Graded Skew-Gentle Algebras

• Graded skew-gentle algebras are Z-graded algebras defined by a quiver with special idempotent loops and gentle-type relations, enabling a link between algebra and geometry.
• They use geometric models based on marked surfaces and orbifolds, allowing the translation of combinatorial dissection data into algebraic invariants like Koszulity and derived categories.
• Key insights include the correspondence between graded arcs and indecomposable modules, computations of Hochschild cohomology, and connections to Bridgeland stability via quadratic differentials.

A graded skew-gentle algebra is a class of finite-dimensional $\mathbb{Z}$-graded algebras generalizing the gentle algebras, incorporating special idempotent loops (so-called "special loops") together with gentle-type combinatorics. This class is notable for the intricate interplay between its homological/algebraic properties and geometric models based on marked surfaces or orbifolds. Central invariants include representation theory, Koszulity, derived categories, and connections to Bridgeland stability via quadratic differentials.

1. Definition and Structural Properties

Let $\Bbbk$ be an algebraically closed field. A graded skew-gentle algebra is specified by a triple $(Q,I,g)$:

• $Q = (Q_0, Q_1, s, t)$ is a finite quiver.
• $Q_1 = Q'_1 \sqcup \mathcal{S}$, with $Q'_1$ forming the arrow set of an underlying gentle quiver and $$\mathcal{S} = \{\varepsilon_i \in Q_1 \mid s(\varepsilon_i) = t(\varepsilon_i) = i\}$$ the set of special loops.
• $$I = I' \sqcup \{\varepsilon_i^2 - \varepsilon_i \mid \varepsilon_i \in \mathcal{S}\}$$, with $I' \subset \Bbbk Q'_1$ the set of gentle-type (length-two) relations such that $A' = \Bbbk Q'_1 / \langle I' \rangle$ is gentle and any vertex with $\varepsilon_i$ has at most one extra incoming or outgoing arrow.
• $g:Q_1 \to \mathbb{Z}$ is a grading, extended linearly to paths.

The resulting algebra is $A = \Bbbk Q / \langle I \rangle$, equipped with the grading induced by $g$. If $\mathcal{S} = \varnothing$, $A$ reduces to a graded gentle algebra; in the ungraded case, all $g \equiv 0$, $A$ is a classical skew-gentle algebra (Chang et al., 30 Dec 2025, Qiu et al., 2022, Labardini-Fragoso et al., 2020).

An equivalent description interprets graded skew-gentle algebras as Morita-equivalent to such quotients. The addition of special loops with relations $\varepsilon_i^2 = \varepsilon_i$ encodes orbifold points of order two in the corresponding geometric models (Bian et al., 8 Jan 2026).

Koszulity is a fundamental property: every skew-gentle algebra admits a grading with all arrows in degree 1 and relations in degree 2, and is a strong Koszul algebra. The Koszul dual of a graded skew-gentle algebra is again skew-gentle, and both the algebra and dual can be recovered from corresponding orbifold dissection data (Labardini-Fragoso et al., 2020).

2. Geometric and Surface Models

Graded skew-gentle algebras are exemplars of the deep correspondence between algebraic and topological/combinatorial data via marked surfaces and orbifolds. Two major models are fundamental:

• The punctured marked surface $(S^\lambda, A^*)$: $S$ is an oriented surface with marked points on the boundary and interior punctures corresponding to special loops. A full formal closed arc system $A^*$ (compatible with a grading/line field $\lambda$) cuts $S$ into polygons, encoding the quiver and relations.
• The binary surface $S^\lambda_*$: Each puncture is replaced with a boundary component (binary) carrying one open and one closed mark, with Dehn twist relations $D_{*_P}^2 = \mathrm{id}$. This reformulation facilitates a description of morphism spaces using honest intersections of arcs (Qiu et al., 2022, Lu et al., 2023).

For graded skew-gentle algebra $A$, the data of $(S,O,\eta,\Delta)$—with $O$ a set of orbifold points (order two), $\eta$ a line field (grading), and $\Delta$ a compatible dissection—functions as a geometric model. The combinatorics of polygons and arcs (and, in particular, winding numbers around boundary components and orbifold points) mirror the algebraic data of simple modules, bands, and their gradings (Chang et al., 30 Dec 2025, Qiu et al., 2022).

3. Representation Theory and Classification

The representation theory of graded skew-gentle algebras inherits the rich structure of gentle and skew-gentle algebras, augmented by grading data. Indecomposable modules and objects in the (perfect) derived category $\mathrm{per}A$ are parametrized by "strings" (inextensible finite words/graded arcs) and "bands" (periodic words/graded closed curves):

• There is a bijection between indecomposable objects of $\mathrm{per}A$ and graded arcs or closed curves on the associated (orbifold) surface, modulo appropriate local systems for bands.
• Morphisms between indecomposable objects correspond to (graded, oriented) intersections of arcs—formally, for indecomposables $X_\alpha, X_\beta$, a formula $\mathrm{Int}^k(\alpha, \beta) = \dim_\Bbbk \Hom_{\mathrm{per}A}(X_\alpha,X_\beta[k])$ holds (Qiu et al., 2022, Lu et al., 2023, Chang et al., 30 Dec 2025).
• The Auslander–Reiten quiver for $\mathrm{per}A$ can be encoded combinatorially via the geometry of arc systems, with mesh relations arising from polygons in the surface decomposition (Qiu et al., 2022).

$\tau$-tilting finiteness is fully classified: $A$ is $\tau$-tilting finite if and only if it is representation-finite, i.e., if it admits no band modules (unparametrized families) (Chang et al., 30 Dec 2025).

4. Homological and Derived Invariants

Several derived and homological invariants can be computed combinatorially from the surface/orbifold model:

• Graded Cartan matrix $C_A(q)$ records the graded dimensions of projective-indecomposable compositions, with determinant given by a product over interior polygons: $\det C_A(q) = \prod_P (1 - (-q)^{|P|})$, $|P|$ being the number of edges (Labardini-Fragoso et al., 2020).
• Singularity categories $\mathsf{sg}(A)$ decompose as products of periodic (cluster) categories indexed by polygons in the skew-gentle dissection.
• Gorenstein dimension equals $d$, with $d-1$ the maximal number of internal edges in boundary polygons (Labardini-Fragoso et al., 2020).
• Koszul duality: The Koszul dual $A^!$ is again skew-gentle, interpretable via the dual (ribbon graph) on the orbifold surface (Labardini-Fragoso et al., 2020).

Silting-discreteness for $\mathrm{per}A$ is controlled by the topological type (genus zero), the number of orbifold points ($|O|\leq 1$), and winding numbers: silting-discreteness is equivalent to the absence of simple closed curves with zero winding, i.e., no vanishing-winding bands (Chang et al., 30 Dec 2025).

5. Hochschild Cohomology and Gerstenhaber Structure

The Hochschild cohomology of a graded skew-gentle algebra is computed via a bar or CS-resolution, with explicit basis classes corresponding to paths, circuits, and special loops in the quiver. The graded commutative algebra (under the cup product) and Gerstenhaber bracket structures are as follows:

• The algebra $HH^*(A)$ admits generators corresponding to maximal paths, primitive circuits, derivations (off-tree arrows), and their extensions, with product and bracket structures encoded solely by combinatorics of paths and cycles (Bian et al., 8 Jan 2026).
• The Gerstenhaber bracket is governed by the Lie algebra of derivations acting on other classes; all other brackets vanish.
• Geometric interpretation: generators correspond to distinguished arcs/loops on the orbifold surface, with the cup product as concatenation and bracket as infinitesimal monodromy (Bian et al., 8 Jan 2026).

6. Bridgeland Stability and Quadratic Differentials

There is an identification between spaces of Bridgeland stability conditions on the derived category $D^b(A)$ and moduli of meromorphic quadratic differentials on the associated graded marked surfaces with binaries (GMSb). The link is established by:

• Constructing a bijection $X$ between graded closed arcs (modulo appropriate Dehn twists) on $S^\lambda$ and isomorphism classes of indecomposable arc-objects in $D^b(A)$ (Lu et al., 2023).
• The heart of a stability condition corresponds to an S-graph (a collection of pairwise non-intersecting graded arcs); the exchange graph of such hearts is canonically identified with the exchange graph of S-graphs on $S^\lambda$.
• There exists a holomorphic isomorphism between the principal component of the Bridgeland stability manifold and the moduli space of quadratic differentials: $$\operatorname{Stab}^\circ(D^b(A)) \cong \operatorname{FQuad}^\circ(S^\lambda)$$, with periods of the differential corresponding to the central charge (Lu et al., 2023).

7. Examples and Applications

Explicit examples illustrate these structures:

• For the quiver with two vertices each carrying a special loop and an arrow between them, the corresponding skew-gentle algebra is not $\tau$-tilting finite and its surface model has two orbifold points, so the algebra is not silting-discrete.
• If one orbifold point (special loop) is removed and the grading adjusted accordingly, the model becomes a disk with a single orbifold, and the algebra acquires silting-discreteness (Chang et al., 30 Dec 2025).

Computation of Hochschild cohomology, representation type, and derived invariants is enabled by these models, facilitating the study of cluster theory, Fukaya categories of orbifold surfaces, and stability conditions across algebraic and geometric settings (Bian et al., 8 Jan 2026, Lu et al., 2023, Qiu et al., 2022).

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For further technical details, consult (Chang et al., 30 Dec 2025, Qiu et al., 2022, Labardini-Fragoso et al., 2020, Bian et al., 8 Jan 2026, Lu et al., 2023).