On Nichols algebras associated to simple racks

This is a report on the present state of the problem of determining the dimension of the Nichols algebra associated to a rack and a cocycle. This is relevant for the classification of finite-dimensional complex pointed Hopf algebras whose group of group-likes is non-abelian. We deal mainly with simple racks. We recall the notion of rack of type D, collect the known lists of simple racks of type D and include preliminary results for the open cases. This notion is important because the Nichols algebra associated to a rack of type D and any cocycle has infinite dimension. For those racks not of type D, the computation of the cohomology groups is needed. We discuss some techniques for this problem and compute explicitly the cohomology groups corresponding to some conjugacy classes in symmetric or alternating groups of low order.Comment: 26 pages, minor change

Current range characteristics of Swiss stone pine (Pinus cembra L.) along the Carpathians revealed by chloroplast SSR markers.

We investigated the diversity pattern of nine Swiss stone pine (Pinus cembra L.) populations along the Carpathian range including the High Tatras, by using six chloroplast DNA microsatellites (cpSSR). Our aim was to detect genetically distinct regions by clustering of populations, and to tackle possible historical colonization routes. Our analysis referred to an investigated geographical range with the two most distant populations situated at about 500 air km. We found that the most diverse populations are situated at the two edges of the investigated part, in the Retezat Mts. (South Carpathians) and the High Tatras, and diversity decreases towards the populations of the Eastern Carpathians. Hierarchical clustering and NMDS revealed that the populations of the South Carpathians with the Tatras form a distinct cluster, significantly separated from those of the Eastern Carpathians. Moreover, based on the most variable chloroplast microsatellites, the four populations of the two range edges are not significantly different. Our results, supported also by palynological and late glacial macrofossil evidences, indicate refugial territories within the Retezat Mts. that conserved rich haplotype composition. From this refugial territory Pinus cembra might have colonized the Eastern Carpathians, and this was accompanied by a gradual decrease in population diversity. Populations of the High Tatras might have had the same role in the colonizing events of the Carpathians, as positive correlation was detected among populations lying from each other at a distance of 280 km, the maximum distance between neighbouring populations

On Nichols algebras over SL(2,Fq) and GL(2,Fq)

We compute necessary conditions on Yetter-Drinfeld modules over the groups SL(2,Fq) and GL(2,Fq) to generate finite dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with a group of group-likes isomorphic to one of these groups.Comment: Major exposition revision, including referees remarks. To appear in J. Math. Phys. 13 page

Electric dipole and magnetic quadrupole moments of the WW boson via a CP-violating HWWHWW vertex in effective Lagrangians

The possibility of nonnegligible $W$ electric dipole ($\widetilde{\mu}_W$) and magnetic quadrupole ($\widetilde{Q}_W$) moments induced by the most general $HWW$ vertex is examined via the effective Lagrangian technique. It is assumed that new heavy fermions induce an anomalous CP-odd component of the $HWW$ vertex, which can be parametrized by an $SU_L(2)\times U_Y(1)$-invariant dimension-six operator. This anomalous contribution, when combined with the standard model CP-even contribution, lead to CP-odd electromagnetic properties of the $W$ boson, which are characterized by the form factors $\Delta \widetilde{\kappa}$ and $\Delta \widetilde{Q}$. It is found that $\Delta \widetilde{\kappa}$ is divergent, whereas $\Delta \widetilde{Q}$ is finite, which reflects the fact that the latter cannot be generated at the one-loop level in any renormalizable theory. Assuming reasonable values for the unknown parameters, we found that $\widetilde{\mu}_W\sim 3-6\times 10^{-21}$ e-cm, which is eight orders of magnitude larger than the SM prediction and close to the upper bound derived from the neutron electric dipole moment. The estimated size of the somewhat less-studied $\widetilde{Q}_W$ moment is of the order of $-10^{-36}$ e-cm^2, which is fifteen orders of magnitude above the SM contribution.Comment: 7 pages, 6 figures, REVTEX styl

Using reduced representation libraries sequencing methods to identify cpDNA polymorphisms in European beech (Fagus sylvatica L)

Reduced representation genomic libraries (RRLs) are increasingly used to answer diverse questions in evolutionary biology, which remained unresolved otherwise. In the case of European beech (Fagus sylvatica L.), former applications of nuclear and chloroplast genetic markers indicated that most of the beech populations of Central, Eastern, and Northern Europe have a very homogeneous genetic structure. Thus, determining differentiation among populations of F. sylvatica at regional scale in Central Europe has remained a challenge for evolutionary biologists, mainly due to lack of variable genetic markers. In this study, we used the RAD-seq and GBS approaches to identify novel cpDNA polymorphisms that could be helpful to identify population structure of European beech at geographical scale. Our datasets allowed us to indicate a finer population structure and stronger patterns of spatial genetic structure than previous studies. It is expected that the applications of cpDNA-based phylogenetic approaches using whole-genome sequencing are able to provide even more detailed picture of species phylogeography than ever before

Molecular evidence for the natural production of homozygous Cupressus sempervirens L. lines by Cupressus dupreziana seed trees

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A RAPD, AFLP and SSR linkage map, and QTL analysis in European beech (Fagus sylvatica L.)

The genetic linkage map of European beech ( Fagus sylvatica L.) that we report here is the first to our knowledge. Based on a total of 312 markers (28 RAPDs, 274 AFLPs, 10 SSRs) scored in 143 individuals from a F(1) full-sib family. Two maps (one for each parent) were constructed according to a "two-way pseudo-testcross" mapping strategy. In the male map 119 markers could be clustered in 11 major groups (971 cM), while in the female map 132 markers were distributed in 12 major linkage groups (844 cM). In addition, four and one minor linkage groups (doublets and triplets) were obtained for the male and female map respectively. The two maps cover about 82% and 78% of the genome. Based on the position of 15 AFLP and 2 SSR loci segregating in both parents, seven homologous linkage groups could be identified. In the same pedigree we investigated the association with genetic markers of several quantitative traits: leaf area, leaf number and shape in 2 different years, specific leaf area, leaf carbon-isotope discrimination and tree height. A composite interval-mapping approach was used to estimate the number of QTLs, the amount of variation explained by each of them, and their position on the genetic linkage maps. Eight QTLs associated with leaf traits were found that explained between 15% and 35% of the trait variation, five on the female map and three on the male map