Examples of infinite dimensional Hilbert symmetric spaces

1972 ◽  
pp. 115-133
Author(s):  
Pierre de la Harpe
Keyword(s):  
Symmetric Spaces ◽  
Infinite Dimensional

TOWARD SYMMETRIC SPACES OF AFFINE KAC–MOODY TYPE

2006 ◽  
Vol 03 (05n06) ◽  
pp. 881-898 ◽  
Author(s):  
ERNST HEINTZE
Keyword(s):  
Fixed Point ◽  
Lie Groups ◽  
Symmetric Spaces ◽  
Point Group ◽  
Infinite Dimensional ◽  
Simple Lie Groups ◽  
Finite Dimensional ◽  
Expository Article

In this expository article we discuss some ideas and results which might lead to a theory of infinite dimensional symmetric spaces [Formula: see text] where [Formula: see text] is an affine Kac–Moody group and [Formula: see text] the fixed point group of an involution (of the second kind). We point out several striking similarities of these spaces with their finite dimensional counterparts and discuss their geometry. Furthermore we sketch a classification and show that they are essentially in 1 : 1 correspondence with hyperpolar actions on compact simple Lie groups.

JORDAN SYMMETRIC SPACES

2009 ◽  
Vol 02 (03) ◽  
pp. 407-415
Author(s):  
Cho-Ho Chu
Keyword(s):  
Symmetric Spaces ◽  
Hermitian Symmetric Spaces ◽  
Triple Systems ◽  
Infinite Dimensional ◽  
Jordan Triple Systems ◽  
Riemannian Symmetric Spaces

We introduce a class of Riemannian symmetric spaces, called Jordan symmetric spaces, which correspond to real Jordan triple systems and may be infinite dimensional. This class includes the symmetric R-spaces as well as the Hermitian symmetric spaces.

scholarly journals Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

2021 ◽  
pp. 1-50
Author(s):  
BRUNO DUCHESNE ◽  
JEAN LÉCUREUX ◽  
MARIA BEATRICE POZZETTI
Keyword(s):  
Symmetric Spaces ◽  
Hermitian Symmetric Spaces ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Boundary Map ◽  
Density Assumption ◽  
Maximal Representations ◽  
Tube Type ◽  
Maximal Representation ◽  
Hyperbolic Lattices

Abstract We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

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.

Infinite Dimensional Analysis

1994 ◽  
Author(s):  
Charalambos D. Aliprantis ◽  
Kim C. Border
Keyword(s):  
Dimensional Analysis ◽  
Infinite Dimensional ◽  
Infinite Dimensional Analysis
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