Quaternionic and Poisson-Lie structures in 3d gravity: the cosmological constant as deformation parameter

Each of the local isometry groups arising in 3d gravity can be viewed as the group of unit (split) quaternions over a ring which depends on the cosmological constant. In this paper we explain and prove this statement, and use it as a unifying framework for studying Poisson structures associated with the local isometry groups. We show that, in all cases except for Euclidean signature with positive cosmological constant, the local isometry groups are equipped with the Poisson-Lie structure of a classical double. We calculate the dressing action of the factor groups on each other and find, amongst others, a simple and unified description of the symplectic leaves of SU(2) and SL(2,R). We also compute the Poisson structure on the dual Poisson-Lie groups of the local isometry groups and on their Heisenberg doubles; together, they determine the Poisson structure of the phase space of 3d gravity in the so-called combinatorial description.Comment: 34 pages, minor corrections, references adde

The Split-Algebras and Non-compact Hopf Maps

We develop a noncompact version of the Hopf maps based on the split algebras. The split algebras consist of three species: split-complex numbers, split quaternions, and split octonions. They correspond to three noncompact Hopf maps that represent topological maps between hyperboloids in different dimensions with hyperboloid bundle. We realize such noncompact Hopf maps in two ways: one is to utilize the split-imaginary unit, and the other is to utilize the ordinary imaginary unit. Topological structures of the hyperboloid bundles are explored, and the canonical connections are naturally regarded as noncompact gauge field of monopoles.Comment: 40 pages, no figures, + 8 pages, discussions and references added, published versio

Comparison of nesting ecology of three co-existing Atlantic Forest woodpeckers reveals narrow specialization in the Helmeted Woodpecker Celeus galeatus

The Atlantic Forest in southeastern South America is a biodiversity hotspot with high species richness and high forest loss. Three large to medium-sized woodpecker species (Robust Woodpecker Campephilus robustus, Lineated Woodpecker Dryocopus lineatus, and the globally threatened Helmeted Woodpecker Celeus galeatus) coexist in the south of the ecoregion, and show remarkable convergence in plumage patterns and colouration, possibly driven by interspecific interactions. We assessed differentiation in nest tree species, nest substrates, and timing of breeding and evaluated whether nesting ecology may be a factor explaining an association with mature forests in the Helmeted Woodpecker. Robust and Lineated Woodpeckers excavated cavities in nine and eight nest tree species, respectively, whereas Helmeted Woodpeckers specialized narrowly on laurel trees (88% of nest cavities in two species in genus Nectandra of 52 ± 14 cm DBH). Helmeted Woodpecker is further distinguished by a more frequent use of living, partially decayed trees and by more frequent re-use of nest cavities. When nesting in selectively logged forest (vs. old-growth), Robust and Lineated Woodpeckers used trees with smaller DBH and Robust Woodpeckers excavated lower in trees. Helmeted Woodpeckers selected nest trees with similar dimensions in both environments, indicating lesser flexibility. Helmeted Woodpeckers bred 20-23 days later in the year than Lineated and Robust Woodpeckers, potentially allowing them to avoid nesting near the two larger species. The narrow specialization of Helmeted Woodpecker on nesting in Nectandra trees of large sizes at the onset of decay is a possible factor in making it sensitive to selective logging effects. To stop declines of Helmeted Woodpecker we recommend (1) sparing large trees from selective logging (including Nectandra and tree species used for roosting), and (2) restoring more forested areas to mature forest conditions.Fil: Lammertink, J. Martjan. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Nordeste. Instituto de Biología Subtropical. Instituto de Biología Subtropical - Nodo Posadas | Universidad Nacional de Misiones. Instituto de Biología Subtropical. Instituto de Biología Subtropical - Nodo Posadas; ArgentinaFil: Fernández, Juan Manuel. Universidad Nacional del Nordeste; Argentina. Provincia de Entre Ríos. Centro de Investigaciones Científicas y Transferencia de Tecnología a la Producción. Universidad Autónoma de Entre Ríos. Centro de Investigaciones Científicas y Transferencia de Tecnología a la Producción. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Centro de Investigaciones Científicas y Transferencia de Tecnología a la Producción; ArgentinaFil: Cockle, Kristina Louise. University of British Columbia; Canadá. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Nordeste. Instituto de Biología Subtropical. Instituto de Biología Subtropical - Nodo Posadas | Universidad Nacional de Misiones. Instituto de Biología Subtropical. Instituto de Biología Subtropical - Nodo Posadas; Argentin