35,990 research outputs found
Weighted discrete hypergroups
Weighted group algebras have been studied extensively in Abstract Harmonic
Analysis where complete characterizations have been found for some important
properties of weighted group algebras, namely amenability and Arens regularity.
One of the generalizations of weighted group algebras is weighted hypergroup
algebras. Defining weighted hypergroups, analogous to weighted groups, we study
Arens regularity and isomorphism to operator algebras for them. We also examine
our results on three classes of discrete weighted hypergroups constructed by
conjugacy classes of FC groups, the dual space of compact groups, and
hypergroup structure defined by orthogonal polynomials. We observe some
unexpected examples regarding Arens regularity and operator isomorphisms of
weighted hypergroup algebras.Comment: 27 pages. This version is shorter but still covers all the main
results of the previous on
Linearization of Poisson actions and singular values of matrix products
We prove that the linearization functor from the category of Hamiltonian
K-actions with group-valued moment maps in the sense of Lu, to the category of
ordinary Hamiltonian K-actions, preserves products up to symplectic
isomorphism. As an application, we give a new proof of the Thompson conjecture
on singular values of matrix products and extend this result to the case of
real matrices. We give a formula for the Liouville volume of these spaces and
obtain from it a hyperbolic version of the Duflo isomorphism.Comment: 20 page
Desingularizing isolated conical singularities: Uniform estimates via weighted Sobolev spaces
We define a very general "parametric connect sum" construction which can be
used to eliminate isolated conical singularities of Riemannian manifolds. We
then show that various important analytic and elliptic estimates, formulated in
terms of weighted Sobolev spaces, can be obtained independently of the
parameters used in the construction. Specifically, we prove uniform estimates
related to (i) Sobolev Embedding Theorems, (ii) the invertibility of the
Laplace operator and (iii) Poincare' and Gagliardo-Nirenberg-Sobolev type
inequalities.
Our main tools are the well-known theories of weighted Sobolev spaces and
elliptic operators on "conifolds". We provide an overview of both, together
with an extension of the former to general Riemannian manifolds.
For a geometric application of our results we refer the reader to our paper
"Special Lagrangian conifolds, II: Gluing constructions in C^m".Comment: Minor changes, improved presentation. Final version. To appear in CA
Special Lagrangian conifolds, II: Gluing constructions in C^m
We prove two gluing theorems for special Lagrangian (SL) conifolds in complex
space C^m. Conifolds are a key ingredient in the compactification problem for
moduli spaces of compact SLs in Calabi-Yau manifolds.
In particular, our theorems yield the first examples of smooth SL conifolds
with 3 or more planar ends and the first (non-trivial) examples of SL conifolds
which have a conical singularity but are not, globally, cones. We also obtain:
(i) a desingularization procedure for transverse intersection and
self-intersection points, using "Lawlor necks"; (ii) a construction which
completely desingularizes any SL conifold by replacing isolated conical
singularities with non-compact asymptotically conical (AC) ends; (iii) a proof
that there is no upper bound on the number of AC ends of a SL conifold; (iv)
the possibility of replacing a given collection of conical singularities with a
completely different collection of conical singularities and of AC ends.
As a corollary of (i) we improve a result by Arezzo and Pacard concerning
minimal desingularizations of certain configurations of SL planes in C^m,
intersecting transversally.Comment: Several new results. Final version. To appear in Proc. LM
Higgs bundles and higher Teichm\"uller spaces
This paper is a survey on the role of Higgs bundle theory in the study of
higher Teichm\"uller spaces. Recall that the Teichm\"uller space of a compact
surface can be identified with a certain connected component of the moduli
space of representations of the fundamental group of the surface into
. Higher Teichm\"uller spaces correspond to
special components of the moduli space of representations when one replaces
by a real non-compact semisimple Lie group of
higher rank. Examples of these spaces are provided by the Hitchin components
for split real groups, and maximal Toledo invariant components for groups of
Hermitian type. More recently, the existence of such components has been proved
for , in agreement with the conjecture of Guichard and
Wienhard relating the existence of higher Teichm\"uller spaces to a certain
notion of positivity on a Lie group that they have introduced. We review these
three different situations, and end up explaining briefly the conjectural
general picture from the point of view of Higgs bundle theory.Comment: arXiv admin note: substantial text overlap with arXiv:1511.0775
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