Quiver Algebras

• Quiver algebras are finite-dimensional associative algebras formed from directed graphs (quivers) via path algebras modulo relations.
• They play a central role in representation theory by encoding modules, symmetries, and combinatorial data of algebraic structures.
• Modern developments include quantum, cellular, and categorified versions that underpin research in mathematical physics and higher category theory.

A quiver algebra is a (usually finite-dimensional) associative algebra constructed from a quiver—a directed graph whose vertices and arrows encode algebra generators and relations—or more generally, a path algebra of a quiver modulo relations. Quiver algebras appear throughout representation theory, geometry, homological algebra, mathematical physics, and encode the data of categories, symmetries, and moduli in highly structured ways.

1. Path Algebras and Quiver Algebras

Given a quiver $Q$ with set of vertices $Q_0$ and set of arrows $Q_1$, the path algebra $kQ$ over a field $k$ is the $k$-vector space with basis given by all paths in $Q$ (including stationary paths at each vertex), with multiplication induced by concatenation of paths (defined to be zero if the tail of the second does not coincide with the head of the first). Quiver algebras are typically defined as quotients $kQ/I$ by an admissible ideal $I$, encoding nontrivial relations between paths.

If $Q$ is finite and admits no oriented cycles, $kQ$ is finite-dimensional and hereditary; $kQ/I$ can model more general classes, such as those with radical or finite global dimension. Important subclasses include:

• Basic algebras: Any finite-dimensional algebra is Morita equivalent to a basic algebra, which arises naturally as a quotient of a path algebra by relations.
• Preprojective and Heisenberg quiver algebras: These are central extensions of the path algebra by relations modeling Auslander–Reiten or other duality properties, and have connections to categorification and dimension formulas (Herschend et al., 13 Feb 2024).

There is a parallel theory involving path coalgebras, the duals of path algebras, which are coassociative coalgebras with multiplication replaced by co-multiplication—deconcatenating a path into pairs of subpaths. Under certain “locally finite” conditions (no cycles and only finitely many arrows between vertices), the path coalgebra coincides with the finite dual of the path algebra, and the correspondence is fully reversible (Dascalescu et al., 2012).

2. Quiver Algebras and Representation Theory

The paper of modules over quiver algebras (i.e., quiver representations) is central in the structure theory of associative algebras:

• Gabriel's Theorem: If $A$ is a finite-dimensional basic algebra over an algebraically closed field, then $A$ is isomorphic to $kQ/I$ for some quiver $Q$ and admissible ideal $I$; the irreducible modules correspond to the vertices (Thompson, 25 Apr 2024).
• The computation and structure of the quiver of a monoid algebra $kM$ relates to the extension groups $\operatorname{Ext}^1_{kM}(U,V)$, which can be algorithmically determined by analyzing derivations modulo inner derivations, the “support lattice” of the monoid, and the representation theory of its maximal subgroups (Margolis et al., 2011):

$$\operatorname{Ext}^1_{kM}(U,V) \cong \frac{\operatorname{Der}(M,\operatorname{Hom}_k(U,V))}{\operatorname{IDer}(M,\operatorname{Hom}_k(U,V))}$$

• Categories of modules: Representations of $kQ$ correspond to collections of vector spaces (one per vertex) and linear maps (one per arrow). Projective and injective representations, indecomposability, and homological invariants are naturally controlled by the quiver and the relations.

A more detailed combinatorial invariant is the full quiver, which refines the classical quiver of a finite-dimensional algebra by recording, in a chosen Wedderburn block realization, all gluing between blocks—via scalar or Frobenius (field automorphism) identifications—of the matrix components, both for vertices and arrows (Belov-Kanel et al., 2011). This data is intrinsic: every representable algebra admits a faithful representation whose structure is completely described by such a full quiver.

3. Quiver Algebras in Geometry and Moduli Spaces

Quiver algebras appear as coordinate rings or convolution algebras on spaces of representations of quivers (quiver varieties), in the geometry of moduli spaces:

• Nakajima quiver varieties (with multiplicities): The construction attaches to a quiver $(Q,d)$, with a positive integer multiplicity $d_i$ at each vertex, a symplectic quotient manifold, generalizing the usual quiver variety. The representation space is upgraded to include “$d$-valued” data:

$$(Q,d)(V) = \bigoplus_{h\in H} \operatorname{Hom}(V_{s(h)}\otimes R_{d_{s(h)}}, V_{t(h)}\otimes R_{d_{t(h)}})$$

where $R_{d_i} = \mathbb{C}[[z]]/z^{d_i}\mathbb{C}[[z]]$, and the moment map depends on parameters $\lambda = (\lambda_i(z))$. When all $d_i=1$, one recovers classical quiver varieties (Yamakawa, 2010).

• Reflection functors generalize simple reflections in the Weyl group of possibly non-symmetric Kac-Moody algebras and are realized as symplectomorphisms on these quiver varieties:

$$F_i : (Q,d)(\lambda, v) \to (Q,d)(r_i(\lambda), s_i(v))$$

where $r_i$ and $s_i$ act on parameters and dimension data. In the case $d_i=1$, this is Nakajima’s original functor.

• Moduli of meromorphic connections: The varieties $(Q,d)(\lambda,v)$ classify moduli spaces of meromorphic connections with fixed formal types on a Riemann sphere, with multiplicities $d_i$ recording pole orders.
• Painlevé equations: The dimension-2 quiver varieties (for star-shaped quivers with appropriate multiplicities) are shown to be (after normalization) moduli spaces whose isomonodromic deformations yield the classical Painlevé equations ($\mathrm{P}_{\mathrm{VI}}$, ..., $\mathrm{P}_{\mathrm{II}}$), with affine Weyl group symmetries realized geometrically by reflection functors.
• Relation to convolution algebras and Yangians: On quiver varieties, one constructs convolution algebras $H_*(Z)$, and, via convolution with a class $c$, defines coproduct-type maps:

$\Delta_c(b) = c^{-1} * b * c$

which geometrically realize algebraic (Drinfeld) coproducts on associated Yangians (Nakajima, 2012).

4. Koszul and Cellular Quiver Algebras; Categorification

Quiver algebras and their graded versions (e.g., KLR (quiver Hecke) algebras and Schur algebras) play a central role in the categorical and combinatorial approach to Lie theory:

• Quiver Hecke/KLR algebras: Arising from the categorification of quantum groups, these are often realized as quiver algebras with additional structure and grading. For instance, cyclotomic Yokonuma–Hecke algebras are shown to be isomorphic to direct sums of cyclotomic quiver Hecke algebras for disjoint cyclic quivers (Rostam, 2016).
• Graded cellular algebras, Koszul duality: The quiver Schur algebras attached to type $A$ quivers are shown to be quasi-hereditary, graded, and Koszul in characteristic zero. The explicit grading matches the KLR and Koszul gradings, and decomposition numbers are given by (parabolic) Kazhdan–Lusztig polynomials (Hu et al., 2011).
• Generalized quiver Hecke algebras: Using the convolution algebra associated with Steinberg varieties, one produces a broad class of algebras encompassing affine nil Hecke algebras, skew group rings, and quiver Hecke algebras, with explicit generators and relations computable via the geometry of flag varieties and divided-difference operators (Sauter, 2013).

5. Modern Developments: Quiver Quantum Algebras, BPS Algebras, and Geometric Representation Theory

Recent advances reveal quiver algebras as the algebraic backbone in physical and geometric models:

• Quiver Yangians, BPS and double quiver algebras: Infinite-dimensional algebras, such as quiver Yangians, quantum toroidal algebras, and their double versions (indexed by quivers with or without potential), underpin BPS state counting in supersymmetric gauge theories and moduli spaces of sheaves on toric Calabi–Yau backgrounds. These algebras admit crystal melting representations, encode the spectrum of BPS (Bogomolnyi–Prasad–Sommerfield) states, and are constructed directly from the combinatorics of the Jeffrey–Kirwan residue formula (Noshita et al., 2021, Galakhov et al., 2021, Li, 2023, Bao et al., 6 Jan 2025).
• The double quiver algebra retains full one-loop determinant information (including all poles) crucial for capturing refined BPS indices and representations corresponding to moduli of supersymmetric vacua.
• Quiver W-algebras and fractional generalizations: In the context of gauge theory and quantum integrable systems, quiver W-algebras generalize deformations of the $W$-algebras to arbitrary quivers (not just simply-laced). The associated current algebras are constructed as commutants of screening operators in a Heisenberg/Fock space and encode generalized (quantum, affine, or even hyperbolic) Lie algebra symmetries. Fractional quiver W-algebras allow non-uniform root length (by assigning "fractional" dimensions or rotation charges to vertices), producing $q$-deformations of non-simply-laced and twisted affine algebras (Kimura et al., 2015, Kimura et al., 2017).

6. Quiver Algebras in Higher Category Theory, Quantum Symmetries, and Further Extensions

Combinatorial data of quivers and associated algebras admits natural extensions in higher category theory:

• Bicategory frameworks relate bounded quivers (with path algebras modulo ideals) and quiver connections—functorially associating vector space data with combinatorial edge correspondences—to basic finite-dimensional algebras and their bimodules, yielding a 2-categorical refinement of Gabriel's theorem (Thompson, 25 Apr 2024).
• This structural upgrade is motivated by the desire to classify quantum symmetries and fusion category actions on non-semisimple algebras, and to extend the combinatorial quiver algebra paradigm to broader categories with fusion or modular data.

A principal role is played by quiver data (splittings of projections to semisimple and radical layers) in reconstructing both the quiver (vertices and edges) and in lifting actions and bimodules to the algebraic side. The equivalence of categories is driven by explicit combinatorial isomorphisms, and is foundational for understanding quantum symmetries and categorical extensions in algebraic settings.

7. Fundamental Formulas and Structures

Key formulas and structures reflecting the theory include:

• The correlation between extension groups and derivations in monoid/quiver algebras:

$$\operatorname{Ext}^1_{kM}(U,V) \cong \frac{\operatorname{Der}(M,\operatorname{Hom}_k(U,V))}{\operatorname{IDer}(M,\operatorname{Hom}_k(U,V))}$$

• The identification of path coalgebras with the finite dual of the path algebra under finiteness and acyclicity conditions:

$KT \cong K[T]^\circ$

• Bond factors, defining relations, and current algebra structures in quantum and toroidal quiver algebras:

$$\varphi^{(i \Rightarrow j)}(z, w) = \prod_{I \in \{j \to i\}} \phi(q_I; z, w) / \prod_{I \in \{i \to j\}} \phi(q_I^{-1}; z, w)$$

$$E_i(z) E_j(w) = (-1)^{|i||j|} \varphi^{(j \Rightarrow i)}(z, w) E_j(w) E_i(z)$$

In all cases, the rich algebraic and combinatorial structure of quiver algebras, their generalizations, and their representations is central to modern developments in representation theory, geometry, category theory, mathematical physics, and quantum algebra.