Real Weighted Quivers

• Real weighted quivers are directed graphs with real-valued weights that extend integer-based quiver structures to continuous, analytic settings.
• Their mutation dynamics use generalized formulas to preserve skew-symmetry, linking the theory to cluster algebras, reflection groups, and geometric models.
• Applications span representation theory, moduli space construction, and quantum algebra, providing unified tools for advances in modern algebra and geometry.

A real weighted quiver is a quiver—a directed graph structure fundamental in representation theory and algebraic geometry—whose edges (arrows) or vertices are endowed with weights taking real values, and whose mutation and representation theory generalize classical integer-weighted settings to real or even analytic parameter spaces. The concept encompasses quivers with real weights on arrows (or, in some frameworks, vertices), admits both arithmetic and topological interpretations, and plays a critical role in the theory of cluster algebras, categorification, Lie theory, and applications across algebra, geometry, and mathematical physics. Recent advancements relate real weighted quivers to geometric representation theory, moduli problems, dynamical systems, and analytic frameworks for both quantum and classical systems.

1. Formal Definition and Variants

A real weighted quiver consists of a quiver $Q = (Q_0, Q_1)$, where $Q_0$ is a set of vertices and $Q_1$ is a set of arrows, together with a weight assignment $w$ that takes real values. The most prevalent settings include:

• Arrow-weighted: A function $w: Q_1 \to \mathbb{R}$ assigns to each arrow a real number, often interpreted as multiplicities, mass parameters, or lengths in geometry; the quiver’s combinatorics are encoded in a real-valued skew-symmetric (or skew-symmetrizable) matrix $B = (b_{ij})$.
• Vertex-weighted (as in some recurrence and dual number frameworks): $w: Q_0 \to \mathbb{R}$.
• Group-valued generalization: Weights may reside in a group or monoid, with path weights computed multiplicatively (Igusa et al., 2018, Kaul et al., 2020).

A crucial subclass arises when the weights take the form $2\cos(\pi m/d)$ for coprime integers $m, d$—the so-called "trigonometric weights"—linked to the geometry of Coxeter groups and reflection groups (Felikson et al., 2019, Felikson et al., 2016).

Weighted quivers are interpreted as:

• Data structures for certain Lie algebras (Pike, 2014);
• Edge-weighted graphs for network analysis and algebraic topology (Kaul et al., 2020);
• Combinatorial models parameterized by real or continuous parameters for quantum field theories (Kimura et al., 2015).

2. Mutation, Dynamics, and Geometric Models

Mutation for real weighted quivers proceeds via direct generalization of Fomin–Zelevinsky quiver mutation. For a skew-symmetric real matrix $B = (b_{ij})$, the mutated matrix at vertex $k$ is given by: $$b_{ij}' = \begin{cases} -b_{ij} & \text{if } i = k \text{ or } j = k, \ b_{ij} + \frac{|b_{ik}|b_{kj} + b_{ik}|b_{kj}|}{2} & \text{otherwise}. \end{cases}$$ This formula preserves skew-symmetry and real-valuedness. In the theory of mutation-finite real quivers—those whose mutation orbit is finite—extensive classification shows that real weights must be of the form $2\cos(\pi m/d)$, realizing mutation classes via geometric reflection groups, root systems, and acute-angled simplices in suitable quadratic spaces (Felikson et al., 2019, Felikson et al., 2016, Felikson et al., 2019).

In rank‑3, mutation dynamics are governed by an invariant (the Markov constant) $M(a,b,c) = a^2 + b^2 + c^2 - abc$ for cyclic quivers, giving precise criteria for boundedness and finiteness of mutation classes (Casals et al., 22 May 2025). Quiver mutations translate into geometric operations: reflections (in acyclic types), or rotations (in cyclic types) in spaces (spherical, Euclidean, hyperbolic) determined by the signature of the associated quadratic form.

Some generalized mutations include:

• Covering functors and adjoint functors connecting categories of representations for infinite quivers to those of affine Dynkin type (Pike, 2014),
• Weighted quiver mutation rules that permit the tracking of topological or group-theoretic data via arrow or vertex weights (Igusa et al., 2018, Ovsienko et al., 2017).

3. Representation Theory, Categorification, and Semiring Structures

The representation theory of real weighted quivers exhibits diverse phenomena:

• Quiver–Lie algebra correspondence: For certain Lie algebras $L_\mu$ depending on a real or complex parameter $\mu$, modified enveloping algebras can be realized as path algebras of real weighted quivers modulo specific ideals, transferring Lie-theoretic problems into the combinatorial framework of quivers (Pike, 2014).
• Mutation-finite classification: All finite mutation classes with real weights can be geometrized—higher rank cases are rigidly constrained, with maximal denominators bounded by 5; most non-integer finite mutation classes correspond to reflection group geometries (Felikson et al., 2019).
• Categorification via semirings: Weighted unfoldings (and the Chebyshev semiring $R_+$) enable categorification of non-integer quivers: module, derived, and cluster categories over unfolded quivers inherit an $R_+$-coefficient structure, permitting a precise mapping from combinations of objects to "real" roots and generalizing cluster combinatorics to non-crystallographic types (Duffield et al., 2022).
• Homological invariants: Weighted quiver homology groups, defined using functor homology and weight-monoidal structures, capture both the connectivity and the weight amalgamation of weighted directed networks. For acyclic quivers, higher homology vanishes, and $H_1$ can be explicitly computed from the "edge–vertex map" (Kaul et al., 2020).
• Quiver W-algebras: In gauge-theory and quantum algebra, weights on arrows are interpreted as mass parameters; they deform the Cartan matrix, and the partition function data can be lifted to operator-valued currents encoding W-algebra structures. This generalizes representation theory to generalized Borcherds–Kac–Moody algebras and their quantum affinizations, which are sensitive to real weight deformations (Kimura et al., 2015).

Properties such as sign-coherence of c-vectors, categorification of tropical seed patterns, and compatibility of c/g-vectors under unfolding/folding have been extended from the integer to the real-weighted setting (Igusa et al., 2018, Duffield et al., 2022).

4. Moduli Spaces, Algebraic Geometry, and Connections to Geometry

Real weighted quivers play a central role in the construction and paper of moduli spaces:

• Moduli of pointed curves as inverse limits: Moduli spaces such as $\overline{M}_{0,n}$ and Losev–Manin spaces $\overline{L}_n$, as well as Hassett spaces for weighted pointed curves, can be constructed as inverse limits of GIT moduli of quiver representations, where the quivers and dimension/stability data reflect the underlying point and weight data. The combinatorics of weights (possibly real or rational in Hassett spaces) encode the wall-and-chamber structure essential in GIT (Blume et al., 2015).
• Generalized surface algebras and topological realizations: The weighted quivers associated to triangulated (and marked) surfaces underlie the class of generalized triangulation quivers, for which the associated algebras are shown to be finite-dimensional, periodic, and symmetric. These structures arise as the algebraic shadow of surface geometry, and real weights appear naturally in gluing rules and relations (Skowroński et al., 2021).
• Cyclically ordered quivers: Endowing a quiver with a cyclic order on the vertex set creates new invariants under mutation, concretely realized through the unipotent companion matrix, its cosquare, and associated Alexander polynomials. This extra structure enhances the discriminative power of mutation class invariants, admits potential generalization to the real-weighted setting, and connects to singularity theory (Fomin et al., 5 Jun 2024).

The paper of thick subcategories in derived categories of weighted projective curves further relates "real weighted quiver" data—through the Ext-quiver of vertex-like collections—to geometric concepts in the representation theory of algebraic varieties (Elagin, 1 Jul 2024).

5. Analytical and Functional Aspects: Norms and Integrations on Weighted Quivers

Recent developments draw bridges between representation theory, analysis, and functional spaces:

• Normed tensor rings and Banach categories: Path (tensor) algebras of weight quivers can be equipped with norms (parameterized by a basis and weight function), making them normed vector spaces or Banach spaces. Similarly, (A,B)-bimodules can be normed compatibly with actions, allowing analysis-style operations in a categorical context (Liu, 9 Jul 2025).
• Abstract integration and series expansions as morphisms: The category $\mathscr{A}^p_{\varsigma}$ of normed bimodules admits an initial object, whose universal property realizes classical constructions—Lebesgue, Daniell, Bochner integration; power series and Fourier expansions—as unique (A,B)-homomorphisms in the category. This provides a unifying, categorified framework for functional analysis on spaces built from real weighted quivers, establishing analytic concepts as intrinsic algebraic invariants (Liu, 9 Jul 2025).
• Kernel methods and machine learning: Weighted quiver homology is employed to construct invariant graph kernels suitable for machine learning on directed, weighted networks (including real-valued weights). The homological summaries form features distinguishing both network topology and weight amalgamation; practical success is documented in node embeddings and community detection tasks (Kaul et al., 2020).

6. Applications, Broader Implications, and Open Directions

Real weighted quivers permeate several domains:

• Cluster algebras and Lie theory: Their mutation-dynamics underpin the theory of cluster algebras with real coefficients, extendable to non-crystallographic settings (types $H_3$, $H_4$, $I_2(2n+1)$), and play a key role in categorification and non-integer cluster combinatorics (Duffield et al., 2022).
• Quantum field theory and integrable systems: Quiver W-algebras constructed from weighted quivers encode mass deformations in quantum gauge theories, generalizing classical structures to settings involving generalized Kac–Moody algebras and integrable systems (Kimura et al., 2015).
• Algebraic geometry and moduli problems: Interpretation of moduli spaces via quiver representations with real weight parameters provides a unifying language for compactification, wall-crossing, and stability phenomena in both arithmetic and geometric contexts (Blume et al., 2015).
• Algebraic analysis and noncommutative geometry: The normed representation-theoretic frameworks and their initial objects, which encode analytic integration and approximation, offer a template for translating analytic and measure-theoretic constructions to the context of weighted quiver algebras (Liu, 9 Jul 2025).

Real weighted quivers expose a rich dynamic where combinatorics, geometry, algebra, and analysis interact, generating a range of new invariants, categorical tools, and a fruitful landscape for further research in mathematics and mathematical physics. The area continues to develop, with open directions in higher-rank boundedness, the connection between discrete and continuous representation types, and the exploration of analytic structures in categorical and homological settings.