Overview of (pro-)Lie group structures on Hopf algebra character groups

Character groups of Hopf algebras appear in a variety of mathematical and physical contexts. To name just a few, they arise in non-commutative geometry, renormalisation of quantum field theory, and numerical analysis. In the present article we review recent results on the structure of character groups of Hopf algebras as infinite-dimensional (pro-)Lie groups. It turns out that under mild assumptions on the Hopf algebra or the target algebra the character groups possess strong structural properties. Moreover, these properties are of interest in applications of these groups outside of Lie theory. We emphasise this point in the context of two main examples: The Butcher group from numerical analysis and character groups which arise from the Connes--Kreimer theory of renormalisation of quantum field theories.Comment: 31 pages, precursor and companion to arXiv:1704.01099, Workshop on "New Developments in Discrete Mechanics, Geometric Integration and Lie-Butcher Series", May 25-28, 2015, ICMAT, Madrid, Spai

Weak Poisson structures on infinite dimensional manifolds and hamiltonian actions

We introduce a notion of a weak Poisson structure on a manifold $M$ modeled on a locally convex space. This is done by specifying a Poisson bracket on a subalgebra \cA \subeq C^\infty(M) which has to satisfy a non-degeneracy condition (the differentials of elements of \cA separate tangent vectors) and we postulate the existence of smooth Hamiltonian vector fields. Motivated by applications to Hamiltonian actions, we focus on affine Poisson spaces which include in particular the linear and affine Poisson structures on duals of locally convex Lie algebras. As an interesting byproduct of our approach, we can associate to an invariant symmetric bilinear form $\kappa$ on a Lie algebra \g and a $\kappa$-skew-symmetric derivation $D$ a weak affine Poisson structure on \g itself. This leads naturally to a concept of a Hamiltonian $G$-action on a weak Poisson manifold with a \g-valued momentum map and hence to a generalization of quasi-hamiltonian group actions

MetaBar - a tool for consistent contextual data acquisition and standards compliant submission

Background: Environmental sequence datasets are increasing at an exponential rate; however, the vast majority of them lack appropriate descriptors like sampling location, time and depth/altitude: generally referred to as metadata or contextual data. The consistent capture and structured submission of these data is crucial for integrated data analysis and ecosystems modeling. The application MetaBar has been developed, to support consistent contextual data acquisition. Results: MetaBar is a spreadsheet and web-based software tool designed to assist users in the consistent acquisition, electronic storage, and submission of contextual data associated to their samples. A preconfigured Microsoft Excel spreadsheet is used to initiate structured contextual data storage in the field or laboratory. Each sample is given a unique identifier and at any stage the sheets can be uploaded to the MetaBar database server. To label samples, identifiers can be printed as barcodes. An intuitive web interface provides quick access to the contextual data in the MetaBar database as well as user and project management capabilities. Export functions facilitate contextual and sequence data submission to the International Nucleotide Sequence Database Collaboration (INSDC), comprising of the DNA DataBase of Japan (DDBJ), the European Molecular Biology Laboratory database (EMBL) and GenBank. MetaBar requests and stores contextual data in compliance to the Genomic Standards Consortium specifications. The MetaBar open source code base for local installation is available under the GNU General Public License version 3 (GNU GPL3). Conclusion: The MetaBar software supports the typical workflow from data acquisition and field-sampling to contextual data enriched sequence submission to an INSDC database. The integration with the megx.net marine Ecological Genomics database and portal facilitates georeferenced data integration and metadata-based comparisons of sampling sites as well as interactive data visualization. The ample export functionalities and the INSDC submission support enable exchange of data across disciplines and safeguarding contextual data

Ecogenomic Perspectives on Domains of Unknown Function: Correlation-Based Exploration of Marine Metagenomes

Background: The proportion of conserved DNA sequences with no clear function is steadily growing in bioinformatics databases. Studies of sequence and structural homology have indicated that many uncharacterized protein domain sequences are variants of functionally described domains. If these variants promote an organism's ecological fitness, they are likely to be conserved in the genome of its progeny and the population at large. The genetic composition of microbial communities in their native ecosystems is accessible through metagenomics. We hypothesize the co-variation of protein domain sequences across metagenomes from similar ecosystems will provide insights into their potential roles and aid further investigation. Methodology/Principal findings: We calculated the correlation of Pfam protein domain sequences across the Global Ocean Sampling metagenome collection, employing conservative detection and correlation thresholds to limit results to well-supported hits and associations. We then examined intercorrelations between domains of unknown function (DUFs) and domains involved in known metabolic pathways using network visualization and cluster-detection tools. We used a cautious "guilty-by-association'' approach, referencing knowledge-level resources to identify and discuss associations that offer insight into DUF function. We observed numerous DUFs associated to photobiologically active domains and prevalent in the Cyanobacteria. Other clusters included DUFs associated with DNA maintenance and repair, inorganic nutrient metabolism, and sodium-translocating transport domains. We also observed a number of clusters reflecting known metabolic associations and cases that predicted functional reclassification of DUFs. Conclusion/Significance: Critically examining domain covariation across metagenomic datasets can grant new perspectives on the roles and associations of DUFs in an ecological setting. Targeted attempts at DUF characterization in the laboratory or in silico may draw from these insights and opportunities to discover new associations and corroborate existing ones will arise as more large-scale metagenomic datasets emerge

Individual decision making and the evolutionary roots of institutions

Just as many aspects of individual decisions are sometimes called unconscious or automatic, we know that some institutions have evolved through unconscious, nondeliberative mechanisms. Their function can also be largely nondeliberative, as in the case of some institutions that may structure behavior without requiring any reflection on the part of the participants. On the other hand, political institutions exist for the purpose of bringing deliberative mechanisms to bear on institutions in the hope of changing them for the better. The immense project of building an integrated explanation of institutions from individual brains to nations-has only barely begun. In this chapter, we report on our discussions that attempted to sketch the mechanisms that connect individuals to large-scale institutions. We begin with a discussion of current thought on the design of individual decision making. If institutions regulate behavior, then presumably the mechanisms that have evolved to produce individual behavior will be relevant to the broader enterprise of integrating these two scales of explanation. Then we explore ways in which institutions may have evolved, both as a result of individual decision making and as a result of processes distinct from those that govern individual behavior. We approach this topic from two perspectives. Seen one way, unconscious psychological forces constrain the design of institutions, sometimes powerfully. Seen another way, unconscious population-level processes create functional institutional design that few social architects could conceive of with their individual deliberate faculties

Localization via Automorphisms of the CARs. Local gauge invariance

The classical matter fields are sections of a vector bundle E with base manifold M. The space L^2(E) of square integrable matter fields w.r.t. a locally Lebesgue measure on M, has an important module action of C_b^\infty(M) on it. This module action defines restriction maps and encodes the local structure of the classical fields. For the quantum context, we show that this module action defines an automorphism group on the algebra A, of the canonical anticommutation relations on L^2(E), with which we can perform the analogous localization. That is, the net structure of the CAR, A, w.r.t. appropriate subsets of M can be obtained simply from the invariance algebras of appropriate subgroups. We also identify the quantum analogues of restriction maps. As a corollary, we prove a well-known "folk theorem," that the algebra A contains only trivial gauge invariant observables w.r.t. a local gauge group acting on E.Comment: 15 page