Viktor I. Bekkert, Yuriy A. Drozd
We prove that every finite dimensional algebra over an algebraically closed field is either derived tame or derived wild. The proof is based on the technique of matrix problems (boxes and reduction algorithm). It implies, in particular, that any degeneration of a derived wild algebra is derived wild; respectively, any deformation of a derived tame algebra is derived tame.
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We prove that every finite dimensional algebra over an algebraically closed field is either derived tame or derived wild. We also prove that any deformation of a derived tame algebra is derived tame.
We provide a complete classification of all tame and wild symmetric special multiserial algebras in terms of the underlying Brauer configuration. Our classification contains the symmetric special multiserial algebras of finite representation type.
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.
In this paper, we consider the problem of determining when two tensor networks are equivalent under a heterogeneous change of basis. In particular, to a string diagram in a certain monoidal category (which we call tensor diagrams), we formulate an associated abelian category of representations. Each representation corresponds to a tensor network on that diagram. We then classify which tensor diagrams give rise to categories that are finite, tame, or wild in the traditional sense of representation theory. For those tensor diagrams of finite and tame type, we classify the indecomposable represen…
Rational quiver representations: tame and wild
该研究针对特征0域上的$K$-有理箭图，利用其Brauer障碍平凡的性质开展分析，并构造出野表示型箭图的实例，实现了给定伽罗瓦扩张的全部Brauer障碍，补充了有理箭图表示理论的相关内容。
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".
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.
Tame and wild theorem for the category of filtered by standard modules
本文引入交错弱ditalgebra并利用约化程序，证明了任意有限同调系统下Δ-滤过模范畴的驯服-野性二分性，并在驯服情形下刻画了模与其Auslander-Reiten平移的重合条件。
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 …
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…
$τ$-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。
Entropy of cluster DT transformations and the finite-tame-wild trichotomy of acyclic quivers
本文以簇 DT 变换 τ 的符号稳定性和簇拉伸因子刻画无环箭图的有限–驯–野三分法，并计算相关变异环熵，证明其代数与范畴熵由 Coxeter 矩阵谱半径对数统一给出。
We give a criterion of tameness and wildness for a finite-dimensional Lie algebra over an algebraically closed field.
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.
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…
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 …
We prove that the number of parameters defining a complex of projective modules over a finite dimensional algebra is upper semi-continuous in families of algebras. Supposing that every algebra is either derived tame or derived wild, we get that a degeneration of a derived wild algebra is derived wild too. We also explain why the so-called counter-example of Bruestle is in fact not a counter-example.
Let $Λ$ be a finite-dimensional algebra over an algebraically closed field, then $Λ$ is either tame or wild. Is there any homological description in terms of AR-translations on tameness? Or equivalently, is there any combinatorial description in terms of AR-quivers? The answer is yes. In the present paper we prove the following main theorem: ``$Λ$ is tame if and only if almost all modules are isomorphic to their Auslander-Reiten translations, if and only if they lie in homogeneous tubes". The method used in the paper is bimodule problem, which has been studied by several authors. We treat bimo…
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.…
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.
For the algebra $I_n$ of polynomial integro-differential operators over a field $K$ of characteristic zero, a classification of simple weight and generalized weight (left and right) $I_n$-modules is given. It is proven that the category of weight $I_n$-modules is semisimple. An explicit description of generalized weight $I_n$-modules is given and using it a criterion is obtained for the problem of classification of indecomposable generalized weight $I_n$-modules to be of finite representation type, tame or wild. In the tame case, a classification of indecomposable generalized weight $I_n$-modu…
As the first attempt to classify $τ$-tilting finite two-point algebras, we have determined the $τ$-tilting finiteness for minimal wild two-point algebras and some tame two-point algebras.
Let k be an algebraically closed field of characteristic p> 0. For a basic classical Lie superalgebra, we show its restricted cohomology algebra is a finitely generated algebra. Thus the cohomological support theory can be established. As a consequence, we show that the restricted enveloping algebra of a basic classical Lie superalgebra g is always wild except g=sl(2) or g=osp(1|2) or g=C(2). Moreover, all finite dimensional indecomposable restricted representations of u(osp(1|2)), the restricted enveloping algebra of Lie superalgebra osp(1|2), are determined.
We consider a class of finite-dimensional algebras, the so-called "Staircase algebras" parametrized by Young diagrams. We develop a complete classification of representation types of these algebras and look into finite, tame (concealed) and wild cases in more detail. Our results are translated to the setup of graded nilpotent pairs for which we prove certain finiteness conditions.
We establish representation types (finite, tame or wild) of finite dimensional Munn algebras with semisimple bases. As an application, we establish representation types of finite 0-simple semigroups and their mutually annihilating unions.
On the representation type of higher preprojective algebras of type $A$
本文研究A类高阶预投射代数的表示类型，确定其表示类型的有限-驯服-野三分法，完成τ-倾斜有限性分类，刻画不可分解模为τ-刚性的条件。
Let k be a field, let A a finite-dimensional hereditary k-algebra. We consider the category of all finite-dimensional A-modules. We are going to characterize the representation type of A (tame or wild) in terms of the possible subcategories of all M-static modules, where M is an indecomposable A-module.
We establish a number of results which say, roughly, that interpretation functors preserve algebraic complexity. First we show that representation embeddings between categories of modules of finite-dimensional algebras induce embeddings of lattices of pp formulas and hence are non-decreasing on Krull-Gabriel dimension and uniserial dimension. A consequence is that the category of modules of any wild finite-dimensional algebra has width $\infty$ and hence, if the algebra is countable, there is a superdecomposable pure-injective representation. It is conjectured that a stronger result is true: t…
A generic classification of locally free representations of affine GLS algebras
本文研究仿射 GLS 代数的局部自由表示，通过广义一点扩张构造稳定表示单参数族，并折叠路代数的 Kac 典范分解，建立一般分类并证明其 $E$-驯顺。
Representation Type and Schurian Finiteness of Tits Algebras
该研究确定了代数闭域上有限中心实超平面排列的蒂茨代数的表示类型与舒尔有限性，给出了不同超平面数量对应的分类结果，并通过多种代数方法完成证明，还构造了相关模的嵌入。
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.
A tensor-triangular property for categories of representations of restricted Lie algebras
本文基于上同调支撑与张量三角几何为限制李代数定义了一种新性质，探讨了群概结构变化下极小根式厚张量理想的 Green 环结构，并猜想该性质等价于具有有限或驯顺表示型。
We survey the development and status quo of a subject best described as "generic representation theory of finite dimensional algebras", which started taking shape in the early 1980s. Let $Λ$ be a finite dimensional algebra over an algebraically closed field. Roughly, the theory aims at (a) pinning down the irreducible components of the standard parametrizing varieties for the $Λ$-modules with a fixed dimension vector, and (b) assembling generic information on the modules in each individual component, that is, assembling data shared by all modules in a dense open subset of that component. We pr…
$τ$-Tilting finiteness of group algebras of semidirect products of abelian $p$-groups and abelian $p'$-groups
本文研究群代数的$τ$-倾斜有限性的控制子群问题，证明在特定假设下有限群群代数的该性质由$p$-超聚焦子群控制，并验证了阿贝尔$p$-群与阿贝尔$p'$-群半直积的群代数的相关结论。
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