Let A be an Artin algebra and D(A) its bounded derived category. We recall that D(A) is called discrete and A derived discrete if for any sequence h = (hi)i Î Z of non-negative integers with almost all the hi=0, there are only finitely many isoclasses of indecomposable objects X Î D(A) with length of Hi(X) = hi for all i Î Z.
We prove the following:
The Artin algebra A is not derived discrete if and only if there is
a bounded complex of projective A-modules X = (Xi,dXi) with the
(i) for all i the image of dXi is in the radical of Xi+1;
(ii) X is indecomposable in the homotopy category of complexes;
(iii) there is some j such that Hj (X) has not finite length;
(iv) for all i, Hi (X) has finite length as left E-module, where E is the endomorphism ring of X in the homotopy category of complexes.
A complex as before is called generic complex. In case A is a finite-dimensional algebra over an algebraically close field, we also consider the tame representation type in terms of generic complexes.
Let k be an algebraically closed field of characteristic 2, and let W be the ring of infinite Witt vectors over k. Suppose G is a finite group, and B is a block of kG with dihedral defect group D which is Morita equivalent to the principal 2-modular block of a finite simple group. We determine the universal deformation ring R(G,V) for every kG-module V which belongs to B and has stable endomorphism ring k. It follows that R(G,V) is always isomorphic to a subquotient ring of WD. Moreover, we obtain an infinite series of examples of universal deformation rings which are not complete intersections.
Quasi-stratified algebras are a generalization of standardly stratified algebras for which both the Cartan Determinant Conjecture and its converse hold. This talk will relate quasi-stratified algebras to other classes of algebras: left serial, the Yamagata algebras and gentle algebras.
Joint work with Ahmad Mojiri.
Let g = g(C) be the Kac-Moody Lie algebra associated to a Cartan matrix C and U = Uv(g) its quantum group. A key feature in quantum groups is the presence of several natural bases (like the PBW-basis and the canonical basis). There are different approaches to the construction of the canonical basis: algebraic approach (Lusztig, Kashiwara, Beck-Chari-Pressley, Beck-Nakajima), geometric approach (Lusztig) and Ringel-Hall algebra approach (Ringel, Lin-Xiao-Zhang). In this talk, we will recall algebraic and Ringel-Hall algebra approaches to a PBW basis and a canonical basis of U when C is finite or affine. Meanwhile, the root vectors in Ringel-Hall algebras will be discussed.
Let A be an artin algebra. Using the so-called Auslander-Reiten theory, one can assign to A a quiver GA called the Auslander-Reiten quiver of A which "represents" the indecomposable finitely generated A-modules together with some morphisms between them called irreducible. Unfortunately, GA does not give all the informations on the category mod A of the finitely generated A-modules one could expect because not all morphisms can be re-constructed from the irreducible ones. However, (sum of) compositions of irreducible morphisms can give important informations on mod A.
A morphism f : X ® Y is called irreducible provided it does not split and whenever f = gh, then either h is a split monomorphism or g is a split epimorphism. It is not difficult to see that such an irreducible morphism f belongs to the radical rad(X,Y) but not to its square rad2 (X,Y). Consider now a non-zero composition g = fn ¼f1 : X0 ® Xn of n ³ 2 irreducible morphisms fi¢s. It is not always true that g Î radn (X0, Xn) \radn+1 (X0, Xn). In this talk, we shall discuss some results on the problem of when such a composition does lie in radn (X0, Xn) \radn+1 (X0,Xn). The particular cases n=2,3 will be considered in more details.
Joint work with C. Chaio and S. Trepode (Universidad de Mar del Plata).
Coxeter transformations play an important role in the theory of Lie algebras. Namely, the Weyl group is finite (resp. affine, contains a free subgroup) if the Coxeter elements are periodic (resp. have spectral radius 1, > 1). For a hereditary algebra A = k D associated to a quiver D without oriented cycles, the Coxeter transformation is induced from the Auslander-Reiten equivalence of the derived category Db(mod A) to the Grothendieck group of A. The spectral properties of this transformation are essential to understand the representation theory of A. For canonical algebras A over the complex numbers, spectral properties of the Coxeter transformations are related to the classification of Fuchsian groups and their asociated singularities.
The importance of the relationship between an algebra and its Ext-algebra is well established. On the other hand, little is known about which properties of the algebra or its representations imply, or are implied by the noetherianity of the Ext-algebra. The main thrust of this talk is the study of such properties. Particular attention is given to the case when the algebra is Koszul. Some of the results presented are given below.
We prove that if every module in gr(R) has a finitely generated Ext-module, Ån ³ 0 ExtnR (R/J,R/J), where J is the graded Jacobson radical of a standard graded algebra R, then R is left noetherian.
We prove that if R is a Koszul algebra of finite global dimension, then R being left noetherian is equivalent to every module M = Åi Mi in grR has the property that for some n, the module Mn ÅMn+1 Å¼ is linear.
We discuss the interplay between the preprojective representations of a connected valued quiver, the (+)-admissible sequences of vertices, and the Weyl group. To each preprojective representation corresponds a canonical (+)-admissible sequence. A (+)-admissible sequence is the canonical sequence of some preprojective representation if and only if the product of simple reflections associated to the vertices of the sequence is a reduced word in the Weyl group. As a consequence, for any Coxeter element of the Weyl group associated to an indecomposable symmetrizable generalized Cartan matrix, the group is infinite if and only if the powers of the element are reduced words. The latter strengthens known results of Howlett and Fomin-Zelevinsky.
The talk is based on joint work with Helene R. Tyler and with Allen Pelley.
There is a famous conjecture ("Finitistic conjecture") in Representation theory of artin algebras:
"Given an artin algebra, its finitistic dimension is finite".(The finitistic dimension is the supremum of the projective dimensions of the modules with finite projective dimension.)
This conjecture has more than 45 years, and is one of the most interesting problems at this area. In the talk we explain a new technique to treat the conjecture, using the Igusa-Todorov function. We exhibit a new family of quivers algebras with finite finitistic dimension.
Given an n×n matrix M over a (not necessarily commutative) field F and a candidate inverse M¢, the n2 equations M·M¢=I, if solvable, define an inverse for M in EndF(Fn). For us, it is a small wonder that
Having in mind non connected algebras, like the preprojective algebra, we introduce a generalization of the notion of a noncommutative regular algebra given by Artin and Schelter, we obtain some basic results and apply them to the the polynomial algebra. In order to include the category of finitely presented functors from the finitely generated modules over a finite dimensional K-algebra, to the category of K-vector spaces, we extend the notion of Artin Schelter regular to additive categories. Finally, we give an application to the structure of the Auslander Reiten components.
The results presented here are part of a joint work with Oeyvind Solberg.
In this talk I will present recent results from joint work with Claus Michael Ringel (Bielefeld) on nilpotent linear operators and their invariant subspaces.
Let k be a field. We consider triples (V,U,T) where V is a finite dimensional k-space, U a subspace of V and T : V®V a linear operator with Tn=0 for some n, and such that T(U) Í U. Thus, T is a nilpotent linear operator on V and U is an invariant subspace with respect to T.
If v=dimV and u=dimU then (v,u) is the dimension pair of the triple (V,U,T). It turns out that whenver the nilpotency index n is at most 6, then interesting properties about an indecomposable triple (V,U,T) can be read off from the dimension pair.
The study of the representation theory of skew group algebras was started in the eighties with the works of de la Peña, and Reiten and Riedtmann. Given an algebra A and a group G acting on A, we define the skew group algebra A[G]. It turns out that A[G] often retains many features from A, such as being representation-finite, being hereditary, being tilted or quasitilted, etc.
In this talk, we study the interplay between the skew group algebras and the so-called piecewise hereditary algebras, that is algebras A for which there exist a hereditary abelian category H and a triangle-equivalence between the derived categories of bounded complexes over A and H. Those algebras, first studied by Happel, Rickard and Schofield and later by Happel, Reiten and Smalø, played a decisive role in the classification of selfinjective algebras of finite and tame representation type. We show that, under some assumptions, the skew group algebra A[G] is piecewise hereditary when so is A.
The talk is based on joint work in progress with Julie Dionne and Marcelo Lanzilotta.
We study the Auslander-Reiten components which have a sectional bypass and we characterize them. We show that a bypass defines a new irreducible morphism.
Joint work with Claudia Chaio and Edson Ribeiro Alvares.
In Noether normalizations of some subrings of graphs (Comm. Algebra 29(2001), 5525-5534), Alcántar asked when a standard Noether normalizations of the monomial subring or edge subring of a graph exists. In this talk we will give an answer to the above question.
This is a joint work with Florian Luca.