234 research outputs found
Cluster structures for 2-Calabi-Yau categories and unipotent groups
We investigate cluster tilting objects (and subcategories) in triangulated
2-Calabi-Yau categories and related categories. In particular we construct a
new class of such categories related to preprojective algebras of non Dynkin
quivers associated with elements in the Coxeter group. This class of
2-Calabi-Yau categories contains the cluster categories and the stable
categories of preprojective algebras of Dynkin graphs as special cases. For
these 2-Calabi-Yau categories we construct cluster tilting objects associated
with each reduced expression. The associated quiver is described in terms of
the reduced expression. Motivated by the theory of cluster algebras, we
formulate the notions of (weak) cluster structure and substructure, and give
several illustrations of these concepts. We give applications to cluster
algebras and subcluster algebras related to unipotent groups, both in the
Dynkin and non Dynkin case.Comment: 49 pages. For the third version the presentation is revised,
especially Chapter III replaces the old Chapter III and I
Triangulations of prisms and preprojective algebras of type
We show that indecomposable two-term presilting complexes over , the preprojective algebra of , are in bijection with internal -simplices in the prism , the product of an -simplex with a 1-simplex. We show further that this induces a bijection between triangulations of and two-term silting complexes over such that bistellar flips of triangulations correspond to mutations of two-term silting complexes. These bijections are shown to compatible with the known bijections involving the symmetric group
Mutation in triangulated categories and rigid Cohen-Macaulay modules
We introduce the notion of mutation of -cluster tilting subcategories in a
triangulated category with Auslander-Reiten-Serre duality. Using this idea, we
are able to obtain the complete classifications of rigid Cohen-Macaulay modules
over certain Veronese subrings.Comment: 52 pages. To appear in Invent. Mat
Activation of stress-activated protein kinase in osteoarthritic cartilage: evidence for nitric oxide dependence
AbstractObjective We have demonstrated in bovine chondrocytes that nitric oxide (NO) mediates IL1 dependent apoptosis under conditions of oxidant stress. This process is accompanied by activation of c-Jun NH2-terminal kinase (JNK; also called stress-activated protein kinase). In these studies we examined activation of JNK in explant cultures of human osteoarthritic cartilage obtained at joint replacement surgery and we characterized the role of peroxynitrite to act as an upstream trigger.Design A novel technique to isolate chondrocyte proteins (<10% of total cartilage protein) from cartilage specimens was developed. It was used to analyse JNK activation by a western blot technique. To examine the hypothesis that chondrocyte JNK activation is a result of increased peroxynitrite, in vitro experiments were performed in which cultured chondrocytes were incubated with this oxidant.Results Activated JNK was detected in the cytoplasm of osteoarthritis (OA) affected chondrocytes but not in that of controls. In vitro, chondrocytes produce NO and superoxide anion. IL-1 (48h), which induces nitric oxide synthase, resulted in an activation of JNK; this effect was reversed by N-monomethylarginine (NMA). TNFα treated chondrocytes at 48h produce superoxide anion (EPR method). Exposure of cells to peroxynitrite led to an accumulation of intracellular oxidants, in association with JNK activation and cell death by apoptosis.Conclusion We suggest that JNK activation is among the IL-1 elicited responses that injure articular chondrocytes and this activation of JNK is dependent on intracellular oxidant formation (including NO peroxynitrite). In addition, the extraction technique here described is a novel method that permits the quantitation and study of proteins such as JNK involved in the signaling pathways of chondrocytes within osteoarthritic cartilage
Torsion pairs and rigid objects in tubes
We classify the torsion pairs in a tube category and show that they are in
bijection with maximal rigid objects in the extension of the tube category
containing the Pruefer and adic modules. We show that the annulus geometric
model for the tube category can be extended to the larger category and
interpret torsion pairs, maximal rigid objects and the bijection between them
geometrically. We also give a similar geometric description in the case of the
linear orientation of a Dynkin quiver of type A.Comment: 25 pages, 13 figures. Paper shortened. Minor errors correcte
Non-commutative desingularization of determinantal varieties, I
We show that determinantal varieties defined by maximal minors of a generic
matrix have a non-commutative desingularization, in that we construct a maximal
Cohen-Macaulay module over such a variety whose endomorphism ring is
Cohen-Macaulay and has finite global dimension. In the case of the determinant
of a square matrix, this gives a non-commutative crepant resolution.Comment: 52 pages, 3 figures, all comments welcom
On Morita and derived equivalences for cohomological Mackey algebras
By results of the second author, a source algebra equivalence between two p-blocks of finite groups induces an equivalence between the categories of cohomological Mackey functors associated with these blocks, and a splendid derived equivalence between two blocks induces a derived equivalence between the corresponding categories ofcohomological Mackey functors. The main result of this paper proves a partial converse: an equivalence (resp. Rickard equivalence) between the categories of cohomological Mackey functors of two blocks of finite groups induces a permeable Morita (resp. derived) equivalence between the two block algebras
Torsion pairs and simple-minded systems in triangulated categories
Let T be a Hom-finite triangulated Krull-Schmidt category over a field k.
Inspired by a definition of Koenig and Liu, we say that a family S of pairwise
orthogonal objects in T with trivial endomorphism rings is a simple-minded
system if its closure under extensions is all of T. We construct torsion pairs
in T associated to any subset X of a simple-minded system S, and use these to
define left and right mutations of S relative to X. When T has a Serre functor
\nu, and S and X are invariant under \nu[1], we show that these mutations are
again simple-minded systems. We are particularly interested in the case where T
is the stable module category of a self-injective algebra \Lambda. In this
case, our mutation procedure parallels that introduced by Koenig and Yang for
simple-minded collections in the derived category of \Lambda. It follows that
the mutation of the set of simple \Lambda-modules relative to X yields the
images of the simple \Gamma-modules under a stable equivalence between \Gamma\
and \Lambda, where \Gamma\ is the tilting mutation of \Lambda\ relative to X.Comment: Minor corrections. To appear in Applied Categorical Structures. The
final publication is available at springerlink.com:
http://link.springer.com/article/10.1007%2Fs10485-014-9365-
The GL(2, C) McKay correspondence
In this paper we show that for any affine complete rational surface singularity the quiver of the reconstruction algebra can be determined combinatorially from the dual graph of the minimal resolution. As a consequence the derived category of the minimal resolution is equivalent to the derived category of an algebra whose quiver is determined by the dual graph. Also, for any finite subgroup G of GL(2,C)GL(2,C), it means that the endomorphism ring of the special CM CC [[x, y]]G-modules can be used to build the dual graph of the minimal resolution of C2/GC2/G, extending McKay’s observation (McKay, Proc Symp Pure Math, 37:183–186, 1980) for finite subgroups of SL(2,C)SL(2,C) to all finite subgroups of GL(2,C)GL(2,C)
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