Smash Products of Calabi-Yau Algebras by Hopf Algebras

Let H be a Hopf algebra and A be an H-module algebra. This article investigates when the smash product A#H is (skew) Calabi-Yau, has Van den Bergh duality or is Artin-Schelter regular or Gorenstein. In particular, if A and H are skew Calabi-Yau, then so is A#H and its Nakayama automorphism is expressed using the ones of A and H. This is based on a description of the inverse dualising complex of A#H when A is a homologically smooth dg algebra and H is homologically smooth and with invertible antipode. This description is also used to explain the compatibility of standard constructions of Calabi-Yau dg algebras with taking smash products.Comment: Minor corrections and changes to reflect the published articl

Topological invariants of piecewise hereditary algebras

We investigate the Galois coverings of piecewise algebras and more particularly their behaviour under derived equivalences. Under a technical assumption which is satisfied if the algebra is derived equivalent to a hereditary algebra, we prove that there exists a universal Galois covering whose group of automorphisms is free and depends only on the derived category of the algebra. As a corollary, we prove that the algebra is simply connected if and only if its first Hochschild cohomology vanishes.Comment: The hypotheses of the main theorem were modified: The next now deals mainly with piecewise hereditary algebras which are derived equivalent to a hereditary algebra (instead of all piecewise hereditary algebras in the previous version

On the Morita Reduced Versions of Skew Group Algebras of Path Algebras

Let R be the skew group algebra of a finite group acting on the path algebra of a quiver. This article develops both theoretical and practical methods to do computations in the Morita reduced algebra associated to R. Reiten and Riedtmann proved that there exists an idempotent e of R such that the algebra eRe is both Morita equivalent to R and isomorphic to the path algebra of some quiver which was described by Demonet. This article gives explicit formulas for the decomposition of any element of eRe as a linear combination of paths in the quiver described by Demonet. This is done by expressing appropriate compositions and pairings in a suitable monoidal category which takes into account the representation theory of the finite group

Water and the WTO: Don’t kill the messenger.

It is widely recognized that forecasting future climate shocks at a regional level̶which regions will be flooded, which ones will be under water stress on a year by year basis̶is largely out of reach. In such circumstances, trade gets back a role that has faded away during the last sixty years of relatively stable climatic, economic and political conditions. It is to be the ultimate insurer. Regions under sudden water stress will need to import food products in exceptional quantities, and trade happens to be a cheap (efficient) insurance scheme to face a sudden instability in water resources in some parts of the world. There are thus good reasons to look at whether the world trade regime could provide a strong and sound framework to the international water regime. Not many papers have looked at this issue. They generally see the WTO as a source of problems rather than of solutions. Hence, they argue for specific international agreements on water. But, the climate community experience of the COP15 (the 2009 Copenhagen Summit on Climate Change) is a strong warning signal showing how difficult it is to build a “specific” international regime. In contrast, this paper argues that the basic principles on which the world trade regime is built would be equally useful for the international water regime, and that the WTO rules are flexible enough to address the specific problems raised by water management in a international context. It also argues that, if current international trade mirrors domestic distortions, limiting such trade will cost a lot in terms of water use. Killing the messenger (trade) does not solve the problems (domestic markets).

Covering techniques in Auslander-Reiten theory

Given a finite dimensional algebra over a perfect field the text introduces covering functors over the mesh category of any modulated Auslander-Reiten component of the algebra. This is applied to study the composition of irreducible morphisms between indecomposable modules in relation with the powers of the radical of the module category.Comment: Minor modifications. Final version to appear in the Journal of Pure and Applied Algebr

Degrees of irreducible morphisms and finite-representation type

We study the degree of irreducible morphisms in any Auslander-Reiten component of a finite dimensional algebra over an algebraically closed field. We give a characterization for an irreducible morphism to have finite left (or right) degree. This is used to prove our main theorem: An algebra is of finite representation type if and only if for every indecomposable projective the inclusion of the radical in the projective has finite right degree, which is equivalent to require that for every indecomposable injective the epimorphism from the injective to its quotient by its socle has finite left degree. We also apply the techniques that we develop: We study when the non-zero composite of a path of $n$ irreducible morphisms between indecomposable modules lies in the $n+1$-th power of the radical; and we study the same problem for sums of such paths when they are sectional, thus proving a generalisation of a pioneer result of Igusa and Todorov on the composite of a sectional path.Comment: 20 page

Data Analysis in Multimedia Quality Assessment: Revisiting the Statistical Tests

Assessment of multimedia quality relies heavily on subjective assessment, and is typically done by human subjects in the form of preferences or continuous ratings. Such data is crucial for analysis of different multimedia processing algorithms as well as validation of objective (computational) methods for the said purpose. To that end, statistical testing provides a theoretical framework towards drawing meaningful inferences, and making well grounded conclusions and recommendations. While parametric tests (such as t test, ANOVA, and error estimates like confidence intervals) are popular and widely used in the community, there appears to be a certain degree of confusion in the application of such tests. Specifically, the assumption of normality and homogeneity of variance is often not well understood. Therefore, the main goal of this paper is to revisit them from a theoretical perspective and in the process provide useful insights into their practical implications. Experimental results on both simulated and real data are presented to support the arguments made. A software implementing the said recommendations is also made publicly available, in order to achieve the goal of reproducible research