Finite Quiver Research Articles

Generalised Lat-Igusa-Todorov Algebras and Morita Contexts

AbstractIn this paper we define (special) GLIT classes and (special) GLIT algebras. We prove that GLIT algebras, which generalise Lat-Igusa-Todorov algebras, satisfy the finitistic dimension conjecture and give several properties and examples. In addition we show that special GLIT algebras are exactly those that have finite finitistic dimension. Lastly we study Morita algebras arising from a Morita context and give conditions for them to be (special) GLIT in terms of the algebras and bimodules used in their definition. As a consequence we obtain simple conditions for a triangular matrix algebra to be (special) GLIT and also prove that the tensor product of a GLIT $$\mathbb {K}$$ K -algebra with a path algebra of a finite quiver without oriented cycles is GLIT.

Young walls and graded dimension formulas for finite quiver Hecke algebras of types A2n−1(2), Dn(1) and Bn(1)

In this paper, we establish the graded dimension formulas of finite quiver Hecke algebras using level-[Formula: see text] Fock space representations of quantum affine algebras of types [Formula: see text], [Formula: see text] and [Formula: see text].

The commuting algebra

Let KQ be a path algebra, where Q is a finite quiver and K is a field. We study KQ/C where C is the two-sided ideal in KQ generated by all differences of parallel paths in Q. We show that KQ/C is always finite dimensional and its global dimension is finite. Furthermore, we prove that KQ/C is Morita equivalent to an incidence algebra.The paper starts with a more general setting, where KQ is replaced by KQ/I with I a two-sided ideal in KQ.

The Ext-algebra and associated monomial algebra of a finite dimensional K-algebra

It is well known that if the ext-algebra E(A) of an algebra A is generated in degrees 0 and 1, A must be a Koszul algebra. We seek to generalize this notion. To do so, we set A=KQ/I where Q is a finite quiver and I is an admissible ideal. We use a construction from [13], [1], and [11] to form a family of elements in KQ which yields a projective resolution of A¯, called the AGS resolution. We denote this family {fij} and consider the case where the AGS resolution is minimal. Then we consider the associated monomial algebra of A, found in [8] and [9], which we denote AMON. We prove that if the AGS resolution is minimal and E(AMON) is finitely generated, then E(A) is finitely generated. Next we look at 2-d-determined algebras, as defined in [11]. If A is a 2-d-determined algebra, we prove that if the AGS resolution is minimal, then E(A) is generated in degrees 0,1, and 2. However, if the AGS resolution is not minimal, we prove that E(A) need not be K2.

Universal quantum semigroupoids

We introduce the concept of a universal quantum linear semigroupoid (UQSGd), which is a weak bialgebra that coacts on a (not necessarily connected) graded algebra A universally while preserving grading. We restrict our attention to algebraic structures with a commutative base so that the UQSGds under investigation are face algebras (due to Hayashi). The UQSGd construction generalizes the universal quantum linear semigroups introduced by Manin in 1988, which are bialgebras that coact on a connected graded algebra universally while preserving grading. Our main result is that when A is the path algebra k Q of a finite quiver Q , each of the various UQSGds introduced here is isomorphic to the face algebra attached to Q . The UQSGds of preprojective algebras and of other algebras attached to quivers are also investigated.

Quivers associated with finite rings - a cohomological approach

In this paper we present the ongoing research on connection between digraphs associated to finite (commutative) rings and quiver representations. Digraph associated to a finite ring A has the set of vertices V = A2 and arrows (or edges) E = {(x, y) ? (x+y, xy), x, y ? A}. In another terminology, it is a finite quiver with loops. In addition to previous work to understand these graphs, the main goal of the present work is to introduce some new cohomological and quiver methods. These methods should provide us with better understanding of properties and classification of finite rings.

Tangent Spaces of Orbit Closures for Representations of Dynkin Quivers of Type mathbb {D}

Let Bbbk be an algebraically closed field, Q a finite quiver, and denote by textup {rep}_{Q}^{mathbf {d}} the affine Bbbk -scheme of representations of Q with a fixed dimension vector d. Given a representation M of Q with dimension vector d, the set {mathcal {O}}_{M} of points in Bbbk isomorphic as representations to M is an orbit under an action on textup {rep}^{mathbf {d}}_{Q}Bbbk of a product of general linear groups. The orbit {mathcal {O}}_{M} and its Zariski closure overline {mathcal {O}}_{M}, considered as reduced subschemes of textup {rep}_{Q}^{{mathbf {d}}}, are contained in an affine scheme {mathcal {C}}_{M} defined by suitable rank conditions associated to M. For all Dynkin and extended Dynkin quivers, the sets of points of overline {{mathcal {O}}}_{M} and {mathcal {C}}_{M} coincide, or equivalently, overline {{mathcal {O}}}_{M} is the reduced scheme associated to {mathcal {C}}_{M}. Moreover, overline {mathcal {O}}_{M}={mathcal {C}}_{M} provided Q is a Dynkin quiver of type {mathbb {A}}, and this equality is a conjecture for the remaining Dynkin quivers (of type mathbb {D} and {mathbb {E}}). Let Q be a Dynkin quiver of type mathbb {D} and M a finite dimensional representation of Q. We show that the equality T_{N}overline {mathcal {O}}_{M}=T_{N}{mathcal {C}}_{M} of Zariski tangent spaces holds for any closed point N of overline {mathcal {O}}_{M}. As a consequence, we describe the tangent spaces to overline {mathcal {O}}_{M} in representation theoretic terms.

Noncommutative Mather–Yau theorem and its applications to Calabi–Yau algebras

In this article, we prove that for a finite quiver Q the equivalence class of a potential up to formal change of variables of the complete path algebra \(\widehat{{{\mathbb {C}}}Q}\), is determined by its Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential assuming the Jacobi algebra is finite dimensional. This is a noncommutative analogue of the famous theorem of Mather and Yau on isolated hypersurface singularities. We also prove that the right equivalence class of a potential is determined by its sufficiently high jet assuming the Jacobi algebra is finite dimensional. These two theorems can be viewed as a first step towards the singularity theory of noncommutative power series. As an application, we show that if the Jacobi algebra is finite dimensional then the corresponding complete Ginzburg dg-algebra, and the (topological) generalized cluster category thereof, are determined by the isomorphic type of the Jacobi algebra together with the class in its 0-th Hochschild homology represented by the potential.

Principal Components Along Quiver Representations

Quiver representations arise naturally in many areas across mathematics. Here we describe an algorithm for calculating the vector space of sections, or compatible assignments of vectors to vertices, of any finite-dimensional representation of a finite quiver. Consequently, we are able to define and compute principal components with respect to quiver representations. These principal components are solutions to constrained optimisation problems defined over the space of sections and are eigenvectors of an associated matrix pencil.

Constructions of highest weight modules of double Ringel-Hall algebras via functions

In [26], Zheng studied the bounded derived categories of constructible Q¯l-sheaves on certain algebraic stacks consisting of the representations of a framed quiver and categorified the integrable highest weight modules of the corresponding quantum group by using these categories. In this paper, we generalize Zheng's work and realize highest weight modules of a certain subalgebra of the double Ringel-Hall algebra of a finite quiver without edge loops via spaces of functions on representation varieties.

Algebraic properties of face algebras

Prompted by an inquiry of Manin on whether a coacting Hopf-type structure [Formula: see text] and an algebra [Formula: see text] that is coacted upon share algebraic properties, we study the particular case of [Formula: see text] being a path algebra [Formula: see text] of a finite quiver [Formula: see text] and [Formula: see text] being Hayashi’s face algebra [Formula: see text] attached to [Formula: see text]. This is motivated by the work of Huang, Wicks, Won and the second author, where it was established that the weak bialgebra coacting universally on [Formula: see text] (either from the left, right, or both sides compatibly) is [Formula: see text]. For our study, we define the Kronecker square [Formula: see text] of [Formula: see text], and show that [Formula: see text] as unital graded algebras. Then we obtain ring-theoretic and homological properties of [Formula: see text] in terms of graph-theoretic properties of [Formula: see text] by way of [Formula: see text].

Lie generalized derivations on bound quiver algebras

In this paper, we investigate Lie generalized derivations on bound quiver algebras associated with a finite acyclic quiver. The first main result shows that every bound quiver algebra associated with a finite acyclic quiver has the properness Lie generalized derivation property. The second main result investigates the uniqueness property. Namely, among other things, we show that a bound quiver algebra associated with a finite acyclic quiver E has the uniqueness Lie generalized derivation property if and only if the quiver E does not contain any isolated vertex.

A Natural Hermitian Line Bundle on the Moduli Space of Semistable Representations of a Quiver

This paper describes the construction of a natural Hermitian holomorphic line bundle on the stratified moduli space of complex representations of a finite quiver, which are semistable with respect to a fixed rational weight and have a fixed type. It is shown that the curvature of this Hermitian line bundle on each stratum of the moduli space is essentially the Kahler form of that stratum.

On the existence of recollements of functor categories

A sufficient condition for the existence of recollements of functor categories is provided. Using this criterion, we show that a recollement of rings induces a recollement of their path rings (resp. incidence rings, monomial rings) over a locally finite quiver. Also, we present a covering technique for recollement of derived categories of functor categories.

Representations of Quivers over Artinian Rings

Given a unitary artinian ring R and a finite acyclic quiver Q, let Λ := RQ be the path ring of Q over R. Then Gorenstein-projective Λ-modules are exactly the separated monic representations of Q over R which satisfy the local Gorenstein-projective condition. We denote by smon(Q,R) the category of all finitely generated separated monic representations of Q over R, and $\mathcal {G}p({\varLambda })$ the category of all finitely generated Gorenstein-projective Λ-modules. If R is a selfinjective ring, then $\mathcal {G}p({\varLambda })=\text {smon}{\it (Q, R) }$ . As an application, if R is a commutative uniserial selfinjective ring of length 2 (here it means that as a regular module R is uniserial with length 2), let 0 ≠ a ∈radR and $\bar {R}=R/\text {rad}\textit {R}$ , our main result says that there is a full functor $H: \mathcal {G}p({\varLambda }) \to \text {mod} {\bar {R}Q}$ which induces a bijection between the indecomposable non-projetive Gorenstein-projective Λ-modules and the indecomposable $ \bar {R}Q$ -modules. Moreover, in this case, each Gorenstein-projective Λ-module is strongly Gorenstein-projective.

Exceptional Cycles in the Bounded Derived Categories of Quivers

An exceptional n-cycle in a Horn-finite triangulated category with Serre functor has been recently introduced by Broomhead, Pauksztello and Ploog. When n = 1, it is a spherical object. We explicitly determine all the exceptional cycles in the bounded derived category Db (kQ) of a finite quiver Q without oriented cycles. In particular, if Q is an Euclidean quiver, then the length type of exceptional cycles in Db (kQ) is exactly the tubular type of Q; if Q is a Dynkin quiver of type Em (m = 6, 7, 8), or Q is a wild quiver, then there are no exceptional cycles in Db (kQ); and if Q is a Dynkin quiver of type An or Dn, then the length of an exceptional cycle in Db (kQ) is either h or $$\frac{h}{2}$$, where h is the Coxeter number of Q.

Homology of twisted quiver bundles with relations

We study the Ext modules in the category of left modules over a twisted algebra of a finite quiver over a ringed space (X,OX), allowing for the presence of relations. We introduce a spectral sequence which relates the Ext modules in that category with the Ext modules in the category of OX-modules. Contrary to what happens in the absence of relations, this spectral sequence in general does not degenerate at the second page. We also consider local Ext sheaves. Under suitable hypotheses, the Ext modules are represented as hypercohomology groups.

Functors and Morphisms Determined by Subcategories

We study the existence and uniqueness of minimal right determiners in various categories. Particularly in a Hom-finite hereditary abelian category with enough projectives, we prove that the Auslander-Reiten-Smal{\o}-Ringel formula of the minimal right determiner still holds. As an application, we give a formula of minimal right determiners in the category of finitely presented representations of strongly locally finite quivers.

Projective objects in the category of pointwise finite dimensional representations of an interval finite quiver

Abstract For an interval finite quiver Q, we introduce a class of flat representations. We classify the indecomposable projective objects in the category rep ⁢ ( Q ) {\mathrm{rep}(Q)} of pointwise finite dimensional representations. We show that an object in rep ⁢ ( Q ) {\mathrm{rep}(Q)} is projective if and only if it is a direct sum of countably generated flat representations.

Holomorphic aspects of moduli of representations of quivers

This article describes some complex-analytic aspects of the moduli space of the finite-dimensional complex representations of a finite quiver, which are stable with respect to a fixed rational weight. We construct a natural structure of a complex manifold on this moduli space, and a K\"ahler metric on the complex manifold. We then define a Hermitian holomorphic line bundle on the moduli space, and show that its curvature is a rational multiple of the K\"ahler form.