Graded Gentle Algebras: Structure and Topology

• Graded Gentle Algebras are finite-dimensional algebras defined by quivers with relations and an added grading, exhibiting tame representation type and well-structured homological properties.
• They provide a constructive bridge where graded marked surfaces with line fields serve as topological models for their derived categories and partially wrapped Fukaya categories.
• Their homological attributes, such as Koszul duality and derived equivalences, are classified by concrete invariants including boundary winding numbers and AAG-invariants.

Graded gentle algebras are a class of finite-dimensional, typically tame, associative algebras defined by quivers with relations and an additional grading, distinguished by their close connection to surface topology, derived categories, and homological invariants. Their paper lies at the intersection of representation theory, homological algebra, mirror symmetry, and Fukaya categories, where they serve as algebraic models for the partially wrapped Fukaya categories of graded surfaces.

1. Definition, Structure, and Characterization

A graded gentle algebra is a finite-dimensional algebra $A = kQ/I$ where $Q$ is a finite quiver and $I$ is an admissible ideal generated by paths of length two, equipped with a grading by assigning integer degrees to the arrows. The defining conditions on $(Q, I)$ are:

• Each vertex in $Q$ has at most two incoming and at most two outgoing arrows.
• For any arrow $\alpha$, there is at most one arrow $\beta$ such that $\alpha\beta \notin I$, and at most one arrow $\gamma$ such that $\gamma\alpha \notin I$.
• For each arrow $\alpha$, there is at most one arrow $\beta$ with $\alpha\beta \in I$ and at most one $\gamma$ with $\gamma\alpha \in I$.

The grading function $\deg\colon Q_1 \to \mathbb{Z}$ (or sometimes to a finite abelian group $G$) extends to paths by additivity.

These conditions ensure a combinatorial description of modules in terms of strings and bands, and they guarantee a rich but well-behaved homological structure.

2. Surface Models and Geometric Realizations

There is a deep link between graded gentle algebras and surfaces with boundary equipped with line fields (i.e., graded marked surfaces). Every homologically smooth graded gentle algebra $A$ admits a topological model: a triple $(\Sigma_A, \Lambda_A; \eta_A)$, where

• $\Sigma_A$ is an oriented compact surface with boundary,
• $\Lambda_A$ is a finite set of marked points ("stops") on the boundary,
• $\eta_A$ is a line field reflecting the grading data.

The quiver and relations dictating $A$ are encoded combinatorially in a ribbon graph embedded in $\Sigma_A$. The degrees in the grading determine winding numbers of the line field along boundary arcs. The derived category of perfect dg-modules $\operatorname{per}(A)$ is equivalent to the partially wrapped Fukaya category of $(\Sigma_A, \Lambda_A; \eta_A)$ (Lekili et al., 2018, Ikeda et al., 2020).

This geometric realization is constructive: given an admissible dissection of the surface by graded arcs, a graded gentle algebra is associated, and conversely, the algebraic data can be used to reconstruct the surface and line field up to homeomorphism and homotopy.

3. Homological Properties and Koszul Duality

Graded gentle algebras typically exhibit tame representation type and have global dimension at most two. Their (bounded or perfect) derived categories admit a complete classification of indecomposable objects via string and band complexes, extendable to homotopy categories with functorial filtrations (Bennett-Tennenhaus, 2016). Each object decomposes (Krull–Remak–Schmidt–Azumaya) uniquely into such indecomposables.

Koszul duality plays a central role: for a homologically smooth graded gentle algebra $A$, the Koszul functor is realized as a half-rotation in the surface model—relating objects represented by arcs to dual arcs after a $180^\circ$ rotation (Li et al., 22 Mar 2024). The intersection dimension formula

$$\operatorname{Int}^p(\gamma, \nu) = \dim \operatorname{Hom}_{\operatorname{per}A}(X(\gamma), K_A(Y(\nu))[p])$$

connects the combinatorics of geometric intersections to morphism spaces in the derived category.

Homological invariants such as Hochschild cohomology, singularity categories, and Tamarkin–Tsygan calculus are explicit and largely determined by combinatorial data. The Hilbert–Poincaré polynomial of the graded tangent space of $HH^*(A)$ is a derived invariant (Chaparro et al., 2023).

4. Derived Equivalences and Invariants

The surface model and its grading provide a complete derived invariant for graded gentle algebras (Jin et al., 2023, Opper, 13 Oct 2025): two homologically smooth graded gentle algebras are derived equivalent if and only if their associated surfaces with boundary, marked points, and homotopy class of line field are homeomorphic in a grading-preserving manner.

Derived equivalence classes can be characterized by explicit numerical invariants:

• Boundary winding numbers,
• AAG-invariants (Avella-Alaminos–Geiss: pairs $(n(b), n(b) - w(b))$ for each boundary component, with $w(b)$ the total winding along $b$),
• Arf or gcd-type invariants (arising from the mapping class group action on line fields).

These invariants generalize to higher gentle algebras, where a combinatorial (higher-dimensional) structure is imposed on the quiver, and singularity categories reduce to those of idempotent gentle subalgebras (McMahon, 2019).

5. Recollements, Localization, and Pinched Gentle Algebras

Derived categories of graded gentle algebras admit recollement structures through localization at subcategories generated by spherical band objects. This process corresponds, under the surface model, to contracting simple closed curves—yielding new graded pinched gentle algebras associated to marked surfaces with conical singularities (Bodin, 5 Jul 2024). The operation is algebraically realized by inverting certain morphisms (e.g., a linear combination of arrows corresponding to a band object) and geometrically by quotienting out a closed curve on the surface.

The recollement diagram

$$\mathcal{D}(\Lambda[\omega^{-1}]) \to \mathcal{D}(\Lambda) \to \mathcal{D}(K[x]/(x^2))$$

illustrates the decomposition induced by localizing at a spherical band object, reflecting the stratification of the category by geometric/topological features.

6. Grading, Weak Equivalence, and Universal Groups

The grading on a gentle algebra is not uniquely determined by the group ($\mathbb{Z}$ or a finite abelian group); gradings are considered up to weak equivalence (Gordienko et al., 2018). The universal grading group is constructed as the free group on the support modulo relations given by the multiplication of nonzero components. Invariants such as the structure of graded ideals, radical, and polynomial identities are preserved under weak equivalence, and regrading can be used flexibly for classification purposes.

Graded module categories may also be equipped with structures such as graded-monoidal products (twisted by bicharacters) and graded generalizations of Schur's lemma, especially relevant for induced modules over twisted group algebras (Fuchs et al., 15 Mar 2024). In the $\mathbb{Z}/2\mathbb{Z}$-graded case, this recovers the classical supercategory structures.

7. Broader Classes and Homological Dimensions

Higher gentle algebras extend the gentle notion to $d$-cubical combinatorics, leading to Iwanaga–Gorenstein and Cohen–Macaulay properties, as shown by the presence (or absence) of maximal paths in the quiver and relations (McMahon, 2019, Ford et al., 12 Sep 2024). Explicit combinatorial formulas for (graded) global and injective dimensions, center, prime spectrum, and Hilbert series (with Stanley-type palindromicity criteria for AS-Gorenstein cases) allow precise determination of homological properties of graded gentle (and locally gentle) algebras.

The explicit description of injective resolutions, projective resolutions, and Ext groups follows from the combinatorics of maximal forbidden and permitted paths, with the grading controlling the length and duality properties.

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Graded gentle algebras thus serve as a bridge between quiver-based algebraic combinatorics, surface topology, derived category theory, and mirror-symmetric geometry. Their paper provides explicit models for partially wrapped Fukaya categories, a testbed for derived invariants and autoequivalence groups (Opper, 13 Oct 2025), and a versatile class for understanding recollements, silting theory, and Koszul duality from an integrated algebraic and geometric perspective.