On Generalized Cluster Categories

Cluster categories have been introduced by Buan, Marsh, Reineke, Reiten and Todorov in order to categorify Fomin-Zelevinsky cluster algebras. This survey motivates and outlines the construction of a generalization of cluster categories, and explains different applications of these new categories in representation theory.Comment: survey 54pages, v2: small improvements, published in the proceedings of ICRA XIV "Representations of Algebras and Related Topics", European Mathematical Societ

Cluster categories for algebras of global dimension 2 and quivers with potential

Let $k$ be a field and $A$ a finite-dimensional $k$-algebra of global dimension $\leq 2$. We construct a triangulated category \Cc_A associated to $A$ which, if $A$ is hereditary, is triangle equivalent to the cluster category of $A$. When \Cc_A is \Hom-finite, we prove that it is 2-CY and endowed with a canonical cluster-tilting object. This new class of categories contains some of the stable categories of modules over a preprojective algebra studied by Geiss-Leclerc-Schr{\"o}er and by Buan-Iyama-Reiten-Scott. Our results also apply to quivers with potential. Namely, we introduce a cluster category \Cc_{(Q,W)} associated to a quiver with potential $(Q,W)$. When it is Jacobi-finite we prove that it is endowed with a cluster-tilting object whose endomorphism algebra is isomorphic to the Jacobian algebra \Jj(Q,W).Comment: 46 pages, small typos as it will appear in Annales de l'Institut Fourie

On the structure of triangulated category with finitely many indecomposables

We study the problem of classifying triangulated categories with finite-dimensional morphism spaces and finitely many indecomposables over an algebraically closed field. We obtain a new proof of the following result due to Xiao and Zhu: the Auslander-Reiten quiver of such a category is of the form $\mathbb{Z}\Delta/G$ where $\Delta$ is a disjoint union of simply laced Dynkin diagrams and $G$ a weakly admissible group of automorphisms of $\mathbb{Z}\Delta$. Then we prove that for `most' groups $G$, the category \T is standard, \emph{i.e.} $k$-linearly equivalent to an orbit category \mathcal{D}^b(\modd k\Delta)/\Phi. This happens in particular when \T is maximal $d$-Calabi-Yau with $d\geq2$. Moreover, if \T is standard and algebraic, we can even construct a triangle equivalence between \T and the corresponding orbit category. Finally we give a sufficient condition for the category of projectives of a Frobenius category to be triangulated. This allows us to construct non standard 1-Calabi-Yau categories using deformed preprojective algebras of generalized Dynkin type

Logistical constraints on international trade in the Maghreb

Without a competitive transport industry, the Maghreb countries will not truly benefit from reform aimed at increasing the region's share of international trade. A study of barriers to the region's trade, especially with countries of the European Union, identified more than 30 barriers, in four categories: barriers to imports, to exports, of infrastructure and equipment, and of intra-Maghreb trade. These include: 1) direct barriers including: (a) from traditional distortions (price, discriminatory access to markets); (b) nontariff barriers (administrative, regulatory and tax-related restrictions); (c) traffic agreements (protecting national flags); and (d) lack of infrastructure and equipment; and 2) indirect barriers deriving from: (a) trade harmonization (simplified customs procedures and tariffs structures, elimination of quotas, reduction of customs tariffs on transport equipment); and (b) technology lags (telecommunications and handling). The authors quantify barriers in terms of"tariff equivalents,"expressed as a nominal rate of protection based on the freeon board value of the merchandise. But the nominal rate of protection measures only the direct costs of distortions. The effective rate of protection measures both direct and indirect effects, and effective rates are generally twice as high as nominal rates. To reconcile macroeconomic and microeconomic approaches to measuring effective rates, the authors use a partial equilibrium model (SMART model) to estimate the impact on the balance of payments of eliminating excess costs. Most of the corrective policies they recommend concern multimodal transport in the trade between Europe and the Arab Maghreb Union. The challenges are considerable: not only does such a system pave the way for cost and time savings ("just-in-time"transport), but it also adopts the logistics management that the most advanced European enterprises use to orchestrate their raw material purchasing, production and marketing functions. A multimodal transport system allow them to reduce inventories significantly and to respond better to volatile demand. Essential for just-in-time multimodal transport and logistics management include efficient modern transport techniques, efficient communications systems, efficient modern merchandise handling, and appropriate regulations. These conditions are still not fully in place in the Maghreb countries, except partially in some parts of the clothing and textile industry.Economic Theory&Research,Transport and Trade Logistics,Common Carriers Industry,Environmental Economics&Policies,Payment Systems&Infrastructure,Economic Theory&Research,Transport and Trade Logistics,Common Carriers Industry,TF054105-DONOR FUNDED OPERATION ADMINISTRATION FEE INCOME AND EXPENSE ACCOUNT,Environmental Economics&Policies