Theory Of Finite-dimensional Algebras Research Articles

Mutations and Pointing for Brauer Tree Algebras

Brauer tree algebras are important and fundamental blocks in the representation theory of finite dimensional algebras. In this research, we present a combination of two main approaches to the tilting theory of Brauer tree algebras. The first approach is the theory initiated by Rickard, providing a direct link between an ordinary Brauer tree algebra and the Brauer star algebra. This approach was continued by Schaps-Zakay with their theory of pointing the tree. The second approach is the theory developed by Aihara, relating to the sequence of mutations from the ordinary Brauer tree algebra to the Brauer star algebra. Our main purpose in this research is to combine these two approaches. We first find an algorithm based on centers which are all terminal edges, for which we are able to obtain a tilting complex constructed from irreducible complexes of length two [13], which is obtained from a sequence of mutations. In [1], Aihara gave an algorithm for reducing from tree to star by mutations and showed that it gave a two-term tree-to-star complex. We prove that Aihara's complex is obtained from the corresponding completely folded Rickard tree-to-star complex by a permutation of projectives.

Semistable subcategories for tiling algebras

Semistable subcategories were introduced in the context of Mumford’s GIT and interpreted by King in terms of representation theory of finite dimensional algebras. Ingalls and Thomas later showed that for finite dimensional algebras of Dynkin and affine type, the poset of semistable subcategories is isomorphic to the corresponding poset of noncrossing partitions. We show that semistable subcategories defined by tiling algebras, introduced by Coelho Simoes and Parsons, are in bijection with noncrossing tree partitions, introduced by the second author and McConville. Moreover, this bijection defines an isomorphism of the posets on these objects. Our work recovers that of Ingalls and Thomas in Dynkin type A.

Support varieties\u2014an axiomatic approach

We provide an axiomatic approach for studying support varieties of objects in a triangulated category via the action of a tensor triangulated category, where the tensor product is not necessarily symmetric. This is illustrated by examples, taken in particular from the representation theory of finite dimensional algebras.

Borelic pairs for stratified algebras

We determine all values of the parameters for which the cell modules form a standard system, for a class of cellular diagram algebras including partition, Brauer, walled Brauer, Temperley–Lieb and Jones algebras. For this, we develop and apply a general theory of finite dimensional algebras with Borelic pairs. The theory is also applied to give new uniform proofs of the cellular and quasi-hereditary properties of the diagram algebras and to construct quasi-hereditary 1-covers, in the sense of Rouquier, with exact Borel subalgebras, in the sense of König. Another application of the theory leads to a proof that Auslander–Dlab–Ringel algebras admit exact Borel subalgebras.

Extensions in Jacobian algebras and cluster categories of marked surfaces

In the context of representation theory of finite dimensional algebras, string algebras have been extensively studied and most aspects of their representation theory are well-understood. One exception to this is the classification of extensions between indecomposable modules. In this paper we explicitly describe such extensions for a class of string algebras, namely gentle algebras associated to surface triangulations. These algebras arise as Jacobian algebras of unpunctured surfaces. We relate the extension spaces of indecomposable modules to crossings of generalised arcs in the surface and give explicit bases of the extension spaces for indecomposable modules in almost all cases. We show that the dimensions of these extension spaces are given in terms of crossing arcs in the surface.Our approach is new and consists of interpreting snake graphs as indecomposable modules. In order to show that our basis is a spanning set, we need to work in the associated cluster category where we explicitly calculate the middle terms of extensions and give bases of their extension spaces. We note that not all extensions in the cluster category give rise to extensions for the Jacobian algebra.

Infinite dimensional representations of finite dimensional algebras and amenability

We present a novel quantitative approach to the representation theory of finite dimensional algebras motivated by the emerging theory of graph limits. We introduce the rank spectrum of a finite dimensional algebra R over a countable field. The elements of the rank spectrum are representations of the algebra into von Neumann regular rank algebras, and two representations are considered to be equivalent if they induce the same Sylvester rank functions on R-matrices. Based on this approach, we can divide the finite dimensional algebras into three types: finite, amenable and non-amenable representation types. We prove that string algebras are of amenable representation type, but the wild Kronecker algebras are not. Here, the amenability of the rank algebras associated to the limit points in the rank spectrum plays a very important part. We also show that the limit points of finite dimensional representations of algebras of amenable representation type can always be viewed as representations of the algebra in the continuous ring invented by John von Neumann in the 1930’s. As an application in algorithm theory, we introduce and study the notion of parameter testing of modules over finite dimensional algebras, that is analogous to the testing of bounded degree graphs introduced by Goldreich and Ron. We shall see that for string algebras all the reasonable (stable) parameters are testable.

B-Construction and C-Construction

B-construction is a way of obtaining a graded algebra from the triple consisting of an additive category, an object, and an autoequivalence, while C-construction is a way of obtaining an algebra (without unity) from the pair consisting of an additive category and a set of objects. In this article, we study and compare three important classes of algebras in noncommutative algebraic geometry and representation theory of finite dimensional algebras, namely, quantum polynomial algebras, preprojetive algebras and trivial extensions, via these constructions.

Artin–Schelter regular algebras and categories

Motivated by constructions in the representation theory of finite dimensional algebras we generalize the notion of Artin–Schelter regular algebras of dimension n to algebras and categories to include Auslander algebras and a graded analogue for infinite representation type. A generalized Artin–Schelter regular algebra or a category of dimension n is shown to have common properties with the classical Artin–Schelter regular algebras. In particular, when they admit a duality, then they satisfy Serre duality formulas and the Ext -category of nice sets of simple objects of maximal projective dimension n is a finite length Frobenius category.

Representation Theory of Finite Dimensional Algebras

Representation theory of finite dimensional algebras has always been inspired by interactions with other subjects, and Oberwolfach meetings traditionally serve as a forum for such exchange of ideas. The main source of interactions are the many problems in representation theory and in other parts of mathematics which can be formulated in terms of representations of finite dimensional associative algebras. The study of non-semisimple representations took off in the late 20th century with key advances, such as the link to Lie algebras and quantum groups via quivers and Hall algebras, and the use of tilting theory and derived categories to pass from known algebras to new classes of algebras. In modern work, instead of studying an algebra through its category of representations, or derived category, one may study similar but more general categories. Thus the classification of some classes of hereditary abelian categories or Calabi-Yau triangulated categories fits into this setup. Another recent development, which had just started at the time of the last Oberwolfach meeting in February 2005, and is still being played out, is the interaction with cluster algebras. At the workshop, there were 46 participants. Among them, there were experts from neighbouring subjects like commutative algebra, algebraic topology, and combinatorics. Compared to previous meetings, the number of participants was reduced, which made it difficult to include representatives of many other fields with close links to representation theory of finite dimensional algebras. What follows is a quick survey of the main themes of the 23 lectures given at the meeting. \textbf{Cluster combinatorics and Calabi-Yau categories arising from representations of algebras.} Cluster algebras were invented by Fomin and Zelevinsky in 2000 with motivations coming from the study of canonical bases in quantum groups and total positivity in algebraic groups. The combinatorics of these algebras were soon recognized to be closely related to those of tilting theory for hereditary algebras. A collective effort over the last few years has led to a good understanding of these relations for certain classes of cluster algebras. This was made possible by the use of $2$-Calabi-Yau categories constructed from representations of algebras. The introductory talks by Reiten and Iyama were devoted to these developments as well as to the impact of recent important work by Derksen-Weyman-Zelevinsky. In an informal evening presentation, Keller put Derksen-Weyman-Zelevinsky's work into a beautiful homological framework. The talk by Geiss presented cutting-edge results towards the construction of \`dual PBW-bases' in large classes of cluster algebras. The proofs are based on subtle techniques from the study of quasi-hereditary algebras, as demonstrated in Schr\\"oer's talk. Marsh analyzed fine points of the correspondence between cluster variables and rigid indecomposables and disproved a recent conjecture by Fomin-Zelevinsky. A powerful representation-theoretic model for \`higher cluster combinatorics' was presented in the talk by Bin Zhu. \textbf{Categorification via representations.} The method of categorification has been developed and studied successfully in representation theory by Chuang and Rouquier. They constructed $sl\_2$-categorifications for blocks of symmetric groups and used them to establish Brou\\'e's abelian defect group conjecture for the symmetric groups. A similar philosophy led to the categorification of cluster algebras via certain $2$-Calabi-Yau categories, where the multiplication in the cluster algebra is modeled by direct sums. A more recent and very promising approach due to Leclerc was presented by Keller. In this case the multiplication is modeled by the tensor product in certain categories of representations of quantum affine algebras. Categorifications also play an important role in low dimensional topology, thanks to important work of Khovanov. This connection was the motivation for Stroppel's talk on convolution algebras arising from Springer fibres. \textbf{Representation dimension of algebras and complexity of triangulated categories.} The representation dimension of an algebra is a homological invariant which Auslander introduced in 1971 and which remained mysterious for many years thereafter. Some of the modern techniques in representation theory provide now a better understanding. An introductory talk by Ringel discussed the basic ideas and some interesting new phenomena for hereditary algebras. Dimensions of triangulated categories were introduced by Rouquier to obtain lower bounds for representation dimensions and Iyengar presented some new techniques to compute them. The talk of Buchweitz provided a more general perspective for the computation of these dimensions by reviewing the work of Beligiannis and Christensen on projective classes and ghosts in triangulated categories. A description of triangulated structures on additive categories in terms of Hochschild cohomology was presented by Pirashvili. \textbf{Hereditary categories of geometric origin.} Hereditary categories are in some sense the building blocks for many interesting structures in modern representation theory. Typical examples are categories of coherent sheaves which come equipped with some additional geometric structure. Using this extra structure, Lenzing presented a new description of the stable category of vector bundles on a weighted projective line. The talk of Burban discussed an intriguing connection between vector bundles on elleptic curves and solutions of Yang-Baxter equations. A complete classification of abelian 1-Calabi-Yau categories up to derived equivalence was presented by van Roosmalen. \textbf{Representations of quivers.} Quivers and their representations have always played a central role in the representation theory of finite dimensional algebras. They provide the link to Lie theory, either through the theorems of Gabriel and Kac, relating possible dimension vectors of indecomposable representations to positive roots, or more directly via Ringel's construction of quantum groups using Ringel-Hall algebras. Progress since the last meeting includes Hausel's announcement of a positive solution of Kac's conjecture that the constant term of the polynomial counting the number of absolutely indecomposable representations over a finite field is the corresponding root multiplicity. Hausel was invited to the meeting, but sadly in the end it was not possible for him to attend. Hausel's result involves hyper-K\\"ahler geometry, and in his talk Reineke also used geometry, namely the cohomology of moduli spaces of quiver representations, to prove a formula similar to one conjectured by Kontsevich and Soibelman concerning Donaldson-Thomas type invariants. Chapoton and Hille both gave intriguing talks involving tilting modules for quivers, exceptional sequences and braid group actions. Hubery discussed the connections between Hall algebras and cluster algebras and the existence of Hall polynomials for non-simply laced affine diagrams, using species rather than quivers. \textbf{Further aspects of algebras and their representations.} Representation theory of finite dimensional algebras has developed immensely since its origin, and it has now, as demonstrated above, profound connections to many other fields. However, the \`internal' theory of representation theory is still pushed forward: The talk of Skowro\\'nski presented results on algebras with generalized standard almost cyclic coherent Auslander-Reiten components. Representation theory of Lie algebras and algebraic groups is intimately related to finite dimensional algebras which are cellular or quasi-hereditary. These are algebras given by a specific filtration of ideals. K\\"onig presented work on how to generalize such a filtration further in order to deal with possibly infinite dimensional building blocks. Benson, Carlson and others developed a theory of support varieties for finitely generated modules over a finite group, and they obtained deep structural information about modular representations of finite groups in terms of the group cohomology ring. These results found their analogous twin results for Lie algebras and Steenrod algebras arising in topology. Similar support varieties have since then been defined for instance for complete intersections, quantum groups and arbitrary finite dimensional algebras. A common denominator for these situations is the presence of a ring of cohomological operations, and in the latter case this is provided by the Hochschild cohomology ring. The talk of Avramov gave an overview over recent results and questions on the Hochschild cohomology ring of an algebra arising in this context. Nakano presented results on the cohomology and support varieties for quantum groups in a quest to find relationships between representations for quantum groups and geometric constructions in complex Lie theory. The format of the workshop has been a combination of introductory survey lectures and more specialized talks on recent progress. In addition there was plenty of time for informal discussions. Thus the workshop provided an ideal atmosphere for fruitful interaction and exchange of ideas. It is a pleasure to thank the administration and the staff of the Oberwolfach Institute for their efficient support and hospitality.

Co-Point Modules Over Frobenius Koszul Algebras

Let A be a Frobenius Koszul algebra such that its Koszul dual A ! is a quantum polynomial algebra. Co-point modules over A were defined as dual notion of point modules over A ! with respect to the Koszul duality. In this article, we will see that various important functors between module categories over A used in representation theory of finite dimensional algebras send co-point modules to co-point modules. As a consequence, we will show that if (E, σ) is a geometric pair associated to A !, then the map σ:E → E is an automorphism of the point scheme E of A !, so that there is a bijection between isomorphism classes of left point modules over A ! and those of right point modules over A !.

On tilting modules over cluster-tilted algebras

In this paper, we show that the tilting modules over a clustertilted algebra A lift to tilting objects in the associated cluster category CH . As a first application, we describe the induced exchange relation for tilting Amodules arising from the exchange relation for tilting object in CH . As a second application, we exhibit tilting A-modules having cluster-tilted endomorphism algebras. Cluster algebras were introduced by Fomin and Zelevinsky [FZ02] in the context of canonical basis of quantized enveloping algebras and total positivity for algebraic groups, but quickly turned out to be related to many other fields in mathematics. In the representation theory of finite dimensional algebras, the so-called cluster categories were introduced in [BMR06] (and also in [CCS06] for the An case) as a natural categorical model for the combinatorics of the corresponding cluster algebras of Fomin and Zelevinsky. The construction is as follows. Let Q be a quiver without oriented cycles. There is then, for a field k, an associated finite dimensional hereditary path algebra H = kQ. Since H has finite global dimension, its bounded derived category D(H) of the finitely generated modules has almost split triangles [Hap88]. Let τ be the corresponding translation functor. Denoting by F the composition τ[1], where [1] is the shift functor in D(H), the cluster category CH was defined as the orbit category D(H)/F , and was shown to be canonically triangulated [Kel05] and have almost split triangles [BMR06]. In this model, the exceptional objects are associated with the cluster variables of [FZ02] while the tilting objects correspond to the clusters. Remarkably, one also defines an exchange relation on the tilting objects in CH , corresponding to the exchange relation on the clusters of [FZ02]. More precisely, an almost complete tilting object T in CH has exactly two nonisomorphic indecomposable complements M and M, and these are related by triangles M g //B f //M //M[1] and M f∗ //B g∗ //M //M [1] where f, g are minimal right addT -approximations and f , g are minimal left addT -approximations (see [BMR06]). In view of the importance of tilting theory in the representation theory of finite dimensional algebras, the (opposite) endomorphism algebras of these tilting objects, called cluster-tilted, were then introduced and studied in [BMR07] (see also [CCS06]). Their module theory was shown to be to a large extent determined by the cluster categories in which they arise. Indeed, given a cluster category CH and a 1991 Mathematics Subject Classification. 16G20, 18E30.

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.

A Note on Periodicity in Representation Theory of Finite Dimensional Algebras

Let Λ be an algebra over an algebraically closed field K of infinite transcendence degree over its prime subfield. We prove in particular that if every s-dimensional indecomposable nonprojective Λ-module is Ω-periodic (resp. DTr-periodic) then there exists a common bound for the Ω-periods (resp. DTr-periods) of s-dimensional indecomposable nonprojective Λ-modules.

Hereditary noetherian categories of positive Euler characteristic

We develop the fundamentals of hereditary noetherian categories with Serre duality over an arbitrary field k, where the category of coherent sheaves over a smooth projective curve over k serves as the prime example and others are coming from the representation theory of finite dimensional algebras. The proper way to view such a category is to think of coherent sheaves on a possibly non-commutative smooth projective curve. We define for each such category notions like function field and Euler characteristic, determine its Auslander-Reiten components and study stable and semistable bundles for an appropriate notion of degree. We provide a complete classification of hereditary noetherian categories for the case of positive Euler characteristic by relating these to finite dimensional representations of (locally bounded) hereditary k-algebras whose underlying valued quiver admits a positive additive function.

Auslander–Reiten sequences, locally free sheaves and Chebysheff polynomials

Let R be the exterior algebra in n + 1 variables over a field K. We study the Auslander–Reiten quiver of the category of linear R-modules, and of certain subcategories of the category of coherent sheaves over Pn. If n > 1, we prove that up to shift, all but one of the connected components of these Auslander–Reiten quivers are translation subquivers of a -type quiver. We also study locally free sheaves over the projective n-space Pn for n > 1 and we show that each connected component contains at most one indecomposable locally free sheaf of rank less than n. Finally, using results from the theory of finite-dimensional algebras, we construct a family of indecomposable locally free sheaves of arbitrary large ranks, where the ranks can be computed using the Chebysheff polynomials of the second kind.

Hecke algebras of classical type and their representation type

The purpose of this article is to determine the representation type for all of the Hecke algebras of classical type. To do this, we combine methods from our previous work, which is used to obtain information on their Gabriel quivers, and recent advances in the theory of finite-dimensional algebras. The principal computation is for Hecke algebras of type B with two parameters. Then, we show that the representation type of Hecke algebras is governed by their Poincare polynomials.

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.

Extensions of Modules over Schur Algebras, Symmetric Groups and Hecke Algebras

We study the relation between the cohomology of general linear and symmetric groups and their respective quantizations, using Schur algebras and standard homological techniques to build appropriate spectral sequences. As our methods fit inside a much more general context within the theory of finite-dimensional algebras, we develop our results first in that general setting, and then specialize to the above situations. From this we obtain new proofs of several known results in modular representation theory of symmetric groups. Moreover, we reduce certain questions about computing extensions for symmetric groups and Hecke algebras to questions about extensions for general linear groups and their quantizations.

Dimension of the Mesh Algebra of a Finite Auslander–Reiten Quiver

Translation quivers appear naturally in the representation theory of finite dimensional algebras; see, for example, Bongartz and Gabriel (Bongartz, K., Gabriel, P., (1982). Covering spaces in representation theory. Invent. Math. 65:331–378.). A translation quiver defines a mesh algebra over any field. A natural question arises as to whether or not the dimension of the mesh algebra depends on the field. The purpose of this note is to show that the dimension of the mesh algebra of a finite Auslander–Reiten quiver over a field is a purely combinatorial invariant of this quiver. Indeed, our proof yields a combinatorial algorithm for computing this dimension. As a further application, one may use then semicontinuity of Hochschild cohomology of algebras as in Buchweitz and Liu (Buchweitz, R.-O., Lui, S. Hochschild cohomology and representation-finite algebras. Preprint.) to conclude that a finite Auslander–Reiten quiver contains no oriented cycle if its mesh algebra over some field admits no outer derivation.

On perpendicular categories of stones over quiver algebras

Let H = kzt be the path algebra of a quiver A, where k denotes an arbitrary field. J will always be assumed to be finite without oriented cycles. We denote by mod H the category of finite dimensional left modules over H. An indecomposable H-module X is called a stone provided Ext}~(X, X) = 0. Note that for a stone X the endomorphism ring End X = k (see for example part 1). For a stone X we denote by X l the full subcategory of mod H of the modules Y with Homn(X, Y) = 0 = Ext,(X, Y). Dually, • consists of those H-modules Y satisfying Homn(Y, X) = 0 = Ext,(Y, X). The categories X • and Xare called the right respectively left perpendicular category of X. They are equivalent to mod Ar respectively mod At, where A r and At are again finite dimensional, hereditary k-algebras, and both have one simple module less than H (compare [GL] and [S1]). We denote by Ar respectively zt t the quivers with kAr = Ar and kJz = A,. Since their introduction to Geigle, Lenzing and Schofield, perpendicular categories have become a powerful tool in the representation theory of finite dimensional algebras. They inherit essential properties from mod H, furnish a reduction procedure and open the possibility for proofs by induction. In general, it is difficult problem to determine the (right or left) perpendicular category of a given stone X. We briefly recall what is known. It is quite easy to see (Proposition 2.1) that the quiver Ar differs from a full subquiver of zl by an admissible change of orientation if and only if X is preprojective or preinjective over H. Moreover, if H is tame, then there are only finitely many regular stones X over H, and the structure of its perpendicular categories is contained in [Rill. Hence we may restrict to the context where H is wild and X is a regular stone. Baer and Straul3 proved [B2], [St] that in this situation the perpendicular categories are wild again, but combining results of Xi IX] and Kerner [K3] it follows that certain wild module categories, namely of those algebras classified in [X], do not arise as perpendicular categories of regular stones over wild algebras. On the other hand, investigations of growth numbers in [K3] prove, that for a fixed hereditary, connected, wild algebra with at least three simple modules, there are infinitely many pairwise non isomorphic algebras Ai such that mod Ai is equivalent to X • for a regular stone X over H.