Truncated Jacobian Algebras: Theory & Applications

• Truncated Jacobian algebras are finite-dimensional factors of Jacobian algebras formed via a combinatorial cut that removes selected arrows.
• They underpin 2-representation-finite algebras with global dimension at most 2 and establish links to higher preprojective theory and derived equivalence.
• Applications span geometric models from planar quivers to surface triangulations and Postnikov diagrams, enabling tilting, cut-mutations, and cluster categorification.

A truncated Jacobian algebra arises as a finite-dimensional factor algebra of a Jacobian algebra associated to a quiver with potential (QP), obtained via a combinatorially defined procedure called a “cut.” This notion is central to higher representation theory, especially in constructing and classifying 2-representation-finite algebras, and underpins explicit constructions in the context of quivers, species, surface-like geometries, and categorifications of cluster algebras.

1. Definition and Construction of Truncated Jacobian Algebras

Given a QP $(Q, W)$ (a quiver $Q$ with a potential $W$), a \emph{cut} $C\subset Q_1$ (the set of arrows) is defined so that $W$ is homogeneous of degree 1 with respect to the grading $g_C$ given by $g_C(a)=1$ for $a\in C$, $g_C(a)=0$ for $a\notin C$. The cut induces a $\mathbb{Z}$-grading on the Jacobian algebra $\mathcal{P}(Q, W)$; the \emph{truncated Jacobian algebra} is the degree-zero part: $\mathcal{P}(Q,W)_C = (\mathcal{P}(Q, W))_0,$ which can be equivalently described as the factor algebra of $\mathcal{P}(Q, W)$ by the two-sided ideal generated by all arrows in $C$ (Herschend et al., 2010).

For a species with potential $(S, W)$, the analogous construction considers a cut $C \subset Q_1$ inducing a grading as above; one forms the truncated Jacobian algebra $\mathcal{P}(S, W, C) = \mathcal{P}(S, W)_0$, where only the degree-zero part remains after imposing the relations derived from the cyclic derivatives with respect to the arrows in $C$ (Söderberg, 6 Oct 2025).

The truncation is combinatorial and is determined by the choice of $C$; in particular, if every arrow of $Q$ appears in $W$, and $W$ is homogeneous of degree one for the grading, the procedure yields a well-defined finite-dimensional algebra.

2. Relationship to 2-Representation-Finite Algebras and Higher Preprojective Theory

A central theorem [(Herschend et al., 2010), Theorem 3.11] asserts that the class of basic 2-representation-finite algebras coincides with the class of truncated Jacobian algebras arising from selfinjective QPs. Specifically:

• For any selfinjective QP $(Q, W)$ and cut $C$, the truncated Jacobian algebra $\mathcal{P}(Q, W)_C$ has global dimension at most 2 and is 2-representation-finite.
• Conversely, every basic 2-representation-finite algebra is (up to Morita equivalence) a truncated Jacobian algebra from some selfinjective QP and an appropriate cut.

In the context of species with potentials, this extends to tensor product constructions of species; for example, the truncated Jacobian algebra $\mathcal{P}(S, W, C)$ for an appropriate cut $C$ and selfinjective $(S, W)$ captures new families of 2-representation-finite $\mathbb{K}$-algebras (Söderberg, 6 Oct 2025).

Notably, the 3-preprojective algebra of a truncated Jacobian algebra $A_C$ recovers the original selfinjective Jacobian algebra $\mathcal{P}(Q, W)$ [(Pasquali, 2017), (6)]: $\Pi_3(A_C) \cong \mathcal{P}(Q, W).$ Thus, truncated Jacobian algebras serve as “2-dimensional shadows” of selfinjective Jacobian (or higher preprojective) algebras.

3. Cut-Mutation, 2-APR Tilting, and Derived Equivalence

The process of \emph{cut-mutation} transforms one cut to another by local “flips” at strict sources or sinks in the underlying quiver, defined as changing the membership of arrows incident to the vertex. If $x$ is a strict source of $(Q, C)$, the mutated cut at $x$ is

$$\mu^+_x(C) = \big(C \setminus \{\text{arrows ending at } x\}\big) \cup \{\text{arrows starting at } x\}$$

(Söderberg, 6 Oct 2025). Repeated mutations allow transitions between all admissible cuts under suitable connectivity/homogeneity conditions.

A key structural result is the bijection between cut-mutations and 2-APR tilting moves: for any cut $C$ and strict sink $k$ in the acyclic quiver $Q_C$ underlying the truncated Jacobian algebra, the 2-APR tilt at $k$ of $\mathcal{P}(S, W, C)$ is isomorphic to the truncated Jacobian algebra associated with the mutated cut at $k$: $$\operatorname{End}_{\mathcal{P}(S, W, C)}(T)^{\operatorname{op}} \cong \mathcal{P}(S, W, \mu_k^-(C))$$ (Söderberg, 6 Oct 2025).

Under suitable assumptions (selfinjectivity, simply connectedness, “enough cuts,” and existence of preprojective cuts), the set of all cuts is transitive under cut-mutation, and all corresponding truncated Jacobian algebras are iterated 2-APR tilts of each other and share strong derived and 2-representation-finiteness properties (Söderberg, 6 Oct 2025, Herschend et al., 2010). The mutual derived equivalence strengthens the interconnection of large families of 2-representation-finite algebras—an articulation of the “mutation theory” for these structures.

4. Geometric and Combinatorial Models: Planar and Surface-Related Algebras

Truncated Jacobian algebras are realized in several geometric situations:

• Planar QPs: If a QP is planar (its CW complex "canvas" is simply connected and embeddable in $\mathbb{R}^2$), and has enough cuts, all truncated Jacobian algebras from it are derived equivalent (iterated 2-APR tilts) (Herschend et al., 2010).
• Surface Triangulations: For marked surfaces (with or without boundary), Jacobian algebras associated to triangulations admit truncation to finite-dimensional gentle (or related) algebras. Derived equivalence induced by flips (in partial triangulations) connects Jacobian and Brauer graph structures via the truncated algebra formalism (Demonet, 2016, Demonet, 2017).
• Postnikov Diagrams and Dimer Models: Symmetric Postnikov diagrams produce selfinjective Jacobian algebras. Truncating with respect to suitable cuts, one obtains 2-representation-finite algebras; cut-mutations correspond to 2-APR tilts, preserving derived equivalence and cluster tilting structure (Pasquali, 2017).
• 2-Cyclic and Loop Quivers: Truncated Jacobian algebras with specific potentials on 2-cyclic quivers (with paired arrows and possible loops) yield finite-dimensional algebras whose modules categorify generalized cluster algebras in various orbifold/surface settings, provided a truncation-forcing (“finite-dimensionality”) condition on the potential is satisfied (Li et al., 19 Aug 2024).

5. Structural Properties: Homological Bounds and Truncation Mechanisms

The truncation process—passing to the degree-zero part or factoring by a set of arrows—ensures finite-dimensionality that may not be present in the full Jacobian algebra. In surface and combinatorial settings, explicit path-length bounds are established; for instance, for spheres with $n$ punctures ($n \geq 5$), all paths of length $\geq 2n + 2$ vanish in the algebra (Trepode et al., 2012). Similarly, in 2-cyclic quiver situations, truncation is enforced via relations arising from the cyclic derivatives of the potential, guaranteeing that any sufficiently long path can be expressed as a linear combination of bounded-length paths (Li et al., 19 Aug 2024).

These truncation mechanisms are critical not only for finite-dimensionality but also for preserving (or enforcing) favorable homological properties. In classical settings, the result is global dimension at most 2, symmetry, or periodic projective resolutions, properties closely tied to tameness or finiteness of the module category (Herschend et al., 2010, Ladkani, 2012, Valdivieso-Díaz, 2013).

6. Examples and Applications

A range of explicit examples underlines the utility of truncated Jacobian algebras:

• Tensor Products of Species: Tensor products of finite type species produce selfinjective Jacobian algebras whose various truncations (via cuts) yield new 2-representation-finite algebras. Mutation graphs (lattices) between cuts explicate the set of iterated 2-APR equivalent structures (Söderberg, 6 Oct 2025).
• Categorification and Cluster Algebras: Truncated Jacobian algebras obtained from 2-cyclic quivers with well-chosen potentials provide the module categories for categorifying Paquette-Schiffler generalized cluster algebras for specific disks and spheres with marked or orbifold points (Li et al., 19 Aug 2024).
• Surface Combinatorics: Algebras of partial triangulations specialize to truncated Jacobian algebras for certain choices of arcs and relations, providing a geometric model for symmetric finite rank (and in some cases tame) algebras derived equivalent under flips (Demonet, 2016, Demonet, 2017).

7. Broader Significance and Extensions

Truncated Jacobian algebras mediate between the topological/combinatorial input (quivers, potentials, surface decompositions) and algebraic output (representation theory, cluster categorification, derived invariants). The cut-based truncation furnishes a systematic method for generating, relating, and classifying broad families of 2-representation-finite and derived-equivalent algebras.

By extending to species with potentials, tensor algebras, and settings with additional symmetry or covering structure, the construction delivers new examples outside the reach of classical path algebra methods. The transitivity of cut-mutation under full compatibility and enough cuts yields large, connected landscapes of derived-equivalent algebras, building a bridge between advanced tilting theory (2-APR tilting, higher Auslander correspondence) and explicit algebra construction (Söderberg, 6 Oct 2025, Herschend et al., 2010).

In summary, truncated Jacobian algebras provide a unifying and flexible framework for the construction and analysis of finite-dimensional, homologically controlled algebras arising from quivers or species with potentials, underpinning higher representation theory and categorical cluster phenomena.