Algebras Of Finite Representation Type Research Articles

Positive cluster complexes and τ-tilting simplicial complexes of cluster-tilted algebras of finite type

In this study, we consider the positive cluster complex, a full subcomplex of a cluster complex the vertices of which are all non-initial cluster variables. In particular, we provide a formula for the difference in face vectors of positive cluster complexes caused by a mutation for finite type. Moreover, we explicitly describe specific positive cluster complexes of finite type and calculate their face vectors. We also provide a method to compute the face vector of an arbitrary positive cluster complex of finite type using these results. Furthermore, we apply our results to the -tilting theory of cluster-tilted algebras of finite representation type using the correspondence between clusters and support -tilting modules.

A Simple Homological Characterization of String Algebras of Finite Representation Type

We prove that among the finite-dimensional algebras of finite representation type those that are string algebras are precisely the ones that have the property that the middle term of an arbitrary extension of indecomposable modules has at most two direct factors. On the other hand, we show that non-domestic string algebras are very far from having that property.

McKay matrix for indecomposable module of finite representation type Hopf algebra

Let H be a Hopf algebra of finite representation type. For the grouplike elements and skew-primitive elements in H, we prove some general results about the eigenvalues and eigenvectors of the McKay matrix in the framework of Green ring. As an example, we investigate the McKay matrix WV of Taft algebra for tensoring with an indecomposable -module Furthermore, we compute the characteristic polynomial of WV in terms of a kind of generalized Fibonacci polynomial, and give some eigenvector(s) of each eigenvalue for WV by relating to the generalized Fibonacci polynomial. At last, we determine the eigenspaces of all eigenvalues.

From subcategories to the entire module categories

Abstract In this paper we show that how the representation theory of subcategories (of the category of modules over an Artin algebra) can be connected to the representation theory of all modules over some algebra. The subcategories dealing with are some certain subcategories of the morphism categories (including submodule categories studied recently by Ringel and Schmidmeier) and of the Gorenstein projective modules over (relative) stable Auslander algebras. These two kinds of subcategories, as will be seen, are closely related to each other. To make such a connection, we will define a functor from each type of the subcategories to the category of modules over some Artin algebra. It is shown that to compute the almost split sequences in the subcategories it is enough to do the computation with help of the corresponding functors in the category of modules over some Artin algebra which is known and easier to work. Then as an application the most part of Auslander–Reiten quiver of the subcategories is obtained only by the Auslander–Reiten quiver of an appropriate algebra and next adding the remaining vertices and arrows in an obvious way. As a special case, when Λ is a Gorenstein Artin algebra of finite representation type, then the subcategories of Gorenstein projective modules over the 2 × 2 {2\times 2} lower triangular matrix algebra over Λ and the stable Auslander algebra of Λ can be estimated by the category of modules over the stable Cohen–Macaulay Auslander algebra of Λ.

Representation Type of Borel-Schur Algebras

In our previous work (Erdmann et al. J. Algebra Appl. 17(2),1850028, 2018), we found all Borel-Schur algebras of finite representation type. In the present article, we determine which Borel-Schur algebras of infinite representation type are tame, and which are wild.

On the First Hochschild Cohomology of Cocommutative Hopf Algebras of Finite Representation Type

Abstract Let $\mathscr{B}_0({\mathcal{G}})\subseteq k\,{\mathcal{G}}$ be the principal block algebra of the group algebra $k\,{\mathcal{G}}$ of an infinitesimal group scheme ${\mathcal{G}}$ over an algebraically closed field $k$ of characteristic ${\operatorname{char}}(k)=:p\geq 3$. We calculate the restricted Lie algebra structure of the first Hochschild cohomology ${\mathcal{L}}:={\operatorname{H}}^1(\mathscr{B}_0({\mathcal{G}}),\mathscr{B}_0({\mathcal{G}}))$ whenever $\mathscr{B}_0({\mathcal{G}})$ has finite representation type. As a consequence, we prove that the complexity of the trivial ${\mathcal{G}}$-module $k$ coincides with the maximal toral rank of ${\mathcal{L}}$.

Integer sequences arising from Auslander–Reiten quivers of some hereditary artin algebras

Categorification of some integer sequences are obtained by enumerating the number of sections in the Auslander–Reiten quiver of algebras of finite representation type.

Representation type of the little q-Schur algebras

The classification of little q-Schur algebras of finite representation type at the odd roots of unity case was given in [15]. In this paper, we give the complete classification of representation type of little q-Schur algebras in the any roots of unity case.

The Nakayama automorphism of a self‐injective preprojective algebra

We give a simple proof, using Auslander-Reiten theory, that the preprojective algebra of a basic hereditary algebra of finite representation type is Frobenius. We then describe its Nakayama automorphism, which is induced by the Nakayama functor on the module category of our hereditary algebra.

A categorification of biclosed sets of strings

We consider a closure space known as biclosed sets of strings of a gentle algebra of finite representation type. Palu, Pilaud, and Plamondon proved that the collection of all biclosed sets of strings forms a lattice, and moreover, that this lattice is congruence-uniform. Many interesting examples of finite congruence-uniform lattices may be represented as the lattice of torsion classes of an associative algebra. We introduce a generalization, the lattice of torsion shadows, and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows when every indecomposable module over the gentle algebra is a brick.Finite congruence-uniform lattices admit an alternate partial order known as the core label order. In many cases, the core label order of a congruence-uniform lattice is isomorphic to a lattice of wide subcategories of an associative algebra. Analogous to torsion shadows, we introduce wide shadows, and prove that the core label order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows.

On the representation dimension and finitistic dimension of special multiserial algebras

For monomial special multiserial algebras, which in general are of wild representation type, we construct radical embeddings into algebras of finite representation type. As a consequence, we show that the representation dimension of monomial and self-injective special multiserial algebras is less or equal to three. This implies that the finitistic dimension conjecture holds for all special multiserial algebras.

Algebras with finitely many conjugacy classes of left ideals versus algebras of finite representation type

Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple A-modules have dimension at least 6.

Maximal green sequences for cluster-tilted algebras of finite representation type

We show that, for any cluster-tilted algebra of finite representation type over an algebraically closed field, the following three definitions of a maximal green sequence are equivalent: (1) the usual definition in terms of Fomin–Zelevinsky mutation of the extended exchange matrix, (2) a complete forward hom-orthogonal sequence of Schurian modules, (3) the sequence of wall crossings of a generic green path. Together with [24], this completes the foundational work needed to support the author’s work with P. J. Apruzzese [1], namely, to determine all lengths of all maximal green sequences for all quivers whose underlying graph is an oriented or unoriented cycle and to determine which are “linear”.

Bilinear Forms on the Green Rings of Finite Dimensional Hopf Algebras

In this paper, we study the Green ring and the stable Green ring of a finite dimensional Hopf algebra by means of bilinear forms. We show that the Green ring of a Hopf algebra of finite representation type is a Frobenius algebra over $\mathbb {Z}$ with a dual basis associated to almost split sequences. On the stable Green ring we define a new bilinear form which is more accurate to determine the bi-Frobenius algebra structure on the stable Green ring. We show that the complexified stable Green algebra is a group-like algebra, and hence a bi-Frobenius algebra, if the bilinear form on the stable Green ring is non-degenerate.

Representation type for block algebras of Hecke algebras of classical type

We find representation type of the cyclotomic quiver Hecke algebras RΛ(β) of level two in affine type A. In particular, we have determined representation type for all the block algebras of Hecke algebras of classical type (except for characteristic two in type D), which has not been known for a long time. As an application of this result, we prove that block algebras of finite representation type are Brauer tree algebras whose Brauer trees are straight lines without exceptional vertex if the Hecke algebras are of classical type in good characteristic. We conjecture that this statement should hold for Hecke algebras of exceptional type.

Desingularization of Quiver Grassmannians via Nakajima Categories

In this paper, we show that generalized Nakajima Categories provide a framework to construct a desingularization of quiver Grassmannians for self-injective algebras of finite representation type. Furthermore, we show that all standard Frobenius models of orbit categories of the bounded derived category considered in Keller, Documenta Math. 10: 551–581, 2005 are equivalent to proj 𝓡, the finitely generated projective modules of the regular Nakajima category 𝓡.

Algebras of finite representation type arising from maximal rigid objects

We give a complete classification of all algebras appearing as endomorphism algebras of maximal rigid objects in standard 2-Calabi–Yau categories of finite type. Such categories are equivalent to certain orbit categories of derived categories of Dynkin algebras. It turns out that with one exception, all the algebras that occur are 2-Calabi–Yau-tilted, and therefore appear in an earlier classification by Bertani-Økland and Oppermann. We explain this phenomenon by investigating the subcategories generated by rigid objects in standard 2-Calabi–Yau categories of finite type.

Degrees in Auslander–Reiten components with almost split sequences of at most two middle termss

We consider A to be an artin algebra. We study the degrees of irreducible morphisms between modules in Auslander–Reiten components Γ having only almost split sequences with at most two indecomposable middle terms, that is, α(Γ) ≤ 2. We prove that if f : X → Y is an irreducible epimorphism of finite left degree with X or Y indecomposable, then there exists a module Z ∈ Γ and a morphism φ ∈ ℜn(Z, X)\ℜn+1(Z, X) for some positive integer n such that fφ = 0. In particular, for such components if A is a finite dimensional algebra over an algebraically closed field and f = (f1, f2)t : X → Y1 ⊕ Y2 is an irreducible epimorphism of finite left degree then we show that dl(f) = dl(f1) + dl(f2). We also characterize the artin algebras of finite representation type with α(ΓA) ≤ 2 in terms of a finite number of irreducible morphisms with finite degree.

On selfinjective algebras of finite representation type with maximal almost split sequences

We investigate the structure of finite dimensional selfinjective algebras of finite representation type over an arbitrary field having almost split sequences of modules whose middle term admits the maximal possible number of indecomposable direct summands.

Stable Calabi–Yau dimension of self-injective algebras of finite type

We give an equivalent definition of the stable Calabi–Yau dimension in terms of bimodule syzygies and so-called stably inner automorphisms. Using it, we complete the computation of the stable Calabi–Yau dimensions of the self-injective algebras of finite representation type which was started by K. Erdmann, A. Skowroński, J. Białkowski and A. Dugas.