Quiver with Relations in Algebra and Geometry

• Quiver with relations is defined as a finite directed graph together with a set of algebraic relations that impose constraints on path compositions.
• This framework supports the study of moduli spaces and representation theory by encoding complex algebraic conditions into combinatorial structures.
• Applications span algebraic geometry, Lie theory, and mathematical physics, offering computational models for descent algebras, Nakayama algebras, and preprojective algebras.

A quiver with relations is a mathematical object formed by a finite directed graph (the quiver) together with a prescribed set of relations, typically specified as a two-sided ideal in the associated path algebra. These relations serve to enforce algebraic constraints—often in the form of equalities among composed paths—that govern the structure of representations and moduli spaces associated to the quiver. The paper of quivers with relations lies at the heart of modern representation theory, algebraic geometry, geometric invariant theory, Lie theory, and mathematical physics, providing a unifying language and framework that connects broad areas via combinatorial, homological, and geometric tools.

1. Definitions and Fundamental Constructions

A quiver $Q$ consists of a set of vertices $Q_0$ and a set of arrows $Q_1$ between these vertices. Representations of $Q$ assign vector spaces (or more general modules or sheaves) to vertices and linear maps to arrows. The path algebra $kQ$ is generated by all possible finite paths in $Q$, with multiplication given by composition when paths are composable and zero otherwise.

A quiver with relations is the pair $(Q, I)$ where $I$ is a two-sided ideal in $kQ$, often generated by specified linear combinations of paths. Relations typically encode requirements such as certain cycles being trivial or particular diagrams commuting. For example, a relation could enforce that the composition of certain arrows equals zero or that two different compositions of paths yield the same morphism.

Key constructions include:

• G-valued representations with relations: Given a Lie group $G$ and a quiver with a set of oriented cycles $R$, one considers markings $f: Q_1 \to G$ such that for every $p\in R$ (a cycle), the condition $f(p) = I$ holds (where $I$ is the identity in $G$). The moduli of such representations modulo the natural gauge group action generalizes the classical character variety $\operatorname{Hom}(\Gamma, G)/G$ of a finitely generated group $\Gamma$ (Florentino et al., 2011).
• Paths with relations in labeled quivers: If arrows are labeled by vector spaces $H$, the path algebra carries a multiplication given by composing labels when the underlying paths concatenate, and relations may equate different compositions that arise in a diagrammatic geometry (Maslovarić et al., 2017).

2. Moduli Spaces and Homological Criteria

The moduli space of representations of a quiver with relations is a central object of paper, providing both algebraic and geometric invariants. For a (possibly twisted or labeled) quiver with relations $(Q,I)$, isomorphism classes of representations correspond to orbits of the action of the gauge group on the space of tuples satisfying the relations.

• GIT Quotients and Character Varieties: For a reductive group $G$, the moduli space $$\mathcal{M}_G(Q, R) := \mathcal{F}_G(Q,R)//\mathcal{G}_G(Q)$$ is the affine GIT quotient; for compact $G$, the analytic orbit space is used. When $Q$ is connected and non-contractible, this moduli space is isomorphic as an algebraic variety to the character variety $\operatorname{Hom}(F_r, G)/G$, with $r = b_1(Q)$ the first Betti number (Florentino et al., 2011).
• Homological Properties and Obstructions: The quasitilted property of an algebra $A=kQ/I$ is characterized by homological dimensions restricted by the structure of the quiver with relations. The presence of bound consecutive relations (specific overlapping structures between relations identified by walks in the underlying quiver) can force the existence of indecomposables with both projective and injective dimension two, obstructing quasitiltedness. Sufficient and necessary conditions for quasitiltedness and their use in the classification of tilted and cluster tilted algebras (e.g., of Dynkin type $E_p$) are given in terms of the absence or controlled overlap of bound consecutive relations (Bordino et al., 2014).

3. Presentations via Generators and Relations

Quivers with relations provide concise presentations of important algebras, encoding both structure and symmetry.

• Descent Algebras of Symmetric Groups: The descent algebra $\Sigma(S_n)$ has a presentation as the path algebra of a quiver $Q_n$ (whose vertices are certain partitions and arrows correspond to restricted composition refinement) modulo an ideal generated by two types of relations: branch relations (commutator-type) and Jacobi-type relations (modeling the Jacobi identity). Computer verification shows these families suffice to generate the kernel for $n \leq 15$. This provides a concrete combinatorial model for the descent algebra (Bishop et al., 2012).
• Nakayama Algebras and Quipu Quivers: Linear Nakayama algebras with almost separate relations (pairwise relation overlaps of at most one arrow) are classified up to derived equivalence by quipu quivers—a class of trees with a main string and cords. An explicit sequence of tilting mutations (CR-swap) relates the presentation of relations in Nakayama algebras to those in quipu quivers, allowing complete classification for small lengths (Fosse, 2023).

4. Quivers with Relations and Lie Theoretic Structures

Interactions between quivers with relations and Lie theory are extensively developed, especially in the context of Cartan matrices and their symmetrizations.

• Symmetrizable Cartan Matrices: The path algebra of a quiver determined by a symmetrizable Cartan matrix $C$ and symmetrizer $D$ is defined by explicit commutation and nilpotency relations at each vertex and along each arrow. The resulting algebra, $H=H(C,D,\Omega)$, is 1-Iwanaga–Gorenstein, and its category of locally free modules models positive roots of the associated Kac–Moody algebra. The generalized preprojective algebra $\Pi(C,D)$ is a tensor algebra over $H$ of a canonical bimodule and encapsulates reflection and Coxeter functor structures (Geiss et al., 2014, Geiß, 2018).
• Semicanonical Bases and Representation Varieties: The constructible function algebra on varieties of locally free modules over these algebras yields, under convolution, a quotient isomorphic to $U(\mathfrak{n}(C))$ (the enveloping algebra for the positive part of the Kac–Moody Lie algebra), with basis elements corresponding to orbits of indecomposable modules and connections to PBW and semicanonical bases (Geiß, 2018).

5. Quiver Grassmannians and Geometric Relations

The geometry of quiver representations with relations also features prominently.

• Quiver Grassmannians and Plücker Relations: The quiver Plücker relations extend the classical determinantal relations for Grassmannians to subrepresentations of quivers. For a fixed dimension vector $e$, the equations

$$E(v, I, J) = \sum_{i\in B_p\setminus I,~j\in J} (-1)^{\epsilon(i,I)+\epsilon(j,J)} m_{v,ji}\Delta_{I\cup\{i\}}\Delta_{J\setminus\{j\}} = 0$$

cut out the quiver Grassmannian as a closed subscheme inside a product of ordinary Grassmannians, encoding both classical and higher-order (path-dependent) relations. All projective schemes arise as quiver Grassmannians, and their paper is critical for moduli questions (Lorscheid et al., 2016).

• Symmetric Polynomials and Invariants: Explicit solutions to the Plücker-type quiver relations identify the path algebra elements as generating symmetric polynomials (e.g., Schur functions) via the Cauchy–Binet and other expansions, bridging representation and algebraic combinatorics (Noshchenko, 2016).
• Tensor Products and Segre Embedding Analogs: The tensor product of quiver representations with relations is defined via a “tensor quiver” and compatible relations, ensuring consistency of path compositions. Under appropriate stability conditions, tensor products of polystable quiver bundles remain polystable, and the construction recovers Segre-type embeddings of products of moduli (or character varieties) as closed subschemes in higher-rank moduli spaces (Numpaque-Roa, 14 Mar 2025).

6. Applications to Moduli, Topology, and Geometry

Quivers with relations serve as the organizing structures for various geometric and physical moduli spaces.

• Moduli Stacks and Noncommutative Geometry: The moduli stack of relations of a quiver parametrizes deformations of the ideal of relations, which in turn correspond—via derived or Morita equivalence—to noncommutative deformations of algebraic varieties (notably, noncommutative projective planes). For quadratic AS-regular algebras of dimension 3, the moduli of relations is realized as an explicit weighted projective plane and corresponds to the moduli of elliptic curves with level 3 structures (Abdelgadir et al., 2014).
• Topology and Retraction: Moduli spaces of $G$-valued quiver representations (and those with relations) admit strong deformation retractions onto their compact analogues by collapsing or pinching arrows and vertices. This provides a topological bridge between complex and compact representation spaces, with implications for Betti number computations and descriptions of homotopy types (Florentino et al., 2011).
• Hilbert Schemes, Quiver Varieties, and Enumerative Geometry: Nested Hilbert schemes on surfaces such as total spaces of line bundles over $\mathbb{P}^1$ (Hirzebruch surfaces) are realized as quiver varieties for enhanced quivers with prescribed relations. This generalizes the ADHM construction and embeds moduli of nested 0-cycles into the context of quiver representation theory with explicit relations—enabling detailed paper of their geometry and applications to moduli of sheaves and instantons (Bruzzo et al., 26 Mar 2024).
• Cluster Algebras, Knot Invariants, and Point Counting: Relations on certain dual quivers associated with plabic graphs encode algebraic-geometric invariants whose point-count polynomials correspond, via conjectural and proven equalities, to the highest-degree coefficients of topological Hilbert–HOMFLY polynomials of associated knots. For classes such as leaf recurrent plabic graphs, this connection is made precise using the recursive structure of the relations (Galashin et al., 2022).

7. Broader Homological and Categorical Frameworks

Advances in homological algebra, model structures, and category theory have further expanded the landscape of quivers with relations.

• Spectral Sequences and Hypercohomology: The category of modules over a twisted quiver algebra with relations admits a spectral sequence relating Ext groups over the quiver algebra to those over the base space, reflecting the added homological depth brought by the relations. In favorable settings, Ext groups are computed as hypercohomology groups of an explicit complex incorporating the relations (Bartocci et al., 2018).
• Model Category Structures: Quivers with relations provide natural examples of self-injective categories amenable to model category structures. Using functorial lifts (as in a generalized Gillespie theorem), one constructs cotorsion pairs and abelian model structures for the derived category, unifying categories such as $N$-complexes, periodic chain complexes, and repetitive quivers with mesh relations under this paradigm (Holm et al., 2019).
• Fusion Quivers: A categorical formalism extends quivers to endofunctors on semisimple categories. Actions of monoidal categories on quivers lead to fusion products—monoidal structures on categories of quiver modules—compatible with a class of relations (including "q-relations") that generalize the classical algebraic and preprojective relations. This provides a framework for modules with duality, trace, and moduli-level graded ring structures, linking to the construction of fusion categories and quantum invariants (Schaumann, 2023).

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In summary, quivers with relations encapsulate a fundamental dialectic between combinatorics and algebra. They govern the structure of algebras, moduli spaces, and representation theories, provide geometric and topological invariants, and serve as the substrate for categorical and homological constructions across algebra, geometry, and mathematical physics. The precise formulation and paper of their relations—whether classical, potential-theoretic, homological, or categorical—unlock deep structural insights and computational tools in a range of contemporary mathematical fields.