Kronecker Quiver Research Articles

Moduli of representations of one-point extensions

We study moduli spaces of (semi-)stable representations of one-point extensions of quivers by rigid representations. This class of moduli spaces unifies Grassmannians of subrepresentations of rigid representations and moduli spaces of representations of generalized Kronecker quivers. With homological methods, we find numerical criteria for non-emptiness and results on basic geometric properties, construct generating semi-invariants, expand the Gel’fand MacPherson correspondence, and derive a formula for the Poincaré polynomial in singular cohomology of these moduli spaces.

Noncommutative Riemannian geometry of Kronecker algebras

We study aspects of noncommutative Riemannian geometry of the path algebra arising from the Kronecker quiver with N arrows. To start with, the framework of derivation based differential calculi is recalled together with a discussion on metrics and bimodule connections compatible with the ⁎-structure of the algebra. As an illustration, these concepts are applied to the noncommutative torus where examples of torsion free and metric (Levi-Civita) connections are given.In the main part of the paper, noncommutative geometric aspects of (generalized) Kronecker algebras are considered. The structure of derivations and differential calculi is explored, and torsion free bimodule connections are studied together with their compatibility with hermitian forms, playing the role of metrics on the module of differential forms. Moreover, for several different choices of Lie algebras of derivations, non-trivial Levi-Civita connections are constructed.

Remarks on 2d unframed quiver gauge theories

We study 2d mathcal{N} = (2, 2) quiver gauge theories without flavor nodes. There is a special class of quivers whose gauge group ranks stay positive in any duality frame. We illustrate this with the Abelian Kronecker quiver and the Abelian Markov quiver as the simplest examples. In the geometric phase, they engineer an infinite sequence of projective spaces and hypersurfaces in Calabi-Yau spaces, respectively. We show that the Markov quiver provides an Abelianization of SU(3) SQCD. Turning on the FI parameters and the θ angles for the Abelian quiver effectively deform SQCD by such parameters. For an Abelian necklace quiver corresponding to SU(k) SQCD, we find evidence for singular loci supporting non-compact Coulomb branches in the Kähler moduli space.

Dimension vectors with the equal kernels property

Let r∈N, Kr be the generalized Kronecker quiver with r arrows γ1,…,γr:1→2 and δ∈Δ+(Kr) be a positive root of Kr. We say that δ has the equal kernels property if for all α∈kr∖{0} and every indecomposable representation M with dimension vector dim_M=δ the k-linear map Mα:=∑i=1rαiM(γi):M1→M2 is injective. We show that δ has the equal kernels property if and only if qKr(δ)+δ2−δ1≥1, where qKr:Z2→Z,(x,y)↦x2+y2−rxy denotes the Tits quadratic form of Kr.

Dimension vectors of elementary modules of generalized Kronecker quivers

Let k be an algebraically closed field. The generalized or n-Kronecker quiver K(n) is the quiver with two vertices, called a source and a sink, and n arrows from source to sink. Given a finite-dimensional module M of the path algebra , we consider its dimension vector . Let , and let . We construct a module X(x, y) of , and we prove it to be elementary. Suppose that . We show that: if M is an elementary module, then , and when , the module M is elementary if and only if M is of the form X(x, y).

Edmonds' problem and the membership problem for orbit semigroups of quiver representations

A central problem in algebraic complexity, posed by J. Edmonds [16] , asks to decide if the span of a given l -tuple W = ( W 1 , … , W l ) of N × N complex matrices contains a non-singular matrix. In this paper, we provide a quiver invariant theoretic approach to this problem. Viewing W as a representation of the l -Kronecker quiver K l , Edmonds' problem can be rephrased as asking to decide if there exists a semi-invariant on the representation space ( C N × N ) l of weight ( 1 , − 1 ) that does not vanish at W . In other words, Edmonds' problem is asking to decide if the weight ( 1 , − 1 ) belongs to the orbit semigroup of W . Let Q be an arbitrary acyclic quiver and W a representation of Q . We study the membership problem for the orbit semi-group of W by focusing on the so-called W -saturated weights. We first show that for any given W -saturated weight σ , checking if σ belongs to the orbit semigroup of W can be done in deterministic polynomial time. Next, let ( Q , R ) be an acyclic bound quiver with bound quiver algebra A = K Q / 〈 R 〉 and assume that W satisfies the relations in R . We show that if A / Ann A ( W ) is a tame algebra then any weight σ in the weight semigroup of W is W -saturated. Our results provide a systematic way of producing families of tuples of matrices for which Edmonds' problem can be solved effectively.

𝚤Hall algebra of the projective line and 𝑞-Onsager algebra

Theı\imathHall algebra of the projective line is by definition the twisted semi-derived Ringel-Hall algebra of the category of11-periodic complexes of coherent sheaves on the projective line. Thisı\imathHall algebra is shown to realize the universalqq-Onsager algebra (i.e.,ı\imathquantum group of split affineA1A_1type) in its Drinfeld type presentation. Theı\imathHall algebra of the Kronecker quiver was known earlier to realize the same algebra in its Serre type presentation. We then establish a derived equivalence which induces an isomorphism of these twoı\imathHall algebras, explaining the isomorphism of theqq-Onsager algebra under the two presentations.

Curvature of the space of stability conditions

Motivated by the study of the autoequivalence group of triangulated categories via isometric actions on metric spaces, we consider curvature properties (CAT(0), Gromov hyperbolic) of the space of Bridgeland stability conditions with the canonical metric defined by Bridgeland. We then prove that the metric is neither CAT(0) nor hyperbolic, and the quotient metric by the natural $\mathbb{C}$-action is not CAT(0) in case of the Kronecker quiver. Moreover, we also show the hyperbolicity of pseudo-Anosov functors defined by Dimitrov-Haiden-Katzarkov-Kontsevich, which yields the lower-bound of entropy by the translation length. Finally, pseudo-Anosov functors in case of curves have been completely classified.

KLR and Schur Algebras for Curves and Semi-Cuspidal Representations

Abstract Given a smooth curve $C$, we define and study analogues of KLR algebras and quiver Schur algebras, where quiver representations are replaced by torsion sheaves on $C$. In particular, they provide a geometric realization for certain affinized symmetric algebras. When $C={\mathbb{P}}^1$, a version of curve Schur algebra turns out to be Morita equivalent to the imaginary semi-cuspidal category of the Kronecker quiver in any characteristic. As a consequence, we argue that one should not expect to have a reasonable theory of parity sheaves for affine quivers.

A Frobenius manifold for $$\ell $$-Kronecker quiver

We construct a Frobenius structure whose intersection form coincides with the generalized Cartan matrix of the \(\ell \)-Kronecker quiver \(K_{\ell }\), and underlying complex manifold is isomorphic to the space of stability conditions for the bounded derived category of finitely generated modules over the path algebra \({{\mathbb {C}}}K_{\ell }\).

Noncommutative Counting Invariants and Curve Complexes

Abstract In our previous paper, viewing $D^b(K(l))$ as a noncommutative curve, where $K(l)$ is the Kronecker quiver with $l$-arrows, we introduced categorical invariants via counting of noncommutative curves. Roughly, these invariants are sets of subcategories in a given category and their quotients. The noncommutative curve-counting invariants are obtained by restricting the subcategories to be equivalent to $D^b(K(l))$. The general definition, however, defines a larger class of invariants and many of them behave properly with respect to fully faithful functors. Here, after recalling the definition, we focus on the examples and extend our studies beyond counting. We enrich our invariants with the following structures: the inclusion of subcategories makes them partially ordered sets and considering semi-orthogonal pairs of subcategories as edges amounts to directed graphs. It turns out that the problem for counting $D^b(A_k)$ in $D^b(A_n)$ has a geometric combinatorial parallel - counting of maps between polygons. Estimating the numbers counting noncommutative curves in $D^b({\mathbb P}^2)$ modulo the group of autoequivalences, we prove finiteness and that the exact determining of these numbers leads to a solution of Markov problem. Via homological mirror symmetry, this gives a new approach to this problem. Regarding the structure of a partially ordered set mentioned above, we initiate intersection theory of noncommutative curves focusing on the case of noncommutative genus zero. The above-mentioned structure of a directed graph (and related simplicial complex) is a categorical analogue of the classical curve complex, introduced by Harvey and Harrer. The paper contains pictures of the graphs in many examples and also presents an approach to Markov conjecture via counting of subgraphs in a graph associated with $D^b({{\mathbb{P}}}^2)$. Some of the results proved here were announced in a previous work.

Moduli spaces of Ulrich bundles on the Fano 3-fold V5

We study moduli spaces of Ulrich bundles of rank r≥2 on the Fano 3-fold V5 of Picard number 1, degree 5 and index 2. We prove that the moduli space of stable Ulrich bundles of rank r on V5 can be identified with a smooth (r2+1)-dimensional open subset of the moduli space of stable quiver representations with dimension vector (r,r) of the Kronecker quiver with 2 vertices and 3 arrows.

Indecomposable Jordan types of Loewy length 2

Let k be an algebraically closed field, char(k)=p≥2 and Er be a elementary abelian p-group of rank r≥2. Let (c,d)∈N2. We show that there exists an indecomposable module of constant Jordan type [1]c[2]d and Loewy length 2 if and only if qΓr(d,d+c)≤1 and c≥r−1, where qΓr(x,y):=x2+y2−rxy denotes the Tits form of the generalized Kronecker quiver Γr.Since p>2 and constant Jordan type [1]c[2]d imply Loewy length ≤2, we get in this case the full classification of Jordan types [1]c[2]d that arise from indecomposable modules.

Essential Dimension and Genericity for Quiver Representations

We study the essential dimension of representations of a fixed quiver with given dimension vector. We also consider the question of when the genericity property holds, i.e., when essential dimension and generic essential dimension agree. We classify the quivers satisfying the genericity property for every dimension vector and show that for every wild quiver the genericity property holds for infinitely many of its Schur roots. We also construct a large class of examples, where the genericity property fails. Our results are particularly detailed in the case of Kronecker quivers.

An expansion formula for type A and Kronecker quantum cluster algebras

We introduce an expansion formula for elements in quantum cluster algebras associated to type A and Kronecker quivers with principal quantization. Our formula is parametrized by perfect matchings of snake graphs as in the classical case. In the Kronecker type, the coefficients are q-powers whose exponents are given by a weight function induced by the lattice of perfect matchings. As an application, we prove that a reflectional symmetry on the set of perfect matchings satisfies Stembridge's q=−1 phenomenon with respect to the weight function. Furthermore, we discuss a relation of our expansion formula to generating functions of BPS states.

Non-Schurian indecomposables via intersection theory

For an acyclic quiver with three vertices, we consider the canonical decomposition of a non-Schurian root and associate certain representations of a generalized Kronecker quiver. These representations correspond to points contained in the intersection of two subvarieties of a Grassmannian and give rise to representations of the original quiver, preserving indecomposability. We show that these subvarieties intersect using Schubert calculus. Provided that it contains a Schurian representation, the dimension of the intersection is what we expect by Kac's Theorem.

Cell decompositions for rank two quiver Grassmannians

We prove that all quiver Grassmannians for exceptional representations of a generalized Kronecker quiver admit a cell decomposition. In the process, we introduce a class of regular representations which arise as quotients of consecutive preprojective representations. Cell decompositions for quiver Grassmannians of these “truncated preprojectives” are also established. We provide two combinatorial labelings for these cells. On the one hand, they are naturally labeled by certain subsets of a so-called 2-quiver attached to a (truncated) preprojective representation. On the other hand, the cells are in bijection with compatible pairs in a maximal Dyck path as predicted by the theory of cluster algebras. The provided bijection between these two labelings gives a geometric explanation for the appearance of Dyck path combinatorics in the theory of quiver Grassmannians.

Bridgeland stability conditions on wild Kronecker quivers

We prove that for l≥3 the space of Bridgeland stability conditions on the Kronecker quiver with l parallel arrows is biholomorphic to C×H as a complex manifold.

Cluster algebras, invariant theory, and Kronecker coefficients II

We prove that the semi-invariant ring of the standard representation space of the l-flagged m-arrow Kronecker quiver is an upper cluster algebra for any l,m∈N. The quiver and cluster are explicitly given. We prove that the quiver with its rigid potential is a polyhedral cluster model. As a consequence, to compute each Kronecker coefficient gμ,νλ with λ at most m parts, we only need to count lattice points in at most m! fiber (rational) polytopes inside the g-vector cone, which is explicitly given.

Linear versus set valued Kronecker representations

ABSTRACTA set valued representation of the Kronecker quiver is nothing but a quiver. We apply the forgetful functor from vector spaces to sets and compare linear with set valued representations of the Kronecker quiver.