scholarly journals Admissibility for a Class of Quasiregular Representations

2007 ◽  
Vol 59 (5) ◽  
pp. 917-942 ◽  
Author(s):  
Bradley N. Currey
Keyword(s):  
Wavelet Transform ◽  
Semidirect Product ◽  
Explicit Construction ◽  
Vector Group ◽  
Almost Everywhere ◽  
Orthogonality Relations ◽  
Completely Reducible ◽  
Multiplicity Free ◽  
Simply Connected

AbstractGiven a semidirect product G = N ⋊ H where N is nilpotent, connected, simply connected and normal in G and where H is a vector group for which ad() is completely reducible and R-split, let τ denote the quasiregular representation of G in L2(N). An element ψ ∈ L2(N) is said to be admissible if the wavelet transform f ⟼ 〈 f, τ(·)ψ 〉 defines an isometry from L2(N) into L2(G). In this paper we give an explicit construction of admissible vectors in the case where G is not unimodular and the stabilizers in H of its action on are almost everywhere trivial. In this situation we prove orthogonality relations and we construct an explicit decomposition of L2(G) into G-invariant, multiplicity-free subspaces each of which is the image of a wavelet transform . We also show that, with the assumption of (almost-everywhere) trivial stabilizers, non-unimodularity is necessary for the existence of admissible vectors.

scholarly journals A Localization Principle for a Class of Analytic Functions

1963 ◽  
Vol 23 ◽  
pp. 207-212
Author(s):  
D. A. Storvick
Keyword(s):  
Analytic Function ◽  
Circular Disk ◽  
Inverse Image ◽  
Limit Values ◽  
Localization Principle ◽  
Radial Limits ◽  
Bounded Analytic Function ◽  
Simply Connected Domain ◽  
Almost Everywhere ◽  
Simply Connected

It has been shown by Kiyoshi Noshiro [8; p. 35] that a bounded analytic function w = f(z) in |z| < 1 having radial limit values of modulus one almost everywhere satisfies a localization principle of the following type. Let (c) be any circular disk: | w − α | < ρ lying inside |w| < 1 whose periphery may be tangent to the circumference |w| = 1. Denote by Δ any component of the inverse image of (c) under w = f(z) and by z = z(ξ) a function which maps |ξ| < 1 onto the simply connected domain Δ in a one-to-one conformal manner. Then, the functionis also a bounded analytic function in | ξ | < 1 with radial limits of modulus one almost everywhere.

Central Extensions of Loop Groups and Obstruction to Pre-Quantization

2013 ◽  
Vol 56 (1) ◽  
pp. 116-126 ◽  
Author(s):  
Derek Krepski
Keyword(s):  
Riemann Surface ◽  
Lie Group ◽  
Moduli Space ◽  
Explicit Construction ◽  
Loop Groups ◽  
Central Extensions ◽  
Simply Connected

AbstractAn explicit construction of a pre-quantumline bundle for themoduli space of flat G-bundles over a Riemann surface is given, where G is any non-simply connected compact simple Lie group. This work helps to explain a curious coincidence previously observed between Toledano Laredo's work classifying central extensions of loop groups LG and the author's previous work on the obstruction to pre-quantization of the moduli space of flat G-bundles.

scholarly journals Multiplicity-free quotient tensor algebras

2001 ◽  
Vol 71 (2) ◽  
pp. 279-298
Author(s):  
G. E. Wall
Keyword(s):  
General Linear Group ◽  
Partial Solution ◽  
Tensor Algebra ◽  
Dimensional Vector ◽  
Dimensional Vector Space ◽  
Infinite Dimensional ◽  
Tensor Algebras ◽  
Completely Reducible ◽  
Multiplicity Free ◽  
Free Quotient

AbstractLet V be an infinite-dimensional vector space ovre a field of characteristic 0. It is well known that the tensor algebra T on V is a completely reducible module for the general linear group G on V. This paper is concerned with those quotient algebras A of T that are at the same time modules for G. A partial solution is given to the problem of determinig those A in which no irreducible constitutent has multiplicity greater thatn 1.

A local wavelet transform on the torus T2

2015 ◽  
Vol 13 (04) ◽  
pp. 1550027 ◽  
Author(s):  
Bao Qin Wang ◽  
Gang Wang ◽  
Xiao-Hui Zhou
Keyword(s):  
Wavelet Transform ◽  
Smooth Surface ◽  
Graphical Representation ◽  
Explicit Construction ◽  
Reconstruction Formula ◽  
Inverse Transform ◽  
Inverse Projection

For the smooth surface M, the explicit construction of local flattening map p is given, where the bijection projection p is the local flattening map from a smooth surface M to a plane. By virtue of the inverse projection p-1, the local wavelet transform on M can be generated from wavelet transform on a plane. Take the torus T2for example, by using the local flattening map p of torus, the construction of the local dilation on the torus is systematically studied, the local wavelet transform formula on the torus is offered and the inverse transform formula of the local wavelet transform, that is, the reconstruction formula is also offered. Finally, we show the graphical representation of the local wavelet on the torus.

scholarly journals Computable Constants for Korn’s Inequalities on Riemannian Manifolds

2021 ◽  
Author(s):  
R. J. Knops
Keyword(s):  
Boundary Conditions ◽  
Minimal Surface ◽  
Riemannian Manifolds ◽  
Dirichlet Boundary ◽  
Explicit Construction ◽  
Dirichlet Boundary Conditions ◽  
Two Dimensional ◽  
Spherical Cap ◽  
Connected Manifold ◽  
Simply Connected

AbstractA method is presented for the explicit construction of the non-dimensional constant occurring in Korn’s inequalities for a bounded two-dimensional Riemannian differentiable simply connected manifold subject to Dirichlet boundary conditions. The method is illustrated by application to the spherical cap and minimal surface.

scholarly journals MODULAR REDUCTION OF THE STEINBERG LATTICE OF THE GENERAL LINEAR GROUP

2008 ◽  
Vol 07 (06) ◽  
pp. 793-807 ◽  
Author(s):  
FERNANDO SZECHTMAN
Keyword(s):  
Linear Group ◽  
General Linear Group ◽  
General Linear ◽  
Completely Reducible ◽  
Natural Filtration ◽  
Multiplicity Free ◽  
Steinberg Representation

Each factor of the natural filtration of the modular reduction of the Steinberg representation of the general linear group is shown to be completely reducible and the entire modular reduction is shown to be multiplicity free.

Left-invariant almost α-coKähler structures on 3D semidirect product Lie groups

2019 ◽  
Vol 16 (01) ◽  
pp. 1950011 ◽  
Author(s):  
Domenico Perrone
Keyword(s):  
Lie Groups ◽  
Semidirect Product ◽  
Three Dimensional ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
The Matrix ◽  
Canonical Foliation ◽  
Matrix Formula ◽  
Simply Connected

The main result of this paper gives a characterization of left-invariant almost [Formula: see text]-coKähler structures on three-dimensional (3D) semidirect product Lie groups [Formula: see text] in terms of the matrix [Formula: see text]. Then, we study the harmonicity of the Reeb vector field [Formula: see text] of a simply connected homogeneous almost [Formula: see text]-coKähler three-manifold, in terms of the Gaussian curvature of the canonical foliation.

On the topology of manifolds with completely integrable geodesic flows

1992 ◽  
Vol 12 (1) ◽  
pp. 109-121 ◽  
Author(s):  
Gabriel P. Paternain
Keyword(s):  
Riemannian Manifold ◽  
Geodesic Flow ◽  
Homogeneous Spaces ◽  
Rational Homotopy ◽  
Completely Integrable ◽  
Finite Dimensional ◽  
Topology Of Manifolds ◽  
Multiplicity Free ◽  
Simply Connected ◽  
Integrable Geodesic Flows

AbstractWe show that if M is a compact simply connected Riemannian manifold whose geodesic flow is completely integrable with periodic integrals, then M is rationally elliptic, i.e. the rational homotopy of M is finite dimensional. We also show that rational ellipticity is shared by simply connected compact manifolds whose cotangent bundle admits a multiplicity free compact action that leaves invariant the Hamiltonian associated with some Riemannian metric. In particular it follows that if M is a Riemannian manifold whose geodesic flow is completely integrable by the Thimm method, then M is rationally elliptic. Other questions concerning the global behaviour of geodesics on homogeneous spaces are discussed.

scholarly journals Integrating -algebras

2008 ◽  
Vol 144 (4) ◽  
pp. 1017-1045 ◽  
Author(s):  
André Henriques
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Explicit Construction ◽  
Classifying Space ◽  
Simply Connected ◽  
Work Done ◽  
Connected Lie Group

AbstractGiven a Lie n-algebra, we provide an explicit construction of its integrating Lie n-group. This extends work done by Getzler in the case of nilpotent $L_\infty $-algebras. When applied to an ordinary Lie algebra, our construction yields the simplicial classifying space of the corresponding simply connected Lie group. In the case of the string Lie 2-algebra of Baez and Crans, we obtain the simplicial nerve of their model of the string group.

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