56 research outputs found
Galois coverings of weakly shod algebras
We investigate the Galois coverings of weakly shod algebras. For a weakly
shod algebra not quasi-tilted of canonical type, we establish a correspondence
between its Galois coverings and the Galois coverings of its connecting
component. As a consequence, we show that a weakly shod algebra is simply
connected if and only if its first Hochschild cohomology group vanishes.Comment: Some references were added. The proof of Lemma 6.5 was modifie
Tilted algebras and short chains of modules
We provide an affirmative answer for the question raised almost twenty years
ago concerning the characterization of tilted artin algebras by the existence
of a sincere finitely generated module which is not the middle of a short
chain
Orlov's Equivalence and Maximal Cohen-Macaulay Modules over the Cone of an Elliptic Curve
We describe a method for doing computations with Orlov's equivalence between
the bounded derived category of certain hypersurfaces and the stable category
of graded matrix factorisations of the polynomials describing these
hypersurfaces. In the case of a smooth elliptic curve over an algebraically
closed field we describe the indecomposable graded matrix factorisations of
rank one. Since every indecomposable Maximal Cohen-Macaulay module over the
completion of a smooth cubic curve is gradable, we obtain explicit descriptions
of all indecomposable rank one matrix factorisations of such potentials.
Finally, we explain how to compute all indecomposable matrix factorisations of
higher rank with the help of a computer algebra system.Comment: 26 page
Derived equivalence classification of the cluster-tilted algebras of Dynkin type E
We obtain a complete derived equivalence classification of the cluster-tilted
algebras of Dynkin type E. There are 67, 416, 1574 algebras in types E6, E7 and
E8 which turn out to fall into 6, 14, 15 derived equivalence classes,
respectively. This classification can be achieved computationally and we
outline an algorithm which has been implemented to carry out this task. We also
make the classification explicit by giving standard forms for each derived
equivalence class as well as complete lists of the algebras contained in each
class; as these lists are quite long they are provided as supplementary
material to this paper. From a structural point of view the remarkable outcome
of our classification is that two cluster-tilted algebras of Dynkin type E are
derived equivalent if and only if their Cartan matrices represent equivalent
bilinear forms over the integers which in turn happens if and only if the two
algebras are connected by a sequence of "good" mutations. This is reminiscent
of the derived equivalence classification of cluster-tilted algebras of Dynkin
type A, but quite different from the situation in Dynkin type D where a
far-reaching classification has been obtained using similar methods as in the
present paper but some very subtle questions are still open.Comment: 19 pages. v4: completely rewritten version, to appear in Algebr.
Represent. Theory. v3: Main theorem strengthened by including "good"
mutations (cf. also arXiv:1001.4765). Minor editorial changes. v2: Third
author added. Major revision. All questions left open in the earlier version
by the first two authors are now settled in v2 and the derived equivalence
classification is completed. arXiv admin note: some text overlap with
arXiv:1012.466
The double Ringel-Hall algebra on a hereditary abelian finitary length category
In this paper, we study the category of semi-stable
coherent sheaves of a fixed slope over a weighted projective curve. This
category has nice properties: it is a hereditary abelian finitary length
category. We will define the Ringel-Hall algebra of and
relate it to generalized Kac-Moody Lie algebras. Finally we obtain the Kac type
theorem to describe the indecomposable objects in this category, i.e. the
indecomposable semi-stable sheaves.Comment: 29 page
Krull Dimension of Tame Generalized Multicoil Algebras
We determine the Krull dimension of the module category of finite dimensional tame generalized multicoil algebras over an algebraically closed field, which are domestic
Categorification of skew-symmetrizable cluster algebras
We propose a new framework for categorifying skew-symmetrizable cluster
algebras. Starting from an exact stably 2-Calabi-Yau category C endowed with
the action of a finite group G, we construct a G-equivariant mutation on the
set of maximal rigid G-invariant objects of C. Using an appropriate cluster
character, we can then attach to these data an explicit skew-symmetrizable
cluster algebra. As an application we prove the linear independence of the
cluster monomials in this setting. Finally, we illustrate our construction with
examples associated with partial flag varieties and unipotent subgroups of
Kac-Moody groups, generalizing to the non simply-laced case several results of
Gei\ss-Leclerc-Schr\"oer.Comment: 64 page
Higher S-dualities and Shephard-Todd groups
Abstract: Seiberg and Witten have shown that in N=2 SQCD with Nf = 2Nc = 4 the S-duality group PSL2\u2124 acts on the flavor charges, which are weights of Spin(8), by triality. There are other N=2 SCFTs in which SU(2) SYM is coupled to strongly-interacting non-Lagrangian matter: their matter charges are weights of E6, E7 and E8 instead of Spin(8). The S-duality group PSL2\u2124 acts on these weights: what replaces Spin(8) triality for the E6, E7, E8root lattices? In this paper we answer the question. The action on the matter charges of (a finite central extension of) PSL2\u2124 factorizes trough the action of the exceptional Shephard-Todd groups G4 and G8 which should be seen as complex analogs of the usual triality group S3 43WeylA2. Our analysis is based on the identification of S-duality for SU(2) gauge SCFTs with the group of automorphisms of the cluster category of weighted projective lines of tubular type. \ua9 2015, The Author(s)
Cycle-finite module categories
We describe the structure of module categories of finite dimensional algebras
over an algebraically closed field for which the cycles of nonzero
nonisomorphisms between indecomposable finite dimensional modules are finite
(do not belong to the infinite Jacobson radical of the module category).
Moreover, geometric and homological properties of these module categories are
exhibited
The GALLEX Project
AbstractThe GALLEX collaboration aims at the detection of solar neutrinos in a radiochemical experiment employing 30 tons of Gallium in form of concentrated aqueous Gallium-chloride solution. The detector is primarily sensitive to the otherwise inaccessible pp-neutrinos. Details of the experiment have been repeatedly described before [1-7]. Here we report the present status of implementation in the Laboratori Nazionali del Gran Sasso (Italy). So far, 12.2 tons of Gallium are at hand. The present status of development allows to start the first full scale run at the time when 30 tons of Gallium become available. This date is expected to be January, 1990
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