A GR-segment for an artin algebra is a sequence of Gabriel-Roiter measures, which is closed under direct predecessors and successors. The number of the GR-segments indexed by natural numbers $\mathbb{N}$ and integers $\mathbb{Z}$ probably relates to the representation types of artin algebras. Let $k$ be an algebraically closed field and $Q$ be a tame quiver (of type $\widetilde{\mathbb{A}}_n$, $\widetilde{\mathbb{D}}_n$, $\widetilde{\mathbb{E}}_6$, $\widetilde{\mathbb{E}}_7$, or $\widetilde{\mathbb{E}}_8$). Let $b$ be the number of the isomorphism classes of the exceptional quasi-simple modules over the path algebra $Λ=kQ$. We show that the number of the $\mathbb{N}$- and $\mathbb{Z}$-indexed GR-segments in the central part for $Q$ is bounded by $b+1$. Therefore, there are at most $b+3$ GR segments.
Stability conditions play an important role in the study of representations of a quiver. In the present paper, we study semistable representations of quivers. In particular, we describe the slopes of semistable representations of a tame quiver for a fixed stability condition.
In this paper we explain how and why the list of Happel-Vossieck of tame concealed algebras is closely related to the list of A. Seven of minimal infinite cluster quivers.
In the present paper we study the derived Hall algebra for the bounded derived category of the nilpotent representations of a tame quiver over a finite field. We show that for any three given objects in the bounded derived category, the associated derived Hall numbers are given by a rational function in the cardinalities of ground fields.
C.M. Ringel defined Hall algebra associated with the category of representations of a quiver of Dynkin type and gave an explicit description of the structure constants of the corresponding Lie algebra. We utilize functorial properties of the Hall algebra to give a simple proof of Ringel's result, and to generalize it to the case of a quiver of affine type.
Let $\mathfrak{g}$ be an affine Kac-Moody algebra with symmetric Cartan datum, $\mathfrak{n^{+}}$ be the maximal nilpotent subalgebra of $\mathfrak{g}$. By the Hall algebra approach, we construct integral bases of the $\mathbb{Z}$-form of the enveloping algebra $U(\mathfrak{n^{+}})$. In particular, the representation theory of tame quivers is essentially used in this paper.
In this paper, we answer the question of when the subcategory of semi-stable representations is the same for two rational vectors for an acyclic quiver. This question has been previously answered by Ingalls, Paquette, and Thomas in the tame case in [10]. Here we take a more invariant theoretic approach, to answer this question in general. We recover the known result in the tame case.
In \cite{CK2005} and \cite{SZ}, the authors constructed the bases of cluster algebras of finite types and of type $\widetilde{A}_{1,1}$, respectively. In this paper, we will deduce $\mathbb{Z}$-bases for cluster algebras of affine types.
We introduce a new class of algebraic varieties which we call frieze varieties. Each frieze variety is determined by an acyclic quiver. The frieze variety is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. We give a new characterization of the finite--tame--wild trichotomy for acyclic quivers in terms of their frieze varieties. We show that an acyclic quiver is representation finite, tame, or wild, re…
In this paper we determine the derived representation type of quadratic string algebras and we prove that every derived tame quadratic string algebra whose quiver has cycles is derived equivalent to some skewed-gentle algebra.
We give a geometric realization of module categories of type $\tilde{A}_n$. We work with oriented arcs to define a translation quiver isomorphic to the Auslander-Reiten quiver of the module category of type $\tilde{A}_n$. To get a description of the module category, we introduce long moves between arcs. These allow us to include the infinite radical in the geometric description. Finally, our results can also be used to describe the corresponding cluster categories by taking unoriented arcs instead.
We study minimal disjoint degenerations for representations of tame quivers. In particular, we prove that their codimensions are bounded by 2. Therefore a quiver is Dynkin resp. Euclidean resp. wild iff the codimensions are 1 resp. bounded by 2 resp. unbounded. We explain also that for tame quivers the complete classification of all minimal disjoint degenerations is a finite problem that can be solved with the help of a computer.
Tame symmetric algebras of period four with small Gabriel quivers
Mathematical Institute, University of Oxford（牛津大学数学研究所）；Faculty of Mathematics and Computer Science, Nicolaus Copernicus University（尼古拉·哥白尼大学数学与计算机科学学院）
本文从新视角研究周期为四的 tame 对称代数，分类了 Gabriel 弦图顶点数不超过5的代数，证明其为广义加权表面代数，验证了相关猜想。
A symmetric quiver $(Q,σ)$ is a finite quiver without oriented cycles $Q=(Q_0,Q_1)$ equipped with a contravariant involution $σ$ on $Q_0\sqcup Q_1$. The involution allows us to define a nondegenerate bilinear form $<,>$ on a representation $V$ of $Q$. We shall say that $V$ is orthogonal if $<,>$ is symmetric and symplectic if $<,>$ is skew-symmetric. Moreover, we define an action of products of classical groups on the space of orthogonal representations and on the space of symplectic representations. So we prove that if $(Q,σ)$ is a symmetric quiver of tame type then the rings of semi-invarian…
We apply the theory of localization for tame and wild coalgebras in order to prove the following theorem: "Let Q be an acyclic quiver. Then any tame admissible subcoalgebra of KQ is the path coalgebra of a quiver with relations".
In this note we are concerned with the notion of amenable representation type as defined in a recent paper by Gábor Elek. Roughly speaking, an algebra is of amenable type if for all $\varepsilon > 0$, every finite-dimensional module has a submodule which is a direct sum of modules which are small with respect to $\varepsilon$ such that the quotient is also small in that respect. We will show that the tame hereditary path algebras of quivers of extended Dynkin type over any field $k$ are of amenable type, thus extending a conjecture in the aforementioned paper to another class of tame algebras.…
In this paper we state and prove the statement that tame hereditary algebras have finitely many m-maximal green sequences using a generalized version of Brüstle-Dupont-Pérotin's argument that tame quivers have finitely many maximal green sequences. Then we extend this theorem to mutation sequences which are partially red.
The $\mathbf{g}$-vector fan of a finite-dimensional algebra is a fan whose rays are the $\mathbf{g}$-vectors of its $2$-term presilting objects. We prove that the $\mathbf{g}$-vector fan of a tame algebra is dense. We then apply this result to obtain a near classification of quivers for which the closure of the cluster $\mathbf{g}$-vector fan is dense or is a half-space, using the additive categorification of cluster algebras by means of Jacobian algebras. As another application, we prove that for quivers with potentials arising from once-punctured closed surfaces, the stability and cluster sc…
Let $Q$ be a tame quiver of type $\widetilde{\mathbb{A}}_n$ and $\Rep(Q)$ the category of finite dimensional representations over an algebraically closed field. A representation is simply called a module. It will be shown that a regular string module has, up to isomorphism, at most two Gabriel-Roiter submodules. The quivers $Q$ with sink-source orientations will be characterized as those, whose central parts do not contain preinjective modules. It will also be shown that there are only finitely many (central) Gabriel-Roiter measures admitting no direct predecessors. This fact will be generaliz…
We show that a finite connected quiver Q with no oriented cycles is tame if and only if for each dimension vector $\mathbf{d}$ and each integral weight $θ$ of Q, the moduli space $\mathcal{M}(Q,\mathbf{d})^{ss}_θ$ of $θ$-semi-stable $\mathbf{d}$-dimensional representations of Q is just a projective space. In order to prove this, we show that the tame quivers are precisely those whose weight spaces of semi-invariants satisfy a certain log-concavity property. Furthermore, we characterize the tame quivers as being those quivers Q with the property that for each Schur root $\mathbf{d}$ of Q, the f…
This document serves as an arXiv entry point for the appendix to the paper [13] (the ancillary file e6_proof.pdf -- ``Proof of the tree module property for exceptional representations of the quiver $\widetilde{\mathbb{E}}_6$'') and the appendix to the paper [12] (the ancillary file d6_proof.pdf -- ``Proof of the tree module property for exceptional representations of the quiver $\widetilde{\mathbb{D}}_6$''). The ancillary files contain the computer generated part of the proofs of the main results in [13] respectively [12], giving a complete and general list of tree representations correspondin…
In this paper, we study the maximal length of maximal green sequences for quivers of type $\widetilde{\mathbf{D}}$ and $\widetilde{\mathbf{E}}$ by using the theory of tilting mutation. We show that the maximal length does not depend on the choice of the orientation, and determine it explicitly. Moreover, we give a program which counts all maximal green sequences by length for a given Dynkin/extended Dynkin quiver.
The celebrated Drozd's theorem asserts that a finite-dimensional basic algebra $Λ$ over an algebraically closed field $k$ is either tame or wild, whereas the Crawley-Boevey's theorem states that given a tame algebra $Λ$ and a dimension $d$, all but finitely many isomorphism classes of indecomposable $Λ$-modules of dimension $d$ are isomorphic to their Auslander-Reiten translations and hence belong to homogeneous tubes. In this paper, we prove the inverse of Crawley-Boevey's theorem, which gives an internal description of tameness in terms of Auslander-Reiten quivers.
Let $A$ be a tame hereditary algebra over a finite field $k$ with $q$ elements, and ${\bar{A}}$ be the duplicated algebra of $A$. In this paper, we investigate the structure of Ringel-Hall algebra $\mathscr{H} (\bar{A})$ and of the corresponding composition algebra $\mathscr{C} (\bar{A})$. As an application, we prove the existence of Hall polynomials $g_{XY}^M$ for any $\bar{A}$-modules $M, X$ and $Y$ with $X$ and $Y$ indecomposable if $A$ is a tame quiver $k$-algebra, then we also obtain some Lie subalgebras induced by $\bar{A}$.
In this paper we study sufficient conditions for the solvability of the first Hochschild cohomology of a finite dimensional algebra as a Lie algebra in terms of its Ext-quiver in arbitrary characteristic. In particular, we show that if the quiver has no parallel arrows and no loops then the first Hochschild cohomology is solvable. For quivers containing loops, we determine easily verifiable sufficient conditions for the solvability of the first Hochschild cohomology. We apply these criteria to show the solvability of the first Hochschild cohomology space for large families of algebras, namely,…
Let $Λ$ be a finite-dimensional basic algebra over an algebraically closed field $k$. The well-known Drozd's theorem asserts, that $Λ$ is either tame or wild. The Crawley-Boevey's Theorem states that for a given tame algebra $Λ$, and for each dimension $d$ almost all isomorphism classes of indecomposable $Λ$-modules of dimension $d$ are isomorphic to their Auslander-Reiten translations and hence belong to homogeneous tubes. In this paper we prove the converse of Crawley-Boevey's Theorem and thus give an internal description of tameness in terms of AR-quivers. This gives a complete answer to a …
A famous result by Drozd says that a finite-dimensional representation-infinite algebra is of either tame or wild representation type. But one has to make assumption on the ground field. The Gabriel-Roiter measure might be an alternative approach to extend these concepts of tame and wild to arbitrary artin algebras. In particular, the infiniteness of the number of GR segments, i.e. sequences of Gabriel-Roiter measures which are closed under direct predecessors and successors, might relate to the wildness of artin algebras. As the first step, we are going to study the wild quiver with three ver…
We give Erdmann-Nakano type theorem for the finite quiver Hecke algebras $R^{Λ_0}(β)$ of affine type $A^{(1)}_{\ell}$. Note that each finite quiver Hecke algebra lies in one parameter family, and the original Erdmann-Nakano theorem studied the finite quiver Hecke algebra at a special parameter value. We study the general case in our paper. Our result shows in particular that their representation type does not depend on the parameter. Moreover, when the parameter value is nonzero, we show that finite quiver Hecke algebras of tame representation type are biserial algebras.
In this paper, the singular Ringel-Hall algebra for a tame quiver is introduced and shown to be isomorphic to the positive part of the quantum extended Kac-Moody algebra. A PBW basis is constructed and a new class of perverse sheaves is shown to have purity property. This allows to construct the canonical bases of the positive part of the quantum extended Kac-Moody algebra. As an application, the existence of Hall polynomials for tame quiver algebras is proved.
In the present paper we prove that Hall polynomial exists for each triple of decomposition sequences which parameterize isomorphism classes of coherent sheaves of a domestic weighted projective line $\mathbb X$ over finite fields. These polynomials are then used to define the generic Ringel--Hall algebra of $\mathbb X$ as well as its Drinfeld double. Combining this construction with a result of Cramer, we show that Hall polynomials exist for tame quivers, which not only refines a result of Hubery, but also confirms a conjecture of Berenstein and Greenstein.
In this paper, the rational Ringel-Hall algebras for tame quivers are introduced and are identified with the positive part of the quantum extended Kac-Moody algebras. By using the rational Ringel-Hall algebras, we show that the existence of Hall polynomials for tame quiver algebras. The PBW bases are constructed and new classes of perverse sheaves are shown to have strong purity property. These allows us to construct the canonical bases of the positive part of the quantum extended Kac-Moody algebras.
We introduce and study the higher tetrahedral algebras, an exotic family of finite-dimensional tame symmetric algebras over an algebraically closed field. The Gabriel quiver of such an algebra is the triangulation quiver associated to the coherent orientation of the tetrahedron. Surprisingly, these algebras occurred in the classification of all algebras of generalised quaternion type, but are not weighted surface algebras. We prove that a higher tetrahedral algebra is periodic if and only if it is non-singular.
For each prime number $s$ we introduce examples of $(s-1)$- and $s$-representation infinite algebras in the sense of Herschend, Iyama and Oppermann, which arise from skew group algebras of some metacyclic groups embedded in $\mathrm{SL}(s,\mathbb{C})$ and $\mathrm{SL}(s+1,\mathbb{C})$. For this purpose, we give a description of the McKay quiver with a superpotential of such groups. Moreover we show that for $s=2$ our examples correspond to the classical tame hereditary algebras of type $\tilde{D}$.
We give an example of a cluster-tilted algebra A with quiver Q, such that the associated cluster algebra has a denominator vector which is not the dimension vector of any indecomposable A-module. This answers a question posed by T. Nakanishi. The relevant example is a cluster-tilted algebra associated with a tame hereditary algebra. We show that for such a cluster-tilted algebra A, we can write any denominator vector as a sum of the dimension vectors of at most three indecomposable rigid A-modules. In order to do this it is necessary, and of independent interest, to first classify the indecomp…
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 occur 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 …
We define algebras of quasi-quaternion type, which are symmetric algebras of tame representation type whose stable module category has certain structure similar to that of the algebras of quaternion type introduced by Erdmann. We observe that symmetric tame algebras that are also 2-CY-tilted are of quasi-quaternion type. We present a combinatorial construction of such algebras by introducing the notion of triangulation quivers. The class of algebras that we get contains Erdmann's algebras of quaternion type on the one hand and the Jacobian algebras of the quivers with potentials associated by …
In this paper, we study algebras of global dimension at most 2 whose generalized cluster category is equivalent to the cluster category of an acyclic quiver which is either a tree or of type $\widetilde{A}$. We are particularly interested in their derived equivalence classification. We prove that each algebra which is cluster equivalent to a tree quiver is derived equivalent to the path algebra of this tree. Then we describe explicitly the algebras of cluster type $\A_n$ for each possible orientation of $\A_n$. We give an explicit way to read off in which derived equivalence class such an alge…
Given a cluster-tilted algebra B, we study its first Hochschild cohomology group HH^1(B) with coefficients in the B-B-bimodule B. If C is a tilted algebra such that B is the relation extension of C, then we show that if C is constrained, or else if B is tame, then HH^1(B) is isomorphic, as a k-vector space, to the direct sum of HH^1(C) with k^{n\_{B,C}}, where n\_{B,C} is an invariant linking the bound quivers of B and C. In the representation-finite case, HH^1(B) can be read off simply by looking at the quiver of B.
We study a family of three-dimensional Lie algebras $L_μ$ that depend on a continuous parameter $μ$. We introduce certain quivers, which we denote by $Q_{m,n}$ $(m,n \in \mathbb{Z})$ and $Q_{\infty \times \infty}$, and prove that idempotented versions of the enveloping algebras of the Lie algebras $L_μ$ are isomorphic to the path algebras of these quivers modulo certain ideals in the case that $μ$ is rational and non-rational, respectively. We then show how the representation theory of the quivers $Q_{m,n}$ and $Q_{\infty\times\infty}$ can be related to the representation theory of quivers of …
Let k be an arbitrary field and Q an acyclic quiver of tame type. Consider the path algebra kQ and the category of finite dimensional right modules Mod kQ. In the first part of the paper we deduce that the Gabriel-Roiter inclusions in preprojective indecomposables and simple homogeneous regulars (with dimension the minimal radical vector) as well as their Gabriel-Roiter measures are field independent. Using this result we can prove in a more general setting a theorem by Bo Chen which states that the Gabriel-Roiter submodule of a simple homogeneous regular (with dimension the minimal radical ve…
We study the category of graded finite-dimensional representations of the polynomial current algebra associated to a simple Lie algebra. We prove that the category has enough injectives and compute the graded character of the injective envelopes of the simple objects as well as extensions between simple objects. The simple objects in the category are parametized by the affine weight lattice. We show that with respect to a suitable refinement of the standard ordering on affine the weight lattice the category is highest weight. We compute the Ext quiver of the algebra of endomorphisms of the inj…
Let $Λ$ be an Artin algebra. A GR segment of $Λ$ is a sequence of GR measures which is closed under direct successors and direct predecessors. The number of the GR segments was conjectured to relate to the representation type of $Λ$. In this paper, let $k$ be an algebraically closed field and $Λ$ be a finite-dimensional hereditary $k$-algebra. We show that $Λ$ admits infinitely many GR segments if and only if $Λ$ is of wild representation type. Thus the finiteness of the number of the GR segments might be an alternative characterization of the tameness of finite dimensional algebras over algeb…
Let $A$ be a finite dimensional hereditary algebra over a field $k$ and $A^{(1)}$ the duplicated algebra of $A$. We first show that the global dimension of endomorphism ring of tilting modules of $A^{(1)}$ is at most 3. Then we investigate embedding tilting quiver $\mathscr{K}(A)$ of $A$ into tilting quiver $\mathscr{K}(A^{(1)})$ of $A^{(1)}$. As applications, we give new proofs for some results of D.Happel and L.Unger, and prove that every connected component in $\mathscr{K}({A})$ has finite non-saturated points if $A$ is tame type, which gives a partially positive answer to the conjecture of…
We show that the representation type of the Jacobian algebra P(Q,S) associated to a 2-acyclic quiver Q with non-degenerate potential S is invariant under QP-mutations. We prove that, apart from very few exceptions, P(Q,S) is of tame representation type if and only if Q is of finite mutation type. We also show that most quivers Q of finite mutation type admit only one non-degenerate potential up to weak right equivalence. In this case, the isomorphism class of P(Q,S) depends only on Q and not on S.
We prove the tame-wild dichotomy conjecture, due to D. Simson, for infinite dimensional algebras and coalgebras. The key part of the approach is proving new representation theoretic characterizations local finiteness. Among other, we show that the Ext quiver of the category ${\rm f.d.-}A$ of finite dimensional representations of an arbitrary algebra $A$ is locally finite (i.e. $\dim(\Ext^1(S,T))<\infty$ for all simple finite dimensional $A$-modules $S,T$) if and only if for every dimension vector $\underline{d}$, the representations of $A$ of dimension vector $\underline{d}$ are all contained …
When $Γ$ is a row-finite di(rected )graph we classify all finite dimensional modules of the Leavitt path algebra $L(Γ)$ via an explicit Morita equivalence given by an effective combinatorial (reduction) algorithm on the digraph $Γ$. The category of (unital) $L(Γ)$-modules is equivalent to a subcategory of quiver representations of $Γ$. However the category of finite dimensional representations of $L(Γ)$ is tame in contrast to the finite dimensional quiver representations of $Γ$ which are almost always wild.
