FDLIST

Information list for representation theory of finite dimensional algebras

- New papers -

• Lidia Angeleri-Huegel, Jan Trlifaj , Direct limits of modules of finite projective dimension . July 4,2002 .
  Abstract: For an arbitrary ring R, we study the category of all direct limits of modules (or of finitely presented modules) of projective dimension at most n by investigating a related cotorsion pair. We then consider some particular situations: the case when n=1 and R is a commutative domain, the case when R is right coherent an the little finitistic dimension is finite, and the case when R is a finite-dimensional algebra studied before by Igusa, Smal\o\ and Todorov.

• Bangming Deng, Jie Du , On bases of quantized enveloping algebras . June 24,2002 .
  Abstract: In this paper we give a systematic description of many monomial bases for a given quantized enveloping algebra and of many integral monomial bases for the associated Lusztig Z-form. The relations between monomial bases, PBW bases and canonical (or crystal) bases are also discussed. In particular, we provide an elementary construction of canonical bases with respect to the same partial order as the one used in the geometric construction.

• Markus Reineke , The quantic monoid and degenerate quantized enveloping algebras . June 11,2002 .
  Abstract: We study a monoid associated to complex semisimple Lie algebras, called the quantic monoid. Its monoid ring is shown to be isomorphic to a degenerate quantized enveloping algebra. Moreover, we provide normal forms and a straightening algorithm for this monoid. All these results are proved by a realization in terms of representations of quivers, namely as the monoid of generic extensions of a quiver with automorphism.

• Robert Marsh, Markus Reineke, Andrei Zelevinsky , Generalized associahedra via quiver representations . May 15,2002 .
  Abstract: We provide a quiver-theoretic interpretation of certain smooth complete simplicial fans associated to arbitrary finite root systems in recent work of S. Fomin and A. Zelevinsky. The main properties of these fans then become easy consequences of the known facts about tilting modules due to K. Bongartz, D. Happel and C. M. Ringel.

• Markus Reineke , The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli . May 7,2002 .
  Abstract: Methods of Harder and Narasimhan from the theory of moduli of vector bundles are applied to moduli of quiver representations. Using the Hall algebra approach to quantum groups, an analog of the Harder-Narasimhan recursion is constructed inside the quantized enveloping algebra of a Kac-Moody algebra. This leads to a canonical orthogonal system, the HN system, in this algebra. Using a resolution of the recursion, an explicit formula for the HN system is given. As an application, explicit formulas for Betti numbers of the cohomology of quiver moduli are derived, generalizing several results on the cohomology of quotients in 'linear algebra type' situations.

• A. Beligiannis and H. Krause , Realizing maps between modules over Tate cohomoogy rings . April 1,2002 .
  Abstract: Let $G$ be a finite group and $k$ be a field. Given two representations $A$ and $B$ of $G$, we investigate when all homomorphisms $\widehat H^*(G,A)\to\widehat H^*(G,B)$ over the Tate cohomology ring $\widehat H^*(G,k)$ are of the form $\widehat H^*(G,\alpha)$ for some morphism $\alpha\colon A\to B$. We construct an extended Milnor sequence which computes the obstruction for homomorphisms $\widehat H^*(G,A)\to\widehat H^*(G,B)$ to be realizable.

• Andrew Hubery , Quiver Representations Respecting a Quiver Automorphism: A Generalisation of a Theorem of Kac . March 19,2002 .
  Abstract: We consider representations of a quiver respecting a quiver automorphism and show that the dimension vectors of the indecomposable such are precisely the positive roots of an associated symmetrisable Kac-Moody Lie algbera. Moreover, every such Lie algebra occurs in this way. We also discuss the relationship with species of valued quivers over finite fields.

• Grzegorz Bobinski and Grzegorz Zwara , Schubert varieties and representations of Dynkin quivers . February 28,2002 .
  Abstract: We show that the types of singularities of Schubert varieties in the flag varieties are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type A. Similarly, we prove that the types of singularities of Schubert varieties in products of two Grassmannians are equivalent to the types of singularities of orbit closures for the representations of Dynkin quivers of type D. As a consequence, we obtain that the orbit closures in representation varieties of Dynkin quivers of type D are normal varieties.

• Henning Krause, and Oeyvind Solberg , Applications of cotorsion pairs . January 13,2002 .
  Abstract: For an artin algebra A, cotorsion pairs are studied for the category mod A of finitely presented A-modules and for the category Mod A of all A-modules. It is shown that every cotorsion pair for mod A induces a cotorsion pair for Mod A. This has some interesting applications, even for the category of finitely presented modules. Another theme of this paper is the interplay between cotorsion and torsion pairs. This leads to a conjecture which is an analogue of the telescope conjecture in stable homotopy theory.

• Bangming Deng and Jie Du , Monomial Bases For Quantum Affine sl_n . December 20,2001 .
  Abstract: We use the idea of generic extensions to investigate the correspondence between the isomorphism classes of nilpotent representations of a cyclic quiver and the orbits in the corresponding representation varieties. We endow the set $\cal M$ of such isoclasses with a monoid structure and identify the submonoid ${\cal M}_c$ generated by simple modules. On the other hand, we use the partial ordering on the orbits (i.e., the Bruhat-Chevalley type ordering) to induce a poset structure on $\cal M$ and describe the poset ideals generated by an element of the submonoid ${\cal M}_c$ in terms of the existence of a certain composition series of the corresponding module. As applications of these results, we generalize some results of Ringel involving special words to results with no restriction on words and obtain a systematic description of many monomial bases for any given quantum affine ${\frak {sl}}_n$.

• Pu Zhang , Skew differential operator algebras of twisted Hopf algebras . December 18,2001 .
  Abstract: By introducing a twisted Hopf algebras with primitive generators we unify several important objects of study. Skew derivations of such an algebra are defined and the corresponding skew differential operator algebras are studied. This generalizes results in the Weyl algebra. Applying this investigation to the twisted Ringel-Hall algebra we get, in particular, a natural realization of the non-negative part of quantized generalized Kac-Moody algebra, by identifying the cannoical generators with some linear, skew differential operators. This also induces some algebras which are quantum-group-like.

• Pu Zhang , Generalized Green Classes . December 18,2001 .
  Abstract: Generalized Green classes are introduced; some basic properties in a generalized Green class are studied. Finally, we apply the results to $\Cal H(A)$, the Ringel-Hall algebra of a finite-dimensional hereditary algebra $A$ over a finite field. In particular, it is proved that $\Cal H(A)$ belong to a suitable Green class, and that there is direct sum decomposition of spaces $\Cal H(A) = \Cal C(A)\oplus J$, where $\Cal C(A)$ is the composition algebra of $A$, and $J$ is a twisted Hopf ideal of $\Cal H(A)$, which is exactly the orthogonal complement of $\Cal C(A)$ respect to the inner product on $\Cal H(A)$ given by J. Green and C. Ringel.

• H.L. Huang, H.X. Chen, and P. Zhang , Generalized Taft algebras and selfinjective Nakayama algebras . December 18,2001 .
  Abstract: By dropping the restriction to the parameter $q$ in Taft's $n^2$-dimensional Hopf algebras, we get the generalized Taft algebras $A_{n, d}(q)$. By using the quiver techniques, we determine the structure and all indecomposable representations of $A_{n, d}(q)$. In particular, it is the product of $n/d$ copies of $d$-truncated, indecomposable, selfinjective Nakayama algebra $kZ_d/J^d$, where $Z_d$ is the cyclic quiver with $d$ vertices.

• E. L. Green, and P. Zhang , Rigids as iterated skew commutators of simples . December 18,2001 .
  Abstract: Let $A$ be a finite-dimensional hereditary algebra of finite or tame representation type over a finite field, and let $M$ be a rigid $A$-module. Then the element $[M]$ in the Ringel-Hall algebra $\Cal H(A)$ is an iterated skew commutator of isoclasses of simple $A$-modules. This gives another characterization of the rigidness of an indecomposable module over a tame hereditary algebra.

• Hebing Rui, Changchang Xi, Cyclotomic Temperley-Lieb algebras. December 10,2001.
  Abstract: A class of associate algebras called cyclotomic Temperley-Lieb algebras is introduced in terms of generators and relations. They are closely related to the group algebra of a complex reflection group on one hand and a generalization of the usual Temperley-Lieb algebras on the other hand. It is shown that these algebras can be defined by means of dotted planar graphs and that they are cellular in the sense of Graham and Lehrer. One thus obtains a description of irreducible representations and quasi-heredity in the sense of Cline, Parshall and Scott. The branching rule for cell modules and the determinants of Gram matrices for certain cell modules are calculated, here the generalized Tchebychev polynomials play an important role for semisimplicity.

• D. Benson, H. Krause, S. Schwede, Realizing modules over Tate cohomology. November 28,2001.
  Abstract: Let k be a field and let G be a finite group. There is a canonical element gamma of degree (3,-1) in the Hochschild cohomology of the Tate cohomology H^*(G,k) with the following property. Given a graded H^*(G,k)-module X, the image of gamma in Ext^{3,-1}_{H^*(G,k)}(X,X) vanishes if and only if X is isomorphic to a direct summand of H^*(G,M) for some kG-module M. The description of the realizability obstruction works in any triangulated category with direct sums. We show that in the case of the derived category of a differential graded algebra A, there is also a canonical element of Hochschild cohomology HH^{3,-1}H^*(A) which is a predecessor for these obstructions.

• Jin Yun Guo and Qiuxian Wu, Selfinjective koszul algebras of finite complexity. November 5,2001.
  Abstract: In this paper, we study selfinjective Koszul algebras of finite complexity. Let $\Lambda$ be a skew algebra of a finite subgroup $G$ of $SL(m,C)$ over the exterior algebra of a $m$ dimensional vector space. We prove that for each $0 \le t \le m$ there exist a family of modules of complexity $t$ parametrized by $G(t,m)$, the Grassmannian of $t$-dimensional subspaces of an $m$-dimensional vectorspace. Our results also give a new approach to the representation theory of a tame symmetric algebras with vanishing radical cube over an algebraically closed field of characteristic $0$ which is related to the tame hereditary algebra.

• Jan Schröer and Alexander Zimmermann, The class of gentle algebras is closed under derived equivalence. October 22,2001.
  Abstract: We prove that the stable endomorphism algebra of a module without self-extensions over a special biserial algebra is a gentle algebra. As a consequence, any algebra which is derived equivalent to a gentle algebra is gentle.

• David Green, John Hunton and Bjoern Schuster, Chromatic characteristic classes in ordinary group cohomology. September 4,2001.
  Abstract: We study a family of subrings, indexed by the natural numbers, of the mod $p$ cohomology of a finite group $G$. These subrings are based on a family of $v_n$-periodic complex oriented cohomology theories and are constructed as rings of generalised characteristic classes. We identify the varieties associated to these subrings in terms of colimits over categories of elementary abelian subgroups of $G$, naturally interpolating between the work of Quillen on Var$(H^*(BG))$, the variety of the whole cohomology ring, and that of Green and Leary on the variety of the Chern subring, Var$(Ch(G))$. Our subrings give rise to a 'chromatic' (co)filtration of Var$(H^*(BG))$ which has both topological and algebraic definitions, and whose final quotient is the variety Var$(Ch(G))$.

• Osamu Iyama, Finiteness of Representation dimension. August 3,2001.
  Abstract: We will show that any module over an artin algebra is a direct summand of some module whose endomorphism ring is quasi-hereditary. As a conclusion, any artin algebra has a finite representation dimension.

• Osamu Iyama, A proof of Solomon's second conjecture on local zeta functions of orders. August 3,2001.
  Abstract: We will prove Solomon's second conjecture on local zeta functions of orders. A key idea of our proof is to consider certain filtration of the category of lattices and use a reduction to smaller categories. Our filtration is an analogy of preprojective partition given by Auslander and Smalo, which was related to quasi-hereditary algebras by Dlab and Ringel.

• Lidia Angeleri-Hügel and Jan Trlifaj, Tilting theory and the finitistic dimension conjectures. July 16,2001.
  Abstract: Let R be a right noetherian ring and $P^\infty$ be the class of all finitely presented modules of finite projective dimension. We prove that the little finitistic dimension of R is finite iff there is an (infinitely generated) tilting module T such that the Ext-orthogonal classes of T and of $P^\infty$ coincide. If R is an artin algebra, then T can be taken finitely generated iff $P^\infty$ is contravariantly finite. By investigating the class Add T, we obtain a criterion for validity of the First Finitistic Dimension Conjecture which generalizes and refines the well-known result of Huisgen-Zimmermann and Smalo. Moreover, the Second Finitistic Dimension Conjecture is equivalent to a statement on certain (infinitely generated) tilting modules of projective dimension at most one. MSC 16E10 16E30 16G10.

• Bangming Deng and Jie Xiao, Ringel-Hall algebras and Lusztig's symmetries. July 11,2001.
  Abstract: Lusztig has introduced certain automorphisms, called symmetries, of the quantized enveloping algebras of a Kac--Moody algebra and shown that they satisfy the braid group relations. The construction of Lusztig can be easily generalized to obtain isomorphisms of the whole double Ringel--Hall algebra of a finite dimensional hereditary algebra. Recently, the Bernstein--Gelfand--Ponomarev--reflection operators of double Ringel-Hall algebras have been applied to the double composition algebras by Sevenhant and Van den Bergh, also by Xiao and Yang, to derive the fundamental properties of Lusztig's symmetries. In the present paper we show that the BGP-reflection operators coincide with Lusztig's symmetries, up to isomorphism, on the whole double Ringel--Hall algebras. As a consequence, we obtain that the BGP-reflection operators satisfy the braid group relations on the whole double Ringel--Hall algebras, too.

• Bangming Deng and Jie Xiao, The Ringel-Hall algebra interpretation to a conjecture of Kac. July 11,2001.
  Abstract: Sevenhant and Van den Bergh have discovered an interesting relation between a conjecture of Kac on representations of quivers and the structure of the Ringel--Hall algebras in the case where the quivers contain no oriented cycle and extended Dynkin subgraph. In the present paper we show that this relation still holds when we deal with arbitrary finite dimensional hereditary $k$-algebra. Furthermore, we present an alternative proof of a weak form of the Kac's theorem by using the Ringel--Hall algebra approach.

• Grzegorz Zwara, Unibranch orbit closures in module varieties. June 28,2001.
  Abstract: Let A be a representation finite algebra over an algebraically closed field. We show that the orbit closures in the associated module varieties are unibranch.

• Apostolos Beligiannis and Idun Reiten, Homological Aspects of Torsion Theories. June 26,2001.
  Abstract: In this paper we study torsion theories in the general setting of pretriangulated categories, an omnipresent class of additive categories which includes abelian, triangulated, stable, and more generally (homotopy categories of) closed model categories in the sense of Quillen, as special cases. We explore the formal analogies of the concept of a torsion theory in the above settings, concentrating our study to the relationship between orthogonal subcategories and the existence of left or right adjoints in connection with their interaction with the structure of left/right triangles and exact sequences. The main focus of our study lies to the investigation of the strong connections and the interplay between torsion theories and tilting theory in abelian, triangulated and stable categories. The proper setting for the formulation of these connections is via closed model structures. We also study (co)homological functors induced by torsion theories, thus generalizing the Tate-Vogel (co)homological functors, and we give applications to the structure of the representations of Artin algebras and Gorenstein rings.

• Anne Henke and Steffen König , Relating polynomial GL(n)-representations in different degrees. June 12,2001.
  Abstract: We construct explicitly isomorphisms between (generalized) Schur algebras in different degrees. This establishes and explains certain repeating patterns in decomposition matrices of general linear and symmetric groups.

• Reynaud Eric, Algebraic fundamental group and simplicial complexes. May 22,2001.
  Abstract: In this paper, we prove that the fundamental group of a simplicial complex is isomorphic to the algebraic fundamental group of its incidence algebra, and we derive some applications.

• G. Bobinski, Ch. Geiss, A. Skowronski, Classification of discrete derived categories. May 16,2001.
  Abstract: We classify the derived categories of finite dimensional algebras which are discrete. Additionally we study properties of the corresponding Euler- and bilinear forms.

• Markus Reineke, The monoid of families of quiver representations. May 16,2001.
  Abstract: A monoid structure on families of representations of a quiver is introduced by taking extensions of representations in families, i.e. subvarieties of the varieties of representations. The study of this monoid leads to interesting interactions between representation theory, algebraic geometry and quantum group theory. For example, it produces a wealth of interesting examples of families of quiver representations, which can be analyzed by representation theoretic and geometric methods. Conversely, results from representation theory, in particular A. Schofield's work on general properties of quiver representations, allow us to relate the monoid to certain degenerate forms of quantized enveloping algebras.

• Markus Reineke, Quivers, desingularizations and canonical bases. May 16,2001.
  Abstract: A class of desingularizations for orbit closures of representations of Dynkin quivers is constructed, which can be viewed as a graded analogue of the Springer resolution. A stratification of the singular fibres is introduced; its geometry and combinatorics are studied. Via the Hall algebra approach, these constructions relate to bases of quantized enveloping algebras. Using Ginzburg's theory of convolution algebras, the base change coefficients of Lusztig's canonical basis are expressed as decomposition numbers of certain convolution algebras.

• Changchang Xi, Dajing Xiang, Cellular algebras and Cartan matrices. April 12,2001.
  Abstract: Let A be a finite dimensional algebra over a field k, and let C be the Cartan matrix of A. Usually, the eigenvalues of C being integers do not implies the semisimplicity of A. However, we prove that a cellular algebra A is semisimple if and only if det(C)=1 and all eigenvalues of C are integers. Moreover, we use Cartan matrices to classify the cellular algebras with property that the determinant of the Cartan matrix equals a given prime p and all eigenvalues are integers. We also give a classification of cellular Nakayama algebras with integral eigenvalues of their Cartan matrices. Finally, we show that if A is a cellular algebra then its trivial extension T(A) is also a cellular algebra. In particular, if a non-simple connected cellular algebra A is quasi-hereditary, then the Cartan matrix of T(A) has at least one non-integral eigenvalue. The main tool used in this paper is the well-known Perron-Frobenius theory on non-negative matrices.

• H. Krause and Ø. Solberg, Filtering modules of finite projective dimension. April 08, 2001.
  Abstract: preprintabstract: For a right artinian ring R we show that for every natural n there exists a pure-injective R-module P(n) such that the R-modules of projective dimension at most n are precisely the direct factors of R-modules having a finite filtration in products of copies of P(n). This is a consequence of a general description of certain contravariantly finite resolving subcategories of Mod R. It leads in addition to a one-to-one correspondence between equivalence classes of (not necessarily finitely generated) cotilting modules and resolving contravariantly finite subcategories of Mod R which are closed under products and admit finite resolutions and special right approximations. As an application it is shown that every finitely presented partial cotilting module over an artin algebra admits a complement.

• William Crawley-Boevey and Jan Schröer, Irreducible components of varieties of modules. March 20, 2001.
  Abstract: We prove some basic results about irreducible components of varieties of modules for an arbitrary finitely generated associative algebra. Our work generalizes results of Kac and Schofield on representations of quivers, but our methods are quite different, being based on deformation theory.

• William Crawley-Boevey, On matrices in prescribed conjugacy classes with no common invariant subspace and sum zero. March 20, 2001.
  Abstract: We determine those k-tuples of conjugacy classes of matrices, from which it is possible to choose matrices which have no common invariant subspace and have sum zero. This is an additive version of the Deligne-Simpson problem. We deduce the result from earlier work of ours on preprojective algebras and the moment map for representations of quivers. Our answer depends on the root system for a Kac-Moody Lie algebra.

• Karin Erdmann and Anne Henke, On Schur algebras, Ringel duality and symmetric groups. March 08, 2001.
  Abstract: In [EH1] we determine precisely the degrees $r$ for which the Schur algebra $S(2,r)$ is its own Ringel dual. Here we study some applications: We classify uniserial Weyl modules, Specht modules, tilting modules, and Young modules labelled by two-part partitions. Moreover we determine extensions for simple modules for the Ringel duals of arbitrary $S(2,r)$. As a consequence we obtain corresponding results on symmetric groups.

• Claude Cibils, Eduardo Marcos, Maria Julia Redondo and Andrea Solotar, Cohomology of split algebras and of trivial extensions. February 26, 2001.
  Abstract: We consider associative algebras $\Lambda$ over a field provided with a direct sum decomposition of a two-sided ideal $M$ and a sub-algebra $A$ -- examples are provided by trivial extensions or triangular type matrix algebras. In this relative and split setting we describe a long exact sequence computing the Hochschild cohomology of $\Lambda$. We study the connecting homomorphism using the cup-product and we infer several results, in particular the first Hochschild cohomology group of a trivial extension never vanishes.

• Sheila Brenner and M.C.R. Butler, Almost periodic algebras and pivoted bimodules: resolutions and Yoneda algebras. January 3, 2001.
  Abstract (dvi, 1 page)    Paper (ps, 22 pages).

• Otto Kerner, Exact structures on the categories of regular modules. December 22, 2000

• Henning Krause, A duality between complexes of right and left modules. November 22, 2000.
  Abstract: A new duality between complexes of right and and left modules over an associative ring R is introduced. Identifying R-modules with complexes concentrated in degree 0, this becomes the duality for endofinite modules studied by Herzog and Crawley-Boevey. The duality has an analogue in stable homotopy theory which is Brown-Comenetz duality. All this fits into a general theory of endofiniteness for compactly generated categories. This is explained and various examples illustrate this point of view.

• Yuriy Drozd, On cubic functors. November 9, 2000.
  

• Yuriy Drozd, Gert-Martin Greuel and Irina Kashuba, On Cohen-Macaulay modules on surface singularities. November 9, 2000.
  

• Jan Trlifaj, Local splitters for bounded cotorsion theories. October 10, 2000.
  

• Jan Trlifaj, Cotorsion theories induced by tilting and cotilting modules. October 10, 2000.
  

• Apostolos Beligiannis, Purity and Almost Split Morphisms in Abstract Homotopy Categories . September 25, 2000.
  Abstract: Our aim in this paper is to develop a theory of purity and to prove in a unified conceptual way the existence of almost split morphisms, almost split sequences and almost split triangles in Abstract Homotopy Categories, a rather omnipresent class of categories of interest in representation theory. Our main tool for doing this is the classical Brown Representability Theorem.

• Apostolos Beligiannis, Homotopy Theory of Modules and Gorenstein Rings. September 25, 2000.
  Abstract: Our aim in this paper is to investigate and compare various algebraic homotopy theories defined in a natural way in a module category. Examples of such homotopy theories include abstract homotopy theory in the sense of Brown, the theory of closed model categories in the sense of Quillen and the homotopy theory in the sense of Eckmann-Hilton. Homologically finite subcategories provide a connecting link between the above theories. We apply our results to Gorenstein rings.

• Christof Geiss and Henning Krause, On the notion of derived tameness. September 22, 2000.
  Abstract: The notion of tameness for the derived category of a finite dimensional algebra is introduced . This is based on classical tameness definitions of Drozd and Crawley-Boevey for the category of finite dimensional representations.

• Changchang Xi, Representation Dimension and Quasi-Hereditary Algebras. September 15, 2000.
  Abstract: We study Auslander's representation dimension of Artin algebras, which is by definition the minimal projective dimension of coherent functors on modules which are both generators and cogenerators. We show the following statements: (1) if an Artin algebra $A$ is stably hereditary, then the representation dimension of $A$ is at most $3$. (2) If two Artin algebras are stably equivalent of Morita type, then they have the same representation dimension. Particularly, if two self-injective algebras are derived equivalent, then they have the same representation dimension. (3) Any incidence algebra of a finite partially ordered set over a field has finite representation dimension. Moreover, we use results on quasi-hereditary algebras to show that (4) the Auslander algebra of a Nakayama algebra has finite representation dimension.

• Claude Cibils and Marc Rosso, Hopf quivers. September 11, 2000.
  Abstract: We classify graded Hopf algebras structures over path coalgebras, that is over free pointed coalgebras, using Hopf quivers which are analogous to Cayley graphs.
  The description involves formulas for the product besides the canonical formulas for the coproduct. This makes explicit the quantum shuffle product making use of natural elements in a Hopf bimodule having a simple "geometrical" interpretation in the quiver sense, rather than working systematically with right or left coinvariants.

• S.Brenner, M.C.R.Butler and A.D.King, Periodic algebras which are almost Koszul. August 15, 2000.
  Abstract: The preprojective algebra and the trivial extension algebra of a Dynkin quiver (in bipartite orientation) are very close to being a Koszul dual pair of algebras. In this case the usual duality theory may be adapted to show that each algebra has a periodic bimodule resolution built using the other algebra and some extra data: an algebra automorphism. A general theory of such `almost Koszul' algebras is developed and some other examples are found.

• Yuriy A. Drozd, Gert-Martin Greuel and Irina Kashuba, On Cohen-Macaulay modules on surface singularities. August 15, 2000.
  Preprint MPI 2000-76, Max-Planck-Institut fuer Mathematik, Bonn (available from the MPI preprint server)

• Henning Krause, Coherent functors and covariantly finite subcategories. August 14, 2000.
  Abstract: We study certain covariantly finite subcategories of a module category Mod R over a ring R. This involves a filtration of the category of coherent functors from Mod R to abelian groups. This filtration seems to be of independent interest. In fact, a similar filtration arises for triangulated categories in recent work of Neeman.

• Jiuzhao Hua and Yingbo Zhang, A q-analogue of the Kac-Weyl denominator identity of type ~B_2. August 7, 2000.
  Abstract: Canonical forms of indecomposable representations of the valued quiver $\tilde B_2$ over a finite field are explicitly given. By counting their numbers with fixed dimension vectors, we deduce a $q$-analogue of the Kac-Weyl denominator identity of type $\TB_2$.

• William Crawley-Boevey, Decomposition of Marsden-Weinstein reductions for representations of quivers. August 4, 2000.
  Abstract: We decompose the Marsden-Weinstein reductions for the moment map associated to representations of a quiver. The decomposition involves symmetric products of deformations of Kleinian singularities, as well as other terms. As a corollary we deduce that the Marsden-Weinstein reductions are irreducible varieties.

• Jiuzhao Hua, Numbers of representations of valued quivers over finite fields. June 14, 2000.
  Abstract: This paper studies the numbers of representations of valued quivers (species) over finite fields. The generating function of the numbers of isoclasses of representations with fixed dimension vectors is factorized. As a consequence, the number of isoclasses of indecomposable representations with a fixed dimension vector is a polynomial in q, which is independent of orientations. It is conjectured that the constant term of this polynomial is equal to the multiplicity of corresponding root. This conjecture can be interpreted as a combinatorial identity.

• Karin Erdmann and Anne Henke, On Ringel duality for Schur algebras. June 9, 2000.
  Abstract: We show that the Schur algebra S(2,r) is Morita equivalent to its Ringel dual S(2,r)', as a quasi-hereditary algebra if and only if either r< p^2, or r is of the form ap^k-2, or ap^k-2 \pm 1, where 2 \leq a \leq p and k \geq 1.

• Karin Erdmann, Thorsten Holm, Nicole Snashall, Twisted bimodules and Hochschild cohomology for self-injective algebras of class A_n, II. May 17, 2000.
  Abstract: Up to derived equivalence, the representation-finite selfinjective algebras of class A_n are divided into the wreath-like algebras (including all Brauer tree algebras) and the Moebius algebras. In [Erdmann-Holm, Forum Math. 11 (1999), 177-201], the ring structure of Hochschild cohomology of wreath-like algebras was determined, the key observation being that kernels in a minimal bimodule resolution of the algebras are twisted bimodules. In this paper we prove that also for Moebius algebras certain kernels in a minimal bimodule resolution carry the structure of a twisted bimodule. As an application we obtain detailed information on subrings of the Hochschild cohomology rings of Moebius algebras.

• Iain Gordon and Alexander Premet, Block representation type of reduced enveloping algebras. May 15, 2000.
  

• Y.Drozd and S.Ovsienko , Coverings of tame boxes and algebras. March 2000.
  

• Thomas Brüstle, Steffen König and Volodymyr Mazorchuk , The coinvariant algebra and representation types of blocks of category O. April 19, 2000.
  Abstract: Let G be a finite dimensional semisimple Lie algebra over the complex numbers. Let A be the finite dimensional algebra of a (regular or singular) block of the BGG-category O. By results of Soergel, A has a combinatorial description in terms of a subalgebra C_0 of the coinvariant algebra C. In an earlier paper by König and Mazorchuk, an embeddding has been constructed from C_0-mod into the category of A-modules having a Verma flag. This is the main tool for our classification of the categories of A-modules having a Verma flag into finite, tame and wild representation type. As a consequence we also obtain a classification of A-mod into finite, tame and wild representation type, thus reproving a recent result of Futorny, Nakano and Pollack.

• P.C.Eklof and J.Trlifaj , Covers induced by Ext. April 17, 2000.
  

• P.C.Eklof and J.Trlifaj , How to make Ext vanish. April 17, 2000.
  

• C. Stroppel, Quivers of Category O. April 6, 2000.
  Abstract: Using results of W. Soergel, an algorithm can be given to calculate quivers of category O. For regular and singular integral blocks these quivers are calculated for all root systems of rank smaller or equal to 2. The quiver for a regular integral block for A3 with a representation (of the quiver) corresponding to the dominant Verma module are presented. In this case we also give a list of the composition factors of primitive quotients.

• E. L. Green, I. Reiten and Ø. Solberg, Dualities on generalized Koszul algebras. March 31, 2000.
  

• Peter Dräxler and Øyvind Solberg, Exact factors of exact categories. March 17, 2000.
  

• Steffen König and Volodymyr Mazorchuk, Enright's completions and injectively copresented modules. March 17, 2000.
  Abstract: It is shown that the category of (absolutely or relatively) complete modules in the sense of Enright is equivalent to a category of injectively copresented modules, and therefore to the category of eAe-modules for some idempotent e in the algebra A associated with the given block of O. This leads to an easy proof of Deodhar's and Bouaziz's theorem (Enright's conjecture) that completion functors satisfy braid relations. Moreover, the same category is equivalent to some category of Harish-Chandra bimodules (studied by Bernstein and Gelfand) and to some parabolic category O. In the second part of the paper, it is shown that the algebra eAe is standardly stratified. It satisfies a double centralizer property similar to Soergel's results for A itself.

• Aslak Bakke Buan, Subcategories of the derived category and cotilting complexes. 2000.
  

• Aslak Bakke Buan, Closed subbifunctors of the extension bifunctor. 2000.
  

• Steffen König, Inger Heidi Slungård and Changchang Xi, Double centralizer properties, dominant dimension and tilting modules. March 17, 2000.
  Abstract: A machine is developped for establishing double centralizer properties from structures in ring theory and representation theory. As applications, new and easy proofs are obtained for both classical and quantized Schur-Weyl duality and for Soergel's double centralizer property for category O.

• Dieter Vossieck, The Algebras with Discrete Derived Category. March 16, 2000.
  Abstract: We classify the finite dimensional algebras over an algebraically closed field whose bounded derived category does not admit an infinite continuous family of pairwise non-isomorphic indecomposable complexes.

• Bernhard Keller, Introduction to A-infinity algebras and modules. May 18, 1999, last modified on March 14, 2000.
  Abstract: These are slightly expanded notes of a minicourse of three lectures given at the Euroconference "Homological Invariants in Representation Theory" in Ioannina, Greece, March 16 to 21, 1999, and of a talk at the Instituto de Matematicas, UNAM, Mexico, on April 28, 1999. They present basic results on A-infinity-algebras and their modules. They are inspired and motivated by a chapter of a course that M. Kontsevich gave in spring 1998 at the Ecole normale superieure in Paris.
  Click to receive a more detailed abstract.

- Papers listed by month -

September 2000, August 2000, July 2000, June 2000, May 2000, April 2000, March 2000, February 2000, January 2000

- Preprint archives -

Groups, representations and cohomology archive (i.e. Benson's archive)
SFB Bielefeld
Max-Planck-Institut Bonn
NTNU Trondheim (Representation theory group)
XXX archive (section representation theory)
XXX archive (section rings and algebras)
Topology archive (i.e. Wilkerson's archive)

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