Representation Theory Of Finite Dimensional Algebras Research Articles

On the vanishing of the theta invariant and a conjecture of Huneke and Wiegand

Huneke and Wiegand conjectured that, if $M$ is a finitely generated, non-free, torsion-free module with rank over a one-dimensional Cohen-Macaulay local ring $R$, then the tensor product of $M$ with its algebraic dual has torsion. This conjecture, if $R$ is Gorenstein, is a special case of a celebrated conjecture of Auslander and Reiten on the vanishing of self extensions that stems from the representation theory of finite-dimensional algebras. If $R$ is a one-dimensional Cohen-Macaulay ring such that $R=S/(f)$ for some local ring $(S, \mathfrak{n})$, and a non zero-divisor $f \in \mathfrak{n}^2$ on $S$, we make use of Hochster's theta invariant and prove that such $R$-modules $M$ which have finite projective dimension over $S$ satisfy the proposed torsion condition of the conjecture. Along the way we give several applications of our argument pertaining to torsion properties of tensor products of modules.

Semibricks

AbstractIn representation theory of finite-dimensional algebras, (semi)bricks are a generalization of (semi)simple modules, and they have long been studied. The aim of this paper is to study semibricks from the point of view of $\tau $-tilting theory. We construct canonical bijections between the set of support $\tau $-tilting modules, the set of semibricks satisfying a certain finiteness condition, and the set of 2-term simple-minded collections. In particular, we unify Koenig–Yang bijections and Ingalls–Thomas bijections generalized by Marks–Št’ovíček, which involve several important notions in the derived categories and the module categories. We also investigate connections between our results and two kinds of reduction theorems of $\tau $-rigid modules by Jasso and Eisele–Janssens–Raedschelders. Moreover, we study semibricks over Nakayama algebras and tilted algebras in detail as examples.

Higher-dimensional cluster combinatorics and representation theory

Higher Auslander algebras were introduced by Iyama generalizing classical concepts from representation theory of finite-dimensional algebras. Recently these higher analogues of classical representation theory have been increasingly studied. Cyclic polytopes are classical objects of study in convex geometry. In particular, their triangulations have been studied with a view towards generalizing the rich combinatorial structure of triangulations of polygons. In this paper, we demonstrate a connection between these two seemingly unrelated subjects. We study triangulations of even-dimensional cyclic polytopes and tilting modules for higher Auslander algebras of linearly oriented type $A$ which are summands of the cluster tilting module. We show that such tilting modules correspond bijectively to triangulations. Moreover mutations of tilting modules correspond to bistellar flips of triangulations. For any $d$-representation finite algebra we introduce a certain $d$-dimensional cluster category and study its cluster tilting objects. For higher Auslander algebras of linearly oriented type $A$ we obtain a similar correspondence between cluster tilting objects and triangulations of a certain cyclic polytope. Finally we study certain functions on generalized laminations in cyclic polytopes, and show that they satisfy analogues of tropical cluster exchange relations. Moreover we observe that the terms of these exchange relations are closely related to the terms occurring in the mutation of cluster tilting objects.

Cluster-additive functions on stable translation quivers

Additive functions on translation quivers have played an important role in the representation theory of finite-dimensional algebras, the most prominent ones are the hammock functions introduced by S. Brenner. When dealing with cluster categories (and cluster-tilted algebras), one should look at a corresponding class of functions defined on stable translation quivers, namely the cluster-additive ones. We conjecture that the cluster-additive functions on a stable translation quiver of Dynkin type $\mathbb{A}_{n}, \mathbb{D}_{n}, \mathbb{E}_{6}, \mathbb {E}_{7}, \mathbb{E}_{8}$ are non-negative linear combinations of cluster-hammock functions (with index set a tilting set). The present paper provides a first study of cluster-additive functions and gives a proof of the conjecture in the case $\mathbb{A}_{n}$ .

On the centre of a triangulated category

AbstractWe discuss some basic properties of the graded centre of a triangulated category and compute examples arising in representation theory of finite-dimensional algebras.

The structure of AS-Gorenstein algebras

In this paper, we define a notion of AS-Gorenstein algebra for N-graded algebras, and show that symmetric AS-regular algebras of Gorenstein parameter 1 are exactly preprojective algebras of quasi-Fano algebras. This result can be compared with the fact that symmetric graded Frobenius algebras of Gorenstein parameter −1 are exactly trivial extensions of finite-dimensional algebras. The results of this paper suggest that there is a strong interaction between classification problems in noncommutative algebraic geometry and those in representation theory of finite-dimensional algebras.

Weighted locally gentle quivers and Cartan matrices

We introduce and study the class of weighted locally gentle quivers. This naturally extends the class of gentle quivers and gentle algebras, which have been intensively studied in the representation theory of finite-dimensional algebras, to a wider class of potentially infinite-dimensional algebras. Weights on the arrows of these quivers lead to gradings on the corresponding algebras. For natural grading by path lengths, any locally gentle algebra is Koszul. The class of locally gentle algebras consists of the gentle algebras together with their Koszul duals. Our main result is a general combinatorial formula for the determinant of the weighted Cartan matrix of a weighted locally gentle quiver. We show that this weighted Cartan determinant is a rational function which is completely determined by the combinatorics of the quiver–more precisely by the number and the weight of certain oriented cycles.

Representation Theory of Finite-Dimensional Algebras

Representation Theory of Finite-Dimensional Algebras

Quadratic bimodules and quadratic orders

In some branches of mathematics, especially the representation theory of Artin algebras and orders, it has been an important problem to study the (geometric) quotient GLn(Λ)\Mn(M) for a (Λ,Λ)-module M (g · m := gmg−1 for g ∈ GLn(Λ) and m ∈ Mn(M)). An effective formulation of such problems is a bimodule problem [4,9], which is a study of the matrix category Mat(C,M) associated to an additive category C and a (C,C)-module M (Section 2.1). The bimodule problems have nice module-theoretical interpretation by means of prinjective modules [22,29,30]. They have already many useful applications in the representation theory of finite-dimensional algebras, artinian rings, orders, vector space categories, and bocses, see [6,7,9,16,19,25,28,29]. In this paper, we introduce the concept of quadratic extensions of rings (Section 1.2), which are closely related to many rings known in representation theory, for example Bass orders [10,17], Backstrom orders [28], Green orders [27], special biserial algebras [31], and clannish algebras [5] (see Sections 1.3, 3.4, and 3.5). Using quadratic extensions, we introduce the concept of quadratic bimodules (Section 3.1), where we can easily formulate self-reproducing systems [21] and clans [5] in terms of quadratic bimodules. Then we study their matrix category Mat(C,M) associated to a quadratic (C,C)-module M , especially we give a description of all indecomposable objects in terms of walks (Sections 3.3, 6.1), which can be regarded as a generalization of Green walks [13] (Section 3.4), strings and bands [5,31] (Section 3.5). As an application, we give a description of finitely generated indecomposable modules over quadratic orders (Section 1.3(2)) in Section 7.3. In forthcoming papers, our results will be applied in construction of large new classes of orders of tame representation type. Some of the results of this paper were presented at the Conference and Workshop Constanta 2000 and were announced in [19] without proof.

Cellular Algebras: Inflations and Morita Equivalences

Cellular algebras have recently been introduced by Graham and Lehrer [5, 6] as a convenient axiomatization of all of the following algebras, each of them containing information on certain classical algebraic or finite groups: group algebras of symmetric groups in any characteristic, Hecke algebras of type A or B (or more generally, Ariki Koike algebras), Brauer algebras, Temperley–Lieb algebras, (q-)Schur algebras, and so on. The problem of determining a parameter set for, or even constructing bases of simple modules, is in this way reduced (but of course not solved in general) to questions of linear algebra. The present paper has two aims. First, we make explicit an inductive construction of cellular algebras which has as input data of linear algebra, and which in fact produces all cellular algebras (but no other ones). This is what we call ‘inflation’. This construction also exhibits close relations between several of the above algebras, as can be seen from the computations in [6]. Among the consequences of the construction is a natural way of generalizing Hochschild cohomology. Another consequence is the construction of certain idempotents which is used in the second part of the paper. The second aim is to study Morita equivalences of cellular algebras. Since the input of many of the constructions of representation theory of finite-dimensional algebras is a basic algebra, it is useful to know whether any finite-dimensional cellular algebra is Morita equivalent to a basic one by a Morita equivalence that preserves the cellular structure. It turns out that the answer is ‘yes’ if the underlying field has characteristic other than 2. However, there are counterexamples in the case of characteristic 2, or more generally for any ring in which 2 is not invertible. This also tells us that the notion of ‘cellular’ cannot be defined only in terms of the module category. However, in any characteristic we find some useful Morita equivalences which are compatible with cellular structures.

Algorithms for weakly nonnegative quadratic forms

Let q: Z n → Z with q( v) = ∑ n i = 1 v( t) 2 + ∑ i < j a ij v( i) v( j) be a unit form. We present an algorithm that allows one to check if q is weakly nonnegative [i.e., q( v) ⩾ 0 for any vector v ∈ N n ]. The algorithm also calculates the set of critical vectors of q. We sketch the relation of this problem to the representation theory of finite-dimensional algebras.

One point extensions of linear quivers and quadratic forms

In representation theory of finite-dimensional algebras over a field k, the idea of associating an integral quadratic form qA to a given algebra A, in order to study the relation between the representation type of A and the sign of qA in the positive integral cone, has its origins in a fundamental paper of Gabriel [4]. In the present paper we examine, for any n E M, one-point extensions of the form