Quiver Algebras Research Articles

Ringel duality for certain strongly quasi-hereditary algebras

We study quasi-hereditary endomorphism algebras defined over a new class of finite dimensional monomial algebras with a special ideal structure. The main result is a uniform formula describing the Ringel duals of these quasi-hereditary algebras. As special cases, we obtain a Ringel duality formula for a family of strongly quasi-hereditary algebras arising from a type A configuration of projective lines in a rational, projective surface as recently introduced by Hille and Ploog, for certain Auslander–Dlab–Ringel algebras, and for Eiriksson and Sauter’s nilpotent quiver algebras when the quiver has no sinks and no sources. We also recover Tan’s result that the Auslander algebras of self-injective Nakayama algebras are Ringel self-dual.

Gorenstein Properties of Simple Gluing Algebras

For given bound quiver algebras A and B, we obtain a new algebra Λ, called the simple gluing algebra, by identifying two vertices. We investigate the Gorenstein homological property, the singularity category, the Gorenstein defect category and the Cohen-Macaulay Auslander algebra of Λ in terms of that of A and B. Finally, we give applications to cluster-tilted algebras.

On Generalized Minors and Quiver Representations

Abstract The cluster algebra of any acyclic quiver can be realized as the coordinate ring of a subvariety of a Kac–Moody group—the quiver is an orientation of its Dynkin diagram, defining a Coxeter element and thereby a double Bruhat cell. We use this realization to connect representations of the quiver with those of the group. We show that cluster variables of preprojective (resp. postinjective) quiver representations are realized by generalized minors of highest-weight (resp. lowest-weight) group representations, generalizing results of Yang–Zelevinsky in finite type. In type $A_{n}^{\!(1)}$ and finitely many other affine types, we show that cluster variables of regular quiver representations are realized by generalized minors of group representations that are neither highest- nor lowest-weight; we conjecture this holds more generally.

Weighted surface algebras

A finite-dimensional algebra A over an algebraically closed field K is called periodic if it is periodic under the action of the syzygy operator in the category of A-A-bimodules. The periodic algebras are self-injective and occurred naturally in the study of tame blocks of group algebras, actions of finite groups on spheres, hypersurface singularities of finite Cohen–Macaulay type, and Jacobian algebras of quivers with potentials. Recently, the tame periodic algebras of polynomial growth have been classified and it is natural to attempt to classify all tame periodic algebras. We introduce the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular tetrahedral algebras, are symmetric tame periodic algebras of period 4. Moreover, we describe the socle deformations of the weighted surface algebras and prove that all these algebras are also symmetric tame periodic algebras of period 4. The main results of this paper form an important step towards a classification of all periodic symmetric tame algebras of non-polynomial growth, and lead to a complete description of all algebras of generalized quaternion type with 2-regular Gabriel quivers [36].

Jordan Derivations and Lie Derivations on Path Algebras

Without the faithful assumption, we prove that every Jordan derivation on a class of path algebras of quivers without oriented cycles is a derivation and that every Lie derivation on such kinds of algebras is of the standard form.

Zero-relation quiver Poisson algebras

Poisson structures on bound quiver algebras $KQ/I$ are considered, where $Q$ is a quiver without oriented cycles and $I$ is an admissible ideal in the path algebra $KQ$ generated only by paths. In particular, inner Poisson algebras on $KQ/I$ are character

Injective representations of bound quiver algebras

Let [Formula: see text] be a ring and [Formula: see text] be a finite quiver. We give an explicit formula for the injective envelopes and projective precovers in the category [Formula: see text] of bound representations of [Formula: see text] by left [Formula: see text]-modules, where [Formula: see text] is a combination of monomial and commutativity relations. Some applications will be provided. In particular, it is shown that if [Formula: see text] is acyclic and [Formula: see text] is an Iwanaga-Gorenstein ring, then so are these bound quiver algebras.

On the definition of multi-Koszul modules

In [8] we introduced the notion of multi-Koszul algebra: it is an extension of the definition of generalized Koszul algebra given by R. Berger in [1] for homogeneous algebras (see also [7]) that can be applied to any nonnegatively graded connected algebra over a field k. The goal of this article is on the one hand to properly extend the notion of multi-Koszul algebra to the case where the base ring K is a product of copies of a field k, which a priori allows us to treat quiver algebras, and on the other hand to introduce the notion of multi-Koszul module such that it extends the usual definition of generalized Koszul module over a generalized Koszul algebra. We show eventually that multi-Koszul algebras and multi-Koszul modules are strongly linked via the notion of one-point extensions, as in the case of generalized Koszul algebras. Moreover, we describe the complete structure of right A∞-module on ExtA•(M,K) over ExtA•(K,K), where M is a multi-Koszul module over a multi-Koszul algebra A, extending a result in [9]. As a corollary, we obtain that the underlying right module structure of ExtA•(M,K) over ExtA•(K,K) is generated by the component of cohomological degree zero, as in the case of generalized Koszul modules over generalized Koszul algebras.

Ringel–Hall algebra construction of quantum Borcherds–Bozec algebras

We give the Ringel–Hall algebra construction of the positive half of quantum Borcherds–Bozec algebras as the generic composition algebras of quivers with loops.

On presentations of Hochschild extension algebras for a class of self-injective Nakayama algebras

For a bound quiver algebra satisfying the condition that the every oriented cycles in the quiver are vanished in the algebra, Fernández and Platzeck determined the bound quiver algebra which is isomorphic to the trivial extension algebra. In this paper, we consider a Hochschild extension algebra which is a generalization of a trivial extension algebra. The purpose of this paper is to determine the bound quiver algebras which are isomorphic to Hochschild extension algebras of some finite dimensional self-injective Nakayama algebras.

Connected quantized Weyl algebras and quantum cluster algebras

For an algebraically closed field K, we investigate a class of noncommutative K-algebras called connected quantized Weyl algebras. Such an algebra has a PBW basis for a set of generators {x1,…,xn} such that each pair satisfies a relation of the form xixj=qijxjxi+rij, where qij∈K⁎ and rij∈K, with, in some sense, sufficiently many pairs for which rij≠0. For such an algebra it turns out that there is a single parameter q such that each qij=q±1. Assuming that q≠±1, we classify connected quantized Weyl algebras, showing that there are two types linear and cyclic. When q is not a root of unity we determine the prime spectra for each type. The linear case is the easier, although the result depends on the parity of n, and all prime ideals are completely prime. In the cyclic case, which can only occur if n is odd, there are prime ideals for which the factors have arbitrarily large Goldie rank.We apply connected quantized Weyl algebras to obtain presentations of two classes of quantum cluster algebras. Let n≥3 be an odd integer. We present the quantum cluster algebra of a Dynkin quiver of type An−1 as a factor of a linear connected quantized Weyl algebra by an ideal generated by a central element. We also consider the quiver Pn+1(1) identified by Fordy and Marsh in their analysis of periodic quiver mutation. When n is odd, we show that the quantum cluster algebra of this quiver is generated by a cyclic connected quantized Weyl algebra in n variables and one further generator. We also present it as the factor of an iterated skew polynomial algebra in n+2 variables by an ideal generated by a central element. For both classes, the quantum cluster algebras are simple noetherian.We establish Poisson analogues of the results on prime ideals and quantum cluster algebras. We determine the Poisson prime spectra for the semiclassical limits of the linear and cyclic connected quantized Weyl algebras and show that, when n is odd, the cluster algebras of An−1 and Pn+1(1) are simple Poisson algebras that can each be presented as a Poisson factor of a polynomial algebra, with an appropriate Poisson bracket, by a principal ideal generated by a Poisson central element.

Non-commutative crepant resolution of minimal nilpotent orbit closures of type A and Mukai flops

In this article, we construct a non-commutative crepant resolution (=NCCR) of a minimal nilpotent orbit closure B(1)‾ of type A, and study relations between an NCCR and crepant resolutions Y and Y+ of B(1)‾. More precisely, we show that the NCCR is isomorphic to the path algebra of the double Beilinson quiver with certain relations and we reconstruct the crepant resolutions Y and Y+ of B(1)‾ as moduli spaces of representations of the quiver. We also study the Kawamata–Namikawa's derived equivalence between crepant resolutions Y and Y+ of B(1)‾ in terms of an NCCR. We also show that the P-twist on the derived category of Y corresponds to a certain operation of the NCCR, which we call multi-mutation, and that a multi-mutation is a composition of Iyama–Wemyss's mutations.

Khovanov–Seidel quiver algebras and Ozsváth–Szabó's bordered theory

In the 2016 preprint “Kauffman states, bordered algebras, and a bigraded knot invariant,” Ozsváth and Szabó introduced a set of algebraic constructions in the spirit of bordered Heegaard Floer homology. Their constructions can be used to compute knot Floer homology algebraically for knots in the 3-sphere. In this paper we investigate a relationship between Ozsváth–Szabó's bordered theory and the algebras and bimodules constructed by Khovanov and Seidel in “Quivers, Floer cohomology, and braid group actions” (2002). Specifically, we show that the Khovanov–Seidel quiver algebras are isomorphic to quotients of idempotent truncations of some of Ozsváth–Szabó's algebras. Furthermore, we show that the dg bimodule associated to a braid generator by Khovanov–Seidel, with the right action restricted using the quotient map, is homotopy equivalent to Ozsváth–Szabó's DA bimodule with the left action induced using the quotient map.

An elementary construction of monomial bases of modified quantum affine gln

AbstractIn 1990 Beilinson, Lusztig and MacPherson provided a geometric realization of modified quantum and its canonical basis. A key step of their work is a construction of a monomial basis. Recently, Du and Fu provided an algebraic construction of the canonical basis for modified quantum affine , which among other results used an earlier construction of monomial bases using Ringel–Hall algebra of the cyclic quiver. In this paper, we give an elementary algebraic construction of a monomial basis for affine Schur algebras and modified quantum affine .

Categorical action of the extended braid group of affine type A

Using a quiver algebra of a double cyclic quiver, we construct a faithful categorical action of the extended braid group of affine type [Formula: see text] on its bounded homotopy category of finitely generated projective modules. The algebra is trigraded and we identify the trigraded dimensions of the space of morphisms of this category with intersection numbers coming from the topological origin of the group.

Finitary 2-categories associated with dual projection functors

We study finitary [Formula: see text]-categories associated to dual projection functors for finite-dimensional associative algebras. In the case of path algebras of admissible tree quivers (which includes all Dynkin quivers of type [Formula: see text]), we show that the monoid generated by dual projection functors is the Hecke–Kiselman monoid of the underlying quiver and also obtain a presentation for the monoid of indecomposable subbimodules of the identity bimodule.

Minimal right determiners of irreducible morphisms in algebras of type [formula omitted

Let Λ be a finite dimensional algebra of type An over a field with the quiver Q and let |Det(Λ)| be the number of the minimal right determiners of all irreducible morphisms between indecomposable left Λ-modules. If Λ is a path algebra, then we have|Det(Λ)|={2n−2,if p=0;2n−p−1,if p≥1, where p=|{i|i is a source in Q with 2≤i≤n−1}|. If Λ is a bound quiver algebra, then we have|Det(Λ)|={2n−2,if r=1;2n−p−q−1,if r≥2, where q is the number of non-zero sink ideals of Λ and r=|{i|i is a sink in Q with 1≤i≤n}|.

Hall algebras of cyclic quivers and q-deformed Fock spaces

Based on the work of Ringel and Green, one can define the (Drinfeld) double Ringel–Hall algebra D(Q) of a quiver Q as well as its highest weight modules. The main purpose of the present paper is to show that the basic representation L(Λ0) of D(Δn) of the cyclic quiver Δn provides a realization of the q-deformed Fock space ⋀∞ defined by Hayashi. This is worked out by extending a construction of Varagnolo and Vasserot. By analysing the structure of nilpotent representations of Δn, we obtain a decomposition of the basic representation L(Λ0) which induces the Kashiwara–Miwa–Stern decomposition of ⋀∞ and a construction of the canonical basis of ⋀∞ defined by Leclerc and Thibon in terms of certain monomial basis elements in D(Δn).

Hochschild Homology Groups of System Quiver Algebras of Maximal Tame Type

Hochschild Homology Groups of System Quiver Algebras of Maximal Tame Type

Quivers with relations for symmetrizable Cartan matrices I: Foundations

We introduce and study a class of Iwanaga-Gorenstein algebras defined via quivers with relations associated with symmetrizable Cartan matrices. These algebras generalize the path algebras of quivers associated with symmetric Cartan matrices. We also define a corresponding class of generalized preprojective algebras. Without any assumption on the ground field, we obtain new representation-theoretic realizations of all finite root systems.