Central extensions of groups of sections

If q : P -> M is a principal K-bundle over the compact manifold M, then any invariant symmetric V-valued bilinear form on the Lie algebra k of K defines a Lie algebra extension of the gauge algebra by a space of bundle-valued 1-forms modulo exact forms. In the present paper we analyze the integrability of this extension to a Lie group extension for non-connected, possibly infinite-dimensional Lie groups K. If K has finitely many connected components we give a complete characterization of the integrable extensions. Our results on gauge groups are obtained by specialization of more general results on extensions of Lie groups of smooth sections of Lie group bundles. In this more general context we provide sufficient conditions for integrability in terms of data related only to the group K.Comment: 54 pages, revised version, to appear in Ann. Glob. Anal. Geo

Unitary Representations of Unitary Groups

In this paper we review and streamline some results of Kirillov, Olshanski and Pickrell on unitary representations of the unitary group \U(\cH) of a real, complex or quaternionic separable Hilbert space and the subgroup \U_\infty(\cH), consisting of those unitary operators $g$ for which g - \1 is compact. The Kirillov--Olshanski theorem on the continuous unitary representations of the identity component \U_\infty(\cH)_0 asserts that they are direct sums of irreducible ones which can be realized in finite tensor products of a suitable complex Hilbert space. This is proved and generalized to inseparable spaces. These results are carried over to the full unitary group by Pickrell's Theorem, asserting that the separable unitary representations of \U(\cH), for a separable Hilbert space \cH, are uniquely determined by their restriction to \U_\infty(\cH)_0. For the $10$ classical infinite rank symmetric pairs $(G,K)$ of non-unitary type, such as (\GL(\cH),\U(\cH)), we also show that all separable unitary representations are trivial.Comment: 42 page

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

Principal 2-bundles and their gauge 2-groups

In this paper we introduce principal 2-bundles and show how they are classified by non-abelian Cech cohomology. Moreover, we show that their gauge 2-groups can be described by 2-group-valued functors, much like in classical bundle theory. Using this, we show that, under some mild requirements, these gauge 2-groups possess a natural smooth structure. In the last section we provide some explicit examples.Comment: 40 pages; v3: completely revised and extended, classification corrected, name changed, to appear in Forum Mat

Guide for Controlling Cotton Insects in the High Plains, Rolling Plains and Trans-Pecos Areas of Texas.

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Full regularity for a C*-algebra of the Canonical Commutation Relations. (Erratum added)

The Weyl algebra,- the usual C*-algebra employed to model the canonical commutation relations (CCRs), has a well-known defect in that it has a large number of representations which are not regular and these cannot model physical fields. Here, we construct explicitly a C*-algebra which can reproduce the CCRs of a countably dimensional symplectic space (S,B) and such that its representation set is exactly the full set of regular representations of the CCRs. This construction uses Blackadar's version of infinite tensor products of nonunital C*-algebras, and it produces a "host algebra" (i.e. a generalised group algebra, explained below) for the \sigma-representation theory of the abelian group S where \sigma(.,.):=e^{iB(.,.)/2}. As an easy application, it then follows that for every regular representation of the Weyl algebra of (S,B) on a separable Hilbert space, there is a direct integral decomposition of it into irreducible regular representations (a known result). An Erratum for this paper is added at the end.Comment: An erratum was added to the original pape

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

Two TRPV1 receptor antagonists are effective in two different experimental models of migraine

Background The capsaicin and heat responsive ion channel TRPV1 is expressed on trigeminal nociceptive neurons and has been implicated in the pathophysiology of migraine attacks. Here we investigate the efficacy of two TRPV1 channel antagonists in blocking trigeminal activation using two in vivo models of migraine. Methods Male Sprague–Dawley rats were used to study the effects of the TRPV1 antagonists JNJ-38893777 and JNJ-17203212 on trigeminal activation. Expression of the immediate early gene c-fos was measured following intracisternal application of inflammatory soup. In a second model, CGRP release into the external jugular vein was determined following injection of capsaicin into the carotid artery. Results Inflammatory up-regulation of c-fos in the trigeminal brain stem complex was dose-dependently and significantly reduced by both TRPV1 antagonists. Capsaicin-induced CGRP release was attenuated by JNJ-38893777 only in higher dosage. JNJ-17203212 was effective in all doses and fully abolished CGRP release in a time and dose-dependent manner. Conclusion Our results describe two TRPV1 antagonists that are effective in two in vivo models of migraine. These results suggest that TRPV1 may play a role in the pathophysiological mechanisms, which are relevant to migraine

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

Search for the exotic Ξ−−(1860)\Xi^{--}(1860) Resonance in 340GeV/c Σ−\Sigma^--Nucleus Interactions

We report on a high statistics search for the $\Xi^{--}(1860)$ resonance in $\Sigma^-$-nucleus collisions at 340GeV/c. No evidence for this resonance is found in our data sample which contains 676000 $\Xi^-$ candidates above background. For the decay channel $\Xi^{--}(1860) \to \Xi^-\pi^-$ and the kinematic range 0.15$<x_F<$0.9 we find a 3$\sigma$ upper limit for the production cross section of 3.1 and 3.5 $\mu$b per nucleon for reactions with carbon and copper, respectively.Comment: 5 pages, 4 figures, modification of ref. 43 and 4