TAME TWO-POINT ALGEBRAS WITHOUT LOOPS

We determine the representation type of the algebras whose quiver has precisely two vertices and admits no loops by listing all minimal wild algebras of this form. It turns out that such an algebra A is tame if and only if A/rad³ A is tame, and in this case A degenerates to a special biserial algebra. Moreover, A is wild if and only if it is controlled wild.     

The Simple Connectedness of Tame Algebras with Separating Almost Cyclic Coherent Auslander–Reiten Components

We study the simple connectedness of the class of finite-dimensional algebras over an algebraically closed field for which the Auslander–Reiten quiver admits a separating family of almost cyclic coherent components. We show that a tame algebra in this class is simply connected if and only if its first Hochschild cohomology space vanishes.

     

Exceptional Components

For a wild acyclic quiver Q, Kerner introduced the notion of exceptional components for the Auslander–Reiten quiver of Q over an algebraically closed field k. He then defined two invariants for these exceptional components and asked whether these invariants coincide for each exceptional component. He showed that for each exceptional component there is a related hereditary factor algebra B of the path algebra kQ. He then proved that B is tame or representation finite and asked whether the representation finite case does occur, at all. We will answer both of Kerner's questions.     

THE REGULAR COMPONENTS OF THE AUSLANDER-REITEN QUIVER OF A TILTED ALGEBRA

Let B be a connected finite-dimensional hereditary algebra of infinite representationtype.It is shown that there exists a regular tilting B-module if and only if B is wild andhas at least three simple modules.In this way,the author determines the possible form ofregular components which arise as a connecting component of the Auslander-Reitenquiver Γ(A)of a tilted algebra A.The second result asserts that for a tilted algebra A,any regular component of Γ(A)which is not a connecting component,is quasi-serial.     

Localization in tame and wild coalgebras

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”.     

Hochschild cohomology of truncated quiver algebras

For a truncated quiver algebra over a field of an arbitrary characteristic, its Hochschild cohomology is calculated. Moreover, it is shown that its Hochschild cohomology algebra is finitedimensional if and only if its global dimension is finite if and only if its quiver has no oriented cycles.     

一个代数的预投射分支和τ-预投射分支

设A是一个Artin代数,Γ_A是A的Auslander-Reiten箭图。我们得到:如果Γ是Γ_A的一个不包含有向循环的预投射分支,那么Γ是Γ_A的一个τ-预投射分支,且对一个拟倾斜代数A,Γ是Γ_A的一个预投射分支当且仅当Γ是Γ_A的一个,τ-预投射分支。     

An algorithm for finding all preprojective components of the Auslander-Reiten quiver

The Auslander-Reiten quiver of a finite-dimensional associative algebra encodes information about the indecomposable finite-dimensional representations of and their homomorphisms. A component of the Auslander-Reiten quiver is called preprojective if it does not admit oriented cycles and each of its modules can be shifted into a projective module using the Auslander-Reiten translation. Preprojective components play an important role in the present research on algebras of finite and tame representation type. We present an algorithm which detects all preprojective components of a given algebra.

     

The number of the Gabriel-Roiter measures admitting no direct predecessors over a wild quiver

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 vertices, labeled by 1, 2 and 3, and one arrow from 1 to 2 and two arrows from 2 to 3. The Gabriel-Roiter submodules of the indecomposable preprojective modules and quasi-simple modules τ⁻ⁱM, i≥0 are described, where M is a Kronecker module and τ=DTr is the Auslander-Reiten translation. Based on these calculations, the existence of infinitely many GR segments will be shown. Moreover, it will be proved that there are infinitely many Gabriel-Roiter measures admitting no direct predecessors.     

Strictly wild algebras with radical square zero

It is proved that a wild algebra with radical square zero is strictly wild if and only if it has a wild hereditary algebra as its factor algebra, which, on the one hand supports one conjecture in [3], on the other hand can be used to construct many non-strictly wild algebras.     

对偶扩张代数的Frobenius态射和固定点代数

设A是由箭图Q和关系I所确定的代数,D(A)是代数A的对偶扩张代数, 对应的箭图Q*和关系I*由Q和I决定．本文证明:带关系箭图(Q*,I*)的自同构由带关系箭图(Q,I)的自同构决定;D(A)的Frobenius态射由A的Frobenius态射完全决定;代数D(A)的固定点代数同构于相应的代数A的固定点代数与A°P的固定点代数的张量积,特别地,当Q为单的箭图时,代数D(A)的固定点代数同构于代数A的固定点代数的对偶扩张代数．     

Pairwise compatibility for 2-simple minded collections

In τ-tilting theory, it is often difficult to determine when a set of bricks forms a 2-simple minded collection. The aim of this paper is to determine when a set of bricks is contained in a 2-simple minded collection for a τ-tilting finite algebra. We begin by extending the definition of mutation from 2-simple minded collections to more general sets of bricks (which we call semibrick pairs). This gives us an algorithm to check if a semibrick pair is contained in a 2-simple minded collection. We then use this algorithm to show that the 2-simple minded collections of a τ-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) are given by pairwise compatibility conditions if and only if every vertex in the corresponding quiver has degree at most 2. As an application, we show that the classifying space of the τ-cluster morphism category of a τ-tilting finite gentle algebra (whose quiver contains no loops or 2-cycles) is an Eilenberg-MacLane space if every vertex in the corresponding quiver has degree at most 2.     

Minimal algebras of infinite representation type with preprojective component

Let A be a finite dimensional, basic and connected algebra (associative, with 1) over an algebraically closed field k. Denote by e₁,...,e_n a complete set of primitive orthogonal idempotents in A and by A_i= A/Ae_iA. A is called a minimal algebra of infinite representation type provided A is itself of infinite representation type,whereas all A_i, 1≤i≤n,are of finite representation type. The main result gives the classification of the minimal algebras having a preprojective component in their Auslander-Reiten quiver. The classification is obtained by realizing that these algebras are essentially given by preprojective tilting modules over tame hereditary algebras.     

Positivity and tameness in rank 2 cluster algebras

We study the relationship between the positivity property in a rank 2 cluster algebra, and the property of such an algebra to be tame. More precisely, we show that a rank 2 cluster algebra has a basis of indecomposable positive elements if and only if it is of finite or affine type. This statement disagrees with a conjecture by Fock and Goncharov.     

A New Characterization of Hereditary Algebras

In this short article, we prove that a finite dimensional algebra is hereditary if and only if there is no loop in its ordinary quiver and every τ-tilting module is tilting.     

用quiver构造拟三角Hopf代数

利用quiver方法确定了一个广义Taft代数具有拟三角Hopf结构当且仅当它是Sweedler 4维Hopf代数．用不同于文[15]的方法,对任意的正整数n,构造出一类拟三角Hopf代数H(n)．     

Minimal representation-infinite coil algebras

For a basic and connected finite dimensional algebra A over an algebraically closed field, we study when the cycles in the category mod A (of finite dimensional modules) are well-behaved. We call A cycle-finite if, for any cycle in mod A, no morphism on the cycle lies in the infinite power of the radical. We show that, in this case, A is tame. We also introduce a natural generalisation of a tube, called a coil, and define A to be a coil algebra if any cycle in mod A lies in a standard coil. We prove that the minimal representation-infinite coil algebras coincide with the tame concealed algebras.     

Stable equivalence preserves representation type

Given two finite dimensional algebras and , it is shown that is of wild representation type if and only if is of wild representation type provided that the stable categories of finite dimensional modules over and $\Gamma$ are equivalent. The proof uses generic modules. In fact, a stable equivalence induces a bijection between the isomorphism classes of generic modules over and , and the result follows from certain additional properties of this bijection. In the second part of this paper the Auslander-Reiten translation is extended to an operation on the category of all modules. It is shown that various finiteness conditions are preserved by this operation. Moreover, the Auslander-Reiten translation induces a homeomorphism between the set of non-projective and the set of non-injective points in the Ziegler spectrum. As a consequence one obtains that for an algebra of tame representation type every generic module remains fixed under the Auslander-Reiten translation. Received: July 24, 1996