Graded Gentle One-Cycle Algebra

• Graded gentle one-cycle algebras are Z-graded, quadratic monomial algebras defined by a unique oriented cycle in a quiver with homogeneous relations, ensuring Koszul properties.
• Their structured grading, Koszul duality, and clock condition enable explicit derived equivalences and triangulated categories linked to geometric models like partially wrapped Fukaya categories.
• The explicit Hall and Lie algebra presentations, coupled with derived invariants such as the Avella–Alaminos–Geiss invariant, facilitate classification and a deeper understanding of their homological and geometric interplay.

A graded gentle one-cycle algebra is a finite-dimensional, $\mathbb{Z}$-graded algebra defined via a quiver with relations such that (1) its underlying gentle algebra possesses a unique oriented cycle, and (2) all relations are homogeneous with respect to the grading. These algebras are quadratic monomial (hence Koszul), arise naturally in the geometric representation theory of surfaces (notably as algebraic models for partially wrapped Fukaya categories of annuli or punctured surfaces), and play a central role in the study of derived categories, Hall algebras, Lie theory, and homological invariants. The interplay of their graded, combinatorial, homological, and geometric features is highly structured and underpins recent advances in categorical, Lie-theoretic, and algebraic frameworks.

1. Definitions and Combinatorial Models

A graded gentle one-cycle algebra $A$ is specified by a quiver $Q$ with a single oriented cycle and a homogeneous ideal $I$ generated by paths of length two, subject to the gentle conditions: every vertex has at most two incoming and two outgoing arrows; for each arrow, there is at most one relation starting or ending at that arrow; each arrow lies in at most one maximal path; and the ideal $I$ is generated by monomial relations. The grading is typically the path-length grading, turning $A$ into a Koszul algebra. The presence of a unique cycle ensures a one-to-one correspondence with combinatorial objects called quasi-diagrams (Editor’s term), which are involutions in the symmetric group $S_n$ representing gluing data and isolated points on an $n$-gon.

A quasi-diagram $a \in S_n$ associated to $A$ encodes the chordal structure of the quiver: 2-cycles correspond to glued chord pairs, and fixed points to isolated vertices. The set of such quasi-diagrams is acted on by the dihedral group $D_n$ (rotations and reflections of the polygon). The algebra $A$ is said to have finite global dimension if and only if all orbits of $a$ under the rotation contain either a designated vertex (e.g., $1$) or an isolated point; this is called regularity of the quasi-diagram (Hu et al., 2024). This combinatorial data directly determines key homological invariants, including the global dimension (see Section 3 below).

2. Grading, Koszul Duality, and Clock Condition

Graded gentle algebras are intrinsically Koszul, and their dual $A^!$ is explicitly constructed by reversing arrows and orthogonally dualizing the relations. Koszul duality acquires a geometric manifestation in recent work: in the geometric model, the Koszul functor corresponds to a half rotation of the underlying arc system on a marked surface. That is, for a graded closed arc $\tilde{\gamma}$ on a surface associated to $A$, the functor $K_A$ maps the indecomposable object associated to $\tilde{\gamma}$ to one associated to its half-rotated image, realizing $K_A$ as a geometric operation (Li et al., 2024). The Koszul dual $A^!$ exists precisely when the quasi-diagram is of maximal type, i.e., when gldim $A = n-1$ (Hu et al., 2024).

The clock condition governs when the bounded derived category orbit category (modulo a power of the shift functor) admits a triangulated structure: for a quadratic monomial algebra, every oriented cycle in the quiver must have an equal number of clockwise and counterclockwise relations. In the one-cycle gentle case, failure of the clock condition is tantamount to failing to be piecewise hereditary, and corresponds to the existence of non-gradable (i.e., non-periodic) differential modules. Explicitly, this precludes the triangulated structure on the orbit category unless the algebra is piecewise hereditary (Stai, 2016).

3. Derived Categories and Orbit Constructions

The perfect derived category $\operatorname{per}(A)$ and finite-dimensional derived category $D_{fd}(A)$ of a graded gentle one-cycle algebra have an explicit description via twisted root categories, i.e., triangulated orbit categories of the homotopy/homological categories of graded module categories modulo a combination of grading and complex-shift functors. The essential construction begins with the category of graded modules, passes to complexes, and takes the orbit category with respect to the autoequivalence $[1] \circ (-1)$ (complex-shift combined with degree-shift) (Kalck et al., 2016, Cheng et al., 17 Oct 2025). The Tot functor identifies complexes whose cohomological and grading shifts combine to the same effect, and this factorization ultimately recovers the triangulated structure in $\operatorname{per}(A)$ as the hull of the orbit category.

For algebras of finite global dimension, $\operatorname{per}(A)$ is triangle equivalent to a twisted root category of representations of a quiver of type $\widetilde{\mathbb{A}}_{p,q}$ (with $p$ and $q$ determined by combinatorial invariants of $A$). In the infinite global dimension case, $D_{fd}(A)$ is triangle equivalent to the root category of an infinite quiver of type $\mathbb{A}_\infty^\infty$ (or similar linear/cyclic type), with the Koszul duality relating the perfect and finite-dimensional derived categories of $A$ and $A^!$.

The triangle structure of these derived categories is shown to be uniquely determined by their underlying additive category (Cheng et al., 17 Oct 2025). Thus, any $k$-linear equivalence between the additive categories underlying two such twisted root categories lifts functorially and uniquely to a triangle equivalence between the derived categories.

4. Hall Algebras, Lie Theory, and Invariants

The derived Hall algebra and Ringel–Hall algebra associated to a graded gentle one-cycle algebra admit explicit presentations in terms of generators (indexed by indecomposable (graded) objects or arcs and their shifts) and explicit relations reflecting the combinatorics and gradings. In particular, for infinite global dimension cases, the derived Hall algebra is generated by elements $x_i^{(k)}$, $z_i^{(k)}$ subject to relations encoding graded dimensions, by analogy with the Poincaré–Birkhoff–Witt type bases (Bobinski et al., 2019, Cheng et al., 17 Oct 2025).

The Lie-theoretic structures arising are closely linked to the module category: for $A = \Lambda(n-1,1,1)$, the Riedtmann Lie algebra $L(A)$ constructed from isomorphism classes of indecomposables and their Euler characteristics is isomorphic to a Lie algebra with a Cartan decomposition indexed by the positive roots of type $BC_n$ (Chen et al., 2022). The Ringel–Hall algebra at $q=1$ recovers $L(A)$, and explicit Hall polynomials exist for all triples of indecomposables, a rare phenomenon outside Dynkin types (Chen et al., 2024).

The (graded) structure of the Hochschild cohomology ring $\mathrm{HH}^*(A)$ is also fully combinatorial, with the top of its radical $$T(A) = \operatorname{rad} \mathrm{HH}^*(A)/\operatorname{rad}^2 \mathrm{HH}^*(A)$$ and its Hilbert–Poincaré polynomial $h_{T(A)}(t)$ yielding derived invariants determined by the ribbon-graph data of $A$ (Chaparro et al., 2023).

5. Geometric and Fukaya-Categorical Realizations

A central theme is the equivalence between $\operatorname{per}(A)$ for graded gentle one-cycle algebras and partially wrapped Fukaya categories of graded marked surfaces, notably annuli or surfaces with a single cycle. The correspondence is constructed via a string model in which indecomposable complexes are modeled by graded arcs on the surface, their intersections encoding the Ext-structure (Ikeda et al., 2020, Chang et al., 2020). The Koszul functor appears as a half-rotation of the geometric arc model, and duality manifests naturally as rotation of the arcs (Li et al., 2024).

Surface-geometric moves, such as the mutation of arcs (flips in the dissection) correspond bijectively to silting (or tilting) mutations and octahedral axiom configurations in the derived category (Chang et al., 2020). Localization by spherical band objects (supported on cycles, or "bands" in the annulus) gives rise to recollement structures: localizing at a band object removes the contribution of the associated simple closed curve on the surface, producing a new category modeled algebraically by a graded pinched gentle algebra and geometrically by a surface with a conical singularity (the contraction of the closed curve) (Bodin, 2024). The bijection between graded pinched gentle algebras and graded marked surfaces with conical singularities generalizes the classical correspondence.

6. Full Derived Invariants and Classification

The derived equivalence (triangle equivalence classes) of graded gentle one-cycle algebras is completely controlled by geometric invariants associated to the surface model: the surface topology, marked points (stops), the winding numbers of the grading line field about boundary components, the genus, and the Arf invariant (in toroidal cases). Two such algebras have triangle equivalent perfect derived categories if and only if their surfaces (with line field data) are homeomorphic and the winding number data matches (Jin et al., 2023, Opper, 13 Oct 2025). The Avella–Alaminos–Geiss invariant, recording the distribution of boundary components and winding numbers, provides a numerical refinement for classification.

The derived Picard group (autoequivalences of the derived category) is explicitly described as a semi-direct product of "unipotent" subgroups (arising from the additive group $\mathbb{G}_a$ and invertible local systems associated to the surface) and the graded mapping class group of the marked surface. When the base field has characteristic zero, the unipotent part is realized as the image of an exponential map from Hochschild cohomology to the derived Picard group; in positive characteristic, deformation theory and formality results are used (Opper, 13 Oct 2025).

7. Invariance and Relationships with Other Constructions

Representation-finiteness and derived-discreteness (rep-finite/discrete categories) are invariant under key algebraic constructions such as BD-algebras and CM-Auslander algebras. That is, for a gentle algebra $A$, the properties of being rep-finite or derived-discrete are shared by its BD-gentle algebra and CM-Auslander algebra, and this extends to the graded gentle one-cycle setting (Zhang et al., 30 Apr 2025). This robustness is a consequence of the persistence of (graded) string and (homotopy) band combinatorics under these functorial operations.

• (Stai, 2016) Differential modules over quadratic monomial algebras
• (Kalck et al., 2016) Derived categories of graded gentle one-cycle algebras
• (Bobinski et al., 2019) Derived Hall algebras of one-cycle gentle algebras
• (Ikeda et al., 2020) Graded decorated marked surfaces: Calabi-Yau-$\mathbb{X}$ categories of gentle algebras
• (Chang et al., 2020) A geometric realization of silting theory for gentle algebras
• (Chen et al., 2022) On Riedtmann's Lie algebra of the gentle one-cycle algebra $\Lambda(n-1,1,1)$
• (Jin et al., 2023) A complete derived invariant and silting theory for graded gentle algebras
• (Chaparro et al., 2023) The Hochschild cohomology and the Tamarkin-Tsygan calculus of gentle algebras
• (Chen et al., 2024) On the Ringel–Hall algebra of the gentle one-cycle algebra $\Lambda(n-1,1,1)$
• (Hu et al., 2024) Quasi-diagrams and gentle algebras
• (Li et al., 2024) A geometric realization of Koszul duality for graded gentle algebras
• (Bodin, 2024) Recollements for graded gentle algebras from spherical band objects
• (Zhang et al., 30 Apr 2025) On BD-algebra and CM-Auslander algebra for a gentle algebra and their representation types
• (Opper, 13 Oct 2025) Autoequivalences of Fukaya categories of surfaces and graded gentle algebras
• (Cheng et al., 17 Oct 2025) The derived Hall algebra of a graded gentle one-cycle algebra I: the triangle structure

Table: Key Features of Graded Gentle One-Cycle Algebras

Feature	Description	Reference
Underlying combinatorics	Quiver with one oriented cycle, quasi-diagram encoding chords and isolated points	(Hu et al., 2024)
Homological structure	Koszul, quadratic monomial; graded structure mirrors path-length	(Kalck et al., 2016Li et al., 2024)
Derived category construction	Twisted root/orbit categories, triangle structure determined by additive category	(Cheng et al., 17 Oct 2025)
Hall/Lie algebras	Explicit generators and relations, Riedtmann and Ringel–Hall algebra isomorphism	(Chen et al., 2024Chen et al., 2022)
Surface/geometry model	Correspondence with graded marked surfaces; arcs model indecomposables, surface mutations ↔ category mutations	(Ikeda et al., 2020Chang et al., 2020)
Derived invariants	Classified by surface topology, winding numbers, line field, combinatorics	(Jin et al., 2023Opper, 13 Oct 2025)