Derived tame local and two-point algebras

On tame strongly simply connected algebras

In this survey we present the criterion for tameness of strongly simply connected algebras due to Brüstle, de la Peña and Skowroński. We recall relevant concepts of representation theory and discuss some applications and connections to other problems.

Derived tame Nakayama algebras

We determine the derived representation type of Nakayama algebras and prove that a derived tame Nakayama algebra without simple projective module is gentle or derived equivalent to some skewed-gentle algebra, and as a consequence, we determine its singularity category.

经典类型 Hecke 代数的驯顺块代数

Tame block algebras of Hecke algebras of classical type

本文在奇特征代数闭域上分类经典类型 Hecke 代数的驯顺块代数，给出完整分类结果。

Ringel-Hall Algebras of Duplicated Tame Hereditary Algebras

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 $\bar{A}$.

On derived Hall numbers for tame 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.

Tame Hereditary Algebras have finitely many m-Maximal Green Sequences

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 No Gap Conjecture for tame hereditary algebras

The "No Gap Conjecture" of Brüstle-Dupont-Pérotin states that the set of lengths of maximal green sequences for hereditary algebras over an algebraically closed field has no gaps. This follows from a stronger conjecture that any two maximal green sequences can be "polygonally deformed" into each other. We prove this stronger conjecture for all tame hereditary algebras over any field.

Algebras with homogeneous module category are tame

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.

Tame concealed algebras and cluster quivers of minimal infinite type

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.

$τ$-倾斜有限性与$\boldsymbol{g}$-驯顺性：偏序集的关联代数与concealed代数

$τ$-tilting finiteness and $\mathbf{g}$-tameness: Incidence algebras of posets and concealed algebras

该研究利用concealed代数理论，刻画了有限偏序集的$τ$-倾斜有限关联代数与单连通偏序集的$\boldsymbol{g}$-驯顺关联代数，提出相关猜想并验证反例稀少，还确定了双曲型concealed代数$τ$-倾斜约化的表示型。

On the Representation Dimension of Tame Cluster Tilted Algebras

The aim of this work is to study the representation dimension of cluster tilted algebras. We prove that the weak representation dimension of tame cluster tilted algebras is equal to three. We construct a generator module that reaches the weak representation dimension, unfortunately this module is not always a cogenerator. We show for which algebras this module gives the representation dimension.

On the Hochschild cohomology of tame Hecke algebras

We explicitly calculate a projective bimodule resolution for a special biserial algebra giving rise to the Hecke algebra H_q(S_4) when q=-1. We then determine the dimensions of the Hochschild cohomology groups.

Moduli spaces of modules of Schur-tame algebras

In this paper, we first show that for an acyclic gentle algebra A, the irreducible components of any moduli space of A-modules are products of projective spaces. Next, we show that the nice geometry of the moduli spaces of modules of an algebra does not imply the tameness of the representation type of the algebra in question. Finally, we place these results in the general context of moduli spaces of modules of Schur-tame algebras. More specifically, we show that for an arbitrary Schur-tame algebra A and theta-stable irreducible component C of a module variety of A-modules, the moduli space of …

tame 余弦代数中原始元素的研究

Primitive elements in Ringel-Hall algebras of tame hereditary algebras

本文研究了tame hereditary代数的Ringel-Hall代数中的原始元素，扩展并改进了Hennecart的结果，并构造了显式基。

Classifying tame blocks and related algebras up to stable equivalences of Morita type

We contribute to the classification of finite dimensional algebras under stable equivalence of Morita type. More precisely we give a classification of the class of Erdmann's algebras of dihedral, semi-dihedral and quaternion type and obtain as byproduct the validity of the Auslander-Reiten conjecture for these classes of algebras.

周期为4的驯顺对称代数的局部结构

Local structure of tame symmetric algebras of period four

本文研究周期为4的驯顺对称代数的双正则Gabriel箭图局部结构，发现其与加权曲面代数的Gabriel箭图局部结构一致，部分推广了广义四元数型代数的已有刻画。

关于 tame 代数上的通用砖块

On generic bricks over tame algebras

研究 tame 代数上通用模块的性质，证明通用模块成为通用砖块的条件，以及 tame 代数存在通用砖块的等价条件。

Affine Lie Algebras and Tame Quivers

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.

Extensions of tame algebras and finite group schemes of domestic representation type

Let k be an algebraically closed field. Given an extension A : B of finite-dimensional k- algebras, we establish criteria ensuring that the representation-theoretic notion of polynomial growth is preserved under ascent and descent. These results are then used to show that principal blocks of finite group schemes of odd characteristic are of polynomial growth if and only if they are Morita equivalent to trivial extensions of radical square zero tame hereditary algebras. In that case, the blocks are of domestic representation type and the underlying group schemes are closely related to binary po…

First Hochschild cohomology group and stable equivalence classification of Morita type of some tame symmetric algebras

We use the dimension and the Lie algebra structure of the first Hochschild cohomology group to distinguish some algebras of dihedral, semi-dihedral and quaternion type up to stable equivalence of Morita type. In particular, we complete the classification of algebras of dihedral type that was mostly determined by Zhou and Zimmermann.

Tame hereditary path algebras and amenability

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.…

On the invariant theory for tame tilted algebras

We show that a tilted algebra $A$ is tame if and only if for each generic root $\dd$ of $A$ and each indecomposable irreducible component $C$ of $\module(A,\dd)$, the field of rational invariants $k(C)^{\GL(\dd)}$ is isomorphic to $k$ or $k(x)$. Next, we show that the tame tilted algebras are precisely those tilted algebras $A$ with the property that for each generic root $\dd$ of $A$ and each indecomposable irreducible component $C \subseteq \module(A,\dd)$, the moduli space $\M(C)^{ss}_θ$ is either a point or just $\mathbb P^1$ whenever $θ$ is an integral weight for which $C^s_θ\neq \emptyse…

Estimate of the number of one-parameter families of modules over a tame algebra

The problem of classifying modules over a tame algebra A reduces to a block matrix problem of tame type whose indecomposable canonical matrices are zero- or one-parameter. Respectively, the set of nonisomorphic indecomposable modules of dimension at most d divides into a finite number f(d,A) of modules and one-parameter series of modules. We prove that the number of m-by-n canonical parametric block matrices with a given partition into blocks is bounded by 4^s, where s is the number of free entries (which is at most mn), and estimate the number f(d,A).

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 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 …

Homogeneity implies Tameness

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 …

周期为四的温顺对称代数

Tame symmetric algebras of period four

本文研究周期为四的温顺对称代数的结构，重点分析 Gabriel 箭图为双序列的情形，给出其箭图必须满足的一系列性质，并证明三角形和正方形自然出现，为后续完整分类奠定基础。

Tame algebras have dense $\mathbf{g}$-vector fans

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…

Tilting Modules over Tame Hereditary Algebras

We give a complete classification of the infinite dimensional tilting modules over a tame hereditary algebra R. We start our investigations by considering tilting modules of the form T=R_U\oplus R_U /R where U is a union of tubes, and R_U denotes the universal localization of R at U in the sense of Schofield and Crawley-Boevey. Here R_U/R is a direct sum of the Prüfer modules corresponding to the tubes in U. Over the Kronecker algebra, large tilting modules are of this form in all but one case, the exception being the Lukas tilting module L whose tilting class Gen L consists of all modules wit…

Representation-tame algebras need not be homologically tame

We show that, also within the class of representation-tame finite dimensional algebras $Λ$, the big left finitistic dimension of $Λ$ may be strictly larger than the little. In fact, the discrepancies $Findim Λ- findim Λ$ need not even be bounded for special biserial algebras which constitute one of the (otherwise) most thoroughly understood classes of tame algebras. More precisely: For every positive integer $r$, we construct a special biserial algebra $Λ$ with the property that $findim Λ= r + 1$, while $Findim Λ= 2r + 1$. In particular, there are infinite dimensional representations of $Λ$ wh…

Baer and Mittag-Leffler modules over tame hereditary algebras

We develop a structure theory for two classes of infinite dimensional modules over tame hereditary algebras: the Baer modules, and the Mittag-Leffler ones.

On the Structure of Tame Graded Basic Hopf Algebras II

In continuation of the article \cite{L} we classify all radically graded basic Hopf algebras of tame type over an algebraically closed field of characteristic 0.

On the zero set of semi-invariants for regular modules over tame canonical algebras

We investigate sets of the common zeros of non-constant semi-invariants for regular modules over canonical algebras. In particular, we show that if the considered algebra is tame then for big enough vectors these sets are complete intersections.

Hochschild cohomology and support varieties for tame Hecke algebras

We give a basis for the Hochschild cohomology ring of tame Hecke algebras. We then show that the Hochschild cohomology ring modulo nilpotence is a finitely generated algebra of Krull dimension 2, and describe the support varieties of modules for these algebras. As a consequence we obtain the result that the Hochschild cohomology ring modulo nilpotence of a Hecke algebra has Krull dimension 1 if the algebra is of finite type and has Krull dimension 2 if the algebra is of tame type.

Tame quivers and affine enveloping algebras

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.

The GR-segments for tame quivers

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 module…

用于驯顺真正主级数与局部志村对应的 Hecke 代数

Hecke algebras for tame genuine principal series and local Shimura correspondence

该研究扩展局部志村对应，利用 Bushnell–Kutzko 类型理论与线性代数群方法刻画真正主级数分量的 Hecke 代数并构造类型，同时以反例说明其与主内窥镜群对应代数未必同构。

On moduli spaces for quasi-tilted algebras

We prove that if a quasi-tilted algebra is tame, then the associated moduli spaces are products of projective spaces. Together with an earlier result of Chindris this gives a geometric characterization of the tame quasi-tilted algebras.

Hecke algebras for tame supercuspidal types

Let $F$ be a non-archimedean local field of residue characteristic $p\neq 2$. Let $G$ be a connected reductive group over $F$ that splits over a tamely ramified extension of $F$. Yu constructed types which are called tame supercuspidal types and conjectured that Hecke algebras associated with these types are isomorphic to Hecke algebras associated with depth-zero types of some twisted Levi subgroups of $G$. In this paper, we prove this conjecture. We also prove that the Hecke algebra associated with a regular supercuspidal type is isomorphic to the group algebra of a certain abelian group.

Non-commutative nodal curves and derived tame algebras

In this paper, we develop a geometric approach to study derived tame finite dimensional associative algebras, based on the theory of non-commutative nodal curves.

Graded tame blocks of group algebras

In this paper we investigate gradings on tame blocks of group algebras whose defect group is dihedral. We classify gradings on an arbitrary dihedral block up to graded Morita equivalence. We do this by computing the group of outer automorphisms that fix the isomorphism classes of simple modules. We also show how to grade these blocks via transfer of gradings via dervied equivalences.

Integral bases of cluster algebras and representations of tame quivers

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.

On minimal disjoint degenerations of modules over tame path algebras

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.

关于驯顺型cluster-tilted代数上不可分解$τ$-刚性模的研究

On indecomposable $τ$-rigid modules over cluster-tilted algebras of tame type

本文证明了驯顺型cluster-tilted代数上不同的不可分解$τ$-刚性模具有不同的维数向量，并以此为基础建立了驯顺型cluster algebras的弱版本分母猜想，同时利用Iyama-Yoshino构造描述了相关的三角范畴subfactors。

完全温和代数与完全特殊双列代数是 $g$-驯顺的

Complete gentle and special biserial algebras are $g$-tame

本文研究了完全 gentle 代数的 $g$-向量锥扇，利用曲面模型和 Dehn 扭转下的渐近行为证明了其 $g$-驯顺性，并由此推出完全 special biserial 代数也是 $g$-驯顺的。

Is tame open?

Is tame open? No answer so far. One may pose the Tame-Open Conjecture: Tame is open. But how to support it? No effective way to date. In this note, the rank of a wild algebra is introduced. The Wild-Rank Conjecture, which implies the Tame-Open Conjecture, is formulated. The Wild-Rank Conjecture is improved to the Basic-Wild-Rank Conjecture. A covering criterion on the rank of a basic wild algebra is given, which can be effectively applied to verify the Basic-Wild-Rank Conjecture for concrete algebras. It makes all conjectures much reliable.

Rigid and Schurian modules over cluster-tilted algebras of tame type

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…

周期为四的 tame 对称代数

Tame symmetric algebras of period four with small Gabriel quivers

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。

Virtual mutations of weighted surface algebras

The finite-dimensional symmetric algebras over an algebraically closed field, based on surface triangulations, motivated by the theory of cluster algebras, have been extensively investigated and applied. In particular, the weighted surface algebras and their deformations were introduced and studied in [16]-[20], and it was shown that all these algebras, except few singular cases, are symmetric tame periodic algebras of period $4$. In this article, using the general form of a weighted surface algebra from [19], we introduce and study so called virtual mutations of weighted surface algebras, whi…

One class of wild but brick-tame matrix problems

We present a class of wild matrix problems (representations of boxes), which are "brick-tame," i.e. only have one-parameter families of \emph{bricks} (representations with trivial endomorphism algebra). This class includes several boxes that arise in study of simple vector bundles on degenerations of elliptic curves, as well as those arising from the coadjoint action of some linear groups.

Socle equivalences of weighted surface algebras

We describe the structure and properties of the finite-dimensional symmetric algebras over an algebraically closed field $K$ which are socle equivalent to the general weighted surface algebras of triangulated surfaces, investigated in Erdmann and Skowroński (J. Algebra 544:170-227, 2020). In particular, we prove that all these algebras are tame periodic algebras of period 4. The main results of this paper form an essential step towards a classification of all symmetric tame periodic algebras of period 4.

Lengths of maximal green sequences for tame path algebras

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.

Geometric characterizations of the representation type of hereditary algebras and of canonical algebras

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…

Equivalences of tame blocks for p-adic linear groups

Let p and $\ell$ be two distinct primes, F a p-adic field and n an integer. We show that any level 0 block of the category of smooth Z $\ell$-valued representations of GL n (F) is equivalent to the unipotent block of an appropriate product of GL n i (F i). To this end, we first show that the level 0 category of GL n (F) is equivalent to a category of " modules " over a certain Z $\ell$-algebra " with many objects " whose definition only involves n and the residue field of F. Then we use fine properties of Deligne-Lusztig cohomology to split this algebra and produce suitable Morita equivalences…

表示驯服的Geiß-Leclerc-Schröer代数及关于根系的一个修正的GLS猜想

Representation-tame Geiß-Leclerc-Schröer algebras and a revised GLS conjecture on root systems

本文分类了表示驯服的GLS代数，研究其最小对称化子情形，通过扩展弦带描述AR箭图，并修正了仿射型GLS猜想，证明秩向量对应正根及非根向量。