived categories via graded curves on orbifolds (Amiot, 2021);
• Derived invariant computations (Cartan determinants, Gorenstein dimension, singularity categories) via polygon counts in the orbifold dissection (Labardini-Fragoso et al., 2020, Chen et al., 2014).

3. Representation Theory and Module Combinatorics

The indecomposable module category of a skew-gentle algebra is described through string and band combinatorics analogous to the gentle case, but with substantial enrichments:

• Indecomposables correspond bijectively to words (strings or bands) in an associated gentle quiver, augmented with special local rules at vertices carrying idempotent special loops (Geiß, 2023, Burban et al., 2017).
• Each indecomposable $\tau$-rigid module is classified via its combinatorial data, with dimension vectors controlled by intersection numbers with a fixed collection of arcs in the surface model (Deng et al., 16 Nov 2025, He et al., 2020).
• Explicit bases for Hom-spaces between indecomposables are parameterized by "h-lines" in the associated quiver, and homological invariants such as the E-invariant and g-vector are computable in terms of combinatorial data involving real h-lines and fringing/kissing operations (Geiß, 2023).

These combinatorics enable translation between representation-theoretic data and tagged arc collections in the surface model, with cluster-tilting and support $\tau$-tilting objects appearing as maximal non-crossing tagged arc systems (He et al., 2022, He et al., 2020).

4. Homological and Derived Properties

Skew-gentle algebras are strong Koszul in all characteristics except possibly $2$, and their Koszul duals are again skew-gentle, with surface/orbifold models for the dual algebra being dual graphs on the same underlying orbifold (Labardini-Fragoso et al., 2020). These algebras are always Gorenstein, satisfy the finitistic dimension conjecture, Auslander–Reiten's conjecture, and admit well-behaved $K$-theory decompositions reflecting their gentle subquotients (Chen, 2022).

A canonical recollement relates the derived categories of a skew-gentle algebra and its gentle quotient via a homological epimorphism, implying that their singularity categories are equivalent. Finite global dimension of the skew-gentle algebra is equivalent to its gentle part having finite global dimension (Chen et al., 2014, Chen, 2022).

The Rouquier dimension of the derived category $D^b(A)$ is at most $1$, and this dimension fully characterizes the representation type and geometric models of derived categories for tame projective curves arising from skew-gentle algebras (Burban et al., 2017).

5. Geometric Classification of Modules and Derived Objects

Indecomposable modules and derived objects in a skew-gentle algebra correspond to geometric entities:

• Indecomposable module categories correspond to permissible tagged arcs and their multisets on the punctured surface models (Deng et al., 16 Nov 2025, He et al., 2020).
• Indecomposable objects in the bounded derived category correspond bijectively to graded curves in the orbifold (open for strings, closed for bands), with gradings determined by local rules at crossings and orbifold points (Labardini-Fragoso et al., 2020, Amiot, 2021).
• The intersection–dimension formula states that the dimension of morphism spaces corresponds to intersection numbers between arcs/curves, generalizing gentle algebra results (Lu et al., 2023, Qiu et al., 2022).

The geometric AR-translation is realized as rotation of tagged permissible curves, leading to a complete dictionary between topological moves on curves and categorical/homological operations (e.g., mutation, $\tau$-tilting, silting mutation) (He et al., 2020, He et al., 2022).

6. Moduli Spaces, Stability, and Cluster-theoretic Aspects

Skew-gentle algebras exhibit especially tractable moduli of representations: every irreducible component of the moduli space of semistable representations is a product of projective spaces, a structure inherited from their gentle and clannish relatives (Gilbert, 2022). The geometry of these moduli spaces is governed by the combinatorics of string and band modules, and their degeneration/closure relations are controlled by the structure of admissible words and tagged arcs.

The exchange graph of support $\tau$-tilting modules coincides with the flip graph of maximal families of tagged permissible arcs on the associated surfaces, is always connected, and forms a combinatorial model for the principal component of the space of Bridgeland stability conditions (He et al., 2022, Lu et al., 2023).

Further, the spaces of stability conditions for the derived categories of graded skew-gentle algebras are connected, and are identified with moduli of quadratic differentials on surfaces/orbifolds, establishing a direct link with meromorphic flat surfaces and wall-crossing phenomena (Lu et al., 2023). These deep connections position skew-gentle algebras at the interface of algebraic, geometric, and topological representation theory.