518 research outputs found
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 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 classical infinite rank
symmetric pairs 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 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 on a Lie algebra
\g and a -skew-symmetric derivation a weak affine Poisson
structure on \g itself. This leads naturally to a concept of a Hamiltonian
-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
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 Resonance in 340GeV/c -Nucleus Interactions
We report on a high statistics search for the resonance in
-nucleus collisions at 340GeV/c. No evidence for this resonance is
found in our data sample which contains 676000 candidates above
background. For the decay channel and the
kinematic range 0.150.9 we find a 3 upper limit for the
production cross section of 3.1 and 3.5 b per nucleon for reactions with
carbon and copper, respectively.Comment: 5 pages, 4 figures, modification of ref. 43 and 4
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