Variety of idempotents in nonassociative algebras

In this paper, we study the variety of all nonassociative (NA) algebras from the idempotent point of view. We are interested, in particular, in the spectral properties of idempotents when algebra is generic, i.e. idempotents are in general position. Our main result states that in this case, there exist at least $n-1$ nontrivial obstructions (syzygies) on the Peirce spectrum of a generic NA algebra of dimension $n$. We also discuss the exceptionality of the eigenvalue $\lambda=\frac12$ which appears in the spectrum of idempotents in many classical examples of NA algebras and characterize its extremal properties in metrised algebras.Comment: 27 pages, 1 figure, submitte

On the representation ring of the polynomial algebra over a perfect field

We consider the tensor product of modules over the polynomial algebra corresponding to the usual tensor product of linear operators. We present a general description of the representation ring in case the ground field k is perfect. It is made explicit in the special cases when k is real closed respectively algebraically closed. Furthermore, we discuss the generalisation of this problem to representations of quivers. In particular the representation ring of quivers of extended Dynkin type A is provided.Comment: 17 page

Global large carnivore conservation and international law

International cooperation, including through international legal instruments, appears important for the conservation of large carnivores worldwide. This is due to, inter alia, the worrying conservation status and population trends of many large carnivore species; the importance of large carnivores for biodiversity conservation at large; their occurrence at low densities, with many populations extending across various countries; and the international nature of particular threats. For the 31 heaviest species in the order Carnivora, this study (i) documents to what extent existing international legal instruments contribute to large carnivore conservation, and (ii) identifies ways of optimizing their contribution in this regard. From this dual perspective, it reviews all global wildlife conservation treaties—Ramsar Wetlands Convention, World Heritage Convention, Convention on Trade in Endangered Species, Convention on Migratory Species (CMS), Convention on Biological Diversity (CBD)—and selected regional instruments, using standard international law research methodology. Results indicate that a substantial body of relevant international law already exists, whereas simultaneously there is clear potential for enhancing the contribution of international law to large carnivore conservation. Avenues for pursuing this include promotion of instruments’ effective implementation; clarification of their precise implications for large carnivore conservation; development of formal guidance; expansion of instruments’ scope in terms of species, sites and countries; and creation of new instruments. The CMS and CBD hold particular potential in some of these respects. The experiences being gained under European legal instruments constitute an interesting ‘laboratory’ regarding human coexistence with expanding large carnivore populations and transboundary cooperation at the (sub)population level