scholarly journals G -Strands on symmetric spaces

2017 ◽  
Vol 473 (2199) ◽  
pp. 20160795 ◽  
Author(s):  
Alexis Arnaudon ◽  
Darryl D. Holm ◽  
Rossen I. Ivanov
Keyword(s):  
Symmetric Space ◽  
Lie Group ◽  
Particle Physics ◽  
Symmetric Spaces ◽  
Sobolev Norm ◽  
Chiral Model ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Broken Symmetries ◽  
Dimensional Group

We study the G -strand equations that are extensions of the classical chiral model of particle physics in the particular setting of broken symmetries described by symmetric spaces. These equations are simple field theory models whose configuration space is a Lie group, or in this case a symmetric space. In this class of systems, we derive several models that are completely integrable on finite dimensional Lie group G , and we treat in more detail examples with symmetric space SU (2)/ S 1 and SO (4)/ SO (3). The latter model simplifies to an apparently new integrable nine-dimensional system. We also study the G -strands on the infinite dimensional group of diffeomorphisms, which gives, together with the Sobolev norm, systems of 1+2 Camassa–Holm equations. The solutions of these equations on the complementary space related to the Witt algebra decomposition are the odd function solutions.

Corwin–Greenleaf multiplicity functions for complex semisimple symmetric spaces

2015 ◽  
Vol 26 (05) ◽  
pp. 1550039
Author(s):  
Salma Nasrin
Keyword(s):  
Symmetric Space ◽  
Lie Algebras ◽  
Lie Group ◽  
Symmetric Spaces ◽  
Coadjoint Orbit ◽  
Natural Projection ◽  
Real Form ◽  
Single Orbit ◽  
Complex Semisimple ◽  
Complex Symmetric

Let Gℂ be a complex simple Lie group, GU a compact real form, and [Formula: see text] the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit [Formula: see text] of GU, the intersection of [Formula: see text] with a coadjoint orbit [Formula: see text] of Gℂ is either an empty set or a single orbit of GU if [Formula: see text] is isomorphic to a complex symmetric space.

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

scholarly journals Symmetry group analysis and invariant solutions of hydrodynamic-type systems

2004 ◽  
Vol 2004 (10) ◽  
pp. 487-534 ◽  
Author(s):  
M. B. Sheftel
Keyword(s):  
Group Analysis ◽  
Type Systems ◽  
Hydrodynamic Type ◽  
Recursion Operators ◽  
Higher Symmetries ◽  
Infinite Dimensional ◽  
Dimensional Group ◽  
Component Systems ◽  
Two Component Systems ◽  
Existence Conditions

We study point and higher symmetries of systems of the hydrodynamic type with and without an explicit dependence ont,x. We consider such systems which satisfy the existence conditions for an infinite-dimensional group of hydrodynamic symmetries which implies linearizing transformations for these systems. Under additional restrictions on the systems, we obtain recursion operators for symmetries and use them to construct infinite discrete sets of exact solutions of the studied equations. We find the interrelation between higher symmetries and recursion operators. Two-component systems are studied in more detail thann-component systems. As a special case, we consider Hamiltonian and semi-Hamiltonian systems of Tsarëv.

scholarly journals PROPER CURVATURE COLLINEATIONS IN NONSTATIC SPHERICALLY SYMMETRIC SPACE–TIMES

2008 ◽  
Vol 23 (05) ◽  
pp. 749-759 ◽  
Author(s):  
GHULAM SHABBIR ◽  
M. RAMZAN
Keyword(s):  
Symmetric Space ◽  
Vector Fields ◽  
Killing Vector ◽  
Spherically Symmetric ◽  
Killing Vector Fields ◽  
Infinite Dimensional ◽  
Riemann Matrix ◽  
Integration Techniques ◽  
Curvature Collineations ◽  
Proper Curvature

A study of nonstatic spherically symmetric space–times according to their proper curvature collineations is given by using the rank of the 6×6 Riemann matrix and direct integration techniques. Studying proper curvature collineations in each case of the above space–times it is shown that when the above space–times admit proper curvature collineations, they turn out to be static spherically symmetric and form an infinite dimensional vector space. In the nonstatic cases curvature collineations are just Killing vector fields.

The Description of a Distributed Parameter Process through a Finite Discrete Dimensional System

2014 ◽  
Vol 657 ◽  
pp. 874-878
Author(s):  
Sever Şerban ◽  
Doina Corina Şerban
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Industrial Process ◽  
Time Discretization ◽  
Distributed Parameter ◽  
Geometrical Parameters ◽  
Dimensional System ◽  
Infinite Dimensional ◽  
Infinite Dimensional Systems ◽  
Distributed Parameters

This article analyses the process of warming a metal by using a walking beam furnace. This process is meant to offer the technologist objective information that may allow him to produce eventual modifications of the temperature references from the furnaces zones. Thus making the metals temperature at the furnaces exit to have an imposed distribution, within precise limits, according to the technological requests. This industrial process has a geometrical parameters distribution, more precisely it can be described through a partial differential equation, by being attached to dynamic infinite dimensional systems (or with distributed parameters). Using a procedure called geometric-time discretization (in the condition of the solutions convergence), we have managed to obtain a representation under the form of a finite discrete dimensional linear system for a process with distributed parameters.

scholarly journals Finite and infinite dimensional Lie group structures on Riordan groups

2017 ◽  
Vol 319 ◽  
pp. 522-566 ◽  
Author(s):  
Gi-Sang Cheon ◽  
Ana Luzón ◽  
Manuel A. Morón ◽  
L. Felipe Prieto-Martinez ◽  
Minho Song
Keyword(s):  
Lie Group ◽  
Infinite Dimensional ◽  
Infinite Dimensional Lie Group ◽  
Group Structures

scholarly journals Hyperconvex representations and exponential growth

2013 ◽  
Vol 34 (3) ◽  
pp. 986-1010 ◽  
Author(s):  
A. SAMBARINO
Keyword(s):  
Symmetric Space ◽  
Fundamental Group ◽  
Lie Group ◽  
Exponential Growth ◽  
Transversality Condition ◽  
Indicator Function ◽  
Equivariant Map ◽  
Negatively Curved ◽  
Funct Anal ◽  
Growth Indicator

AbstractLet $G$ be a real algebraic semi-simple Lie group and $\Gamma $ be the fundamental group of a closed negatively curved manifold. In this article we study the limit cone, introduced by Benoist [Propriétés asymptotiques des groupes linéaires. Geom. Funct. Anal. 7(1) (1997), 1–47], and the growth indicator function, introduced by Quint [Divergence exponentielle des sous-groupes discrets en rang supérieur. Comment. Math. Helv. 77 (2002), 503–608], for a class of representations $\rho : \Gamma \rightarrow G$ admitting an equivariant map from $\partial \Gamma $ to the Furstenberg boundary of the symmetric space of $G, $ together with a transversality condition. We then study how these objects vary with the representation.

scholarly journals Homological Stability for Spaces of Commuting Elements in Lie Groups

2020 ◽  
Author(s):  
Daniel A Ramras ◽  
Mentor Stafa
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Stable Range ◽  
Rational Homology ◽  
Infinite Dimensional ◽  
Fixed Group ◽  
Character Varieties ◽  
Representation Stability ◽  
Equivariant Homology ◽  
Homological Stability

Abstract In this paper, we study homological stability for spaces $\textrm{Hom}({{\mathbb{Z}}}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, $\textrm{Comm}(G)$ and $B_{\textrm{com}} G$, introduced by Cohen–Stafa and Adem–Cohen–Torres-Giese, respectively. In addition, we show that the rational homology of the space of unordered commuting $n$-tuples in a fixed group $G$ stabilizes as $n$ increases. Our proofs use the theory of representation stability—in particular, the theory of $\textrm{FI}_W$-modules developed by Church–Ellenberg–Farb and Wilson. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.

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