scholarly journals SUBMANIFOLD GEOMETRIES IN A SYMMETRIC SPACE OF NON-COMPACT TYPE AND A PSEUDO-HILBERT SPACE

2004 ◽  
Vol 58 (1) ◽  
pp. 167-202 ◽  
Author(s):  
Naoyuki KOIKE
Keyword(s):  
Hilbert Space ◽  
Symmetric Space ◽  
Compact Type

Brownian Motion on a Symmetric Space of Non-Compact Type: Asymptotic Behaviour in Polar Coordinates

1991 ◽  
Vol 43 (5) ◽  
pp. 1065-1085 ◽  
Author(s):  
J. C. Taylor
Keyword(s):  
Brownian Motion ◽  
Symmetric Space ◽  
Asymptotic Behaviour ◽  
Polar Coordinates ◽  
Skew Product ◽  
Compact Type ◽  
Product Representation

AbstractThe results of Orihara [10] and Malliavin2 [7] on the asymptotic behaviour in polar coordinates of Brownian motion on a symmetric space of non-compact type are obtained by means of a skew product representation on K/M x A+of the Brownian motion on the set of regular points of X. Results of Norris, Rogers, and Williams [9] are interpreted in this context.

scholarly journals The fixed point set of a holomorphic isometry, the intersection of two real forms in a Hermitian symmetric space of compact type and symmetric triads

2015 ◽  
Vol 26 (06) ◽  
pp. 1541005 ◽  
Author(s):  
Osamu Ikawa ◽  
Makiko Sumi Tanaka ◽  
Hiroyuki Tasaki
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Hermitian Symmetric Space ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Compact Type ◽  
Fixed Point Set ◽  
Point Set ◽  
Necessary And Sufficient ◽  
Real Forms

We show a necessary and sufficient condition that the fixed point set of a holomorphic isometry and the intersection of two real forms of a Hermitian symmetric space of compact type are discrete and prove that they are antipodal sets in the cases. We also consider some relations between the intersection of two real forms and the fixed point set of a certain holomorphic isometry.

scholarly journals On the least positive eigenvalue of the Laplacian for the compact quotient of a certain Riemannian symmetric space

1980 ◽  
Vol 78 ◽  
pp. 137-152 ◽  
Author(s):  
Hajime Urakawa
Keyword(s):  
Fixed Point ◽  
Euclidean Space ◽  
Symmetric Space ◽  
Lie Group ◽  
Discrete Subgroup ◽  
Riemannian Symmetric Space ◽  
Compact Type ◽  
Identity Component ◽  
Rank One ◽  
Quotient Manifold

Let (, g) be the standard Euclidean space or a Riemannian symmetric space of non-compact type of rank one. Let G be the identity component of the Lie group of all isometries of (, g). Let Γ be a discrete subgroup of G acting fixed point freely on whose quotient manifold MΓ is compact.

scholarly journals Classification of four qubit states and their stabilisers under SLOCC operations

2022 ◽  
Author(s):  
Heiko Dietrich ◽  
Willem A De Graaf ◽  
Alessio Marrani ◽  
Marcos Origlia
Keyword(s):  
Hilbert Space ◽  
Symmetric Space ◽  
Semidirect Product ◽  
Maximal Rank ◽  
Special Case ◽  
Qubit States

Abstract We classify four qubit states under SLOCC operations, that is, we classify the orbits of the group SL(2,C)^4 on the Hilbert space H_4 = (C^2)^{\otimes 4}. We approach the classification by realising this representation as a symmetric space of maximal rank. We first describe general methods for classifying the orbits of such a space. We then apply these methods to obtain the orbits in our special case, resulting in a complete and irredundant classification of SL(2,C)^4-orbits on H_4. It follows that an element of H_4 is conjugate to an element of precisely 87 classes of elements. Each of these classes either consists of one element or of a parametrised family of elements, and the elements in the same class all have equal stabiliser in SL(2,C)^4. We also present a complete and irredundant classification of elements and stabilisers up to the action of the semidirect product Sym_4\ltimes\SL(2,C)^4 where Sym_4 permutes the four tensor factors of H_4.

scholarly journals Dynamics of the heat semigroup on symmetric spaces

2009 ◽  
Vol 30 (2) ◽  
pp. 457-468 ◽  
Author(s):  
LIZHEN JI ◽  
ANDREAS WEBER
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Chaotic Behavior ◽  
Heat Semigroup ◽  
Euclidean Spaces ◽  
Compact Type ◽  
Heat Semigroups

AbstractThe aim of this paper is to show that the dynamics of Lp heat semigroups (p>2) on a symmetric space of non-compact type is very different from the dynamics of the Lp heat semigroups if 1<p≤2. To see this, we show that certain shifts of the Lp heat semigroups have a chaotic behavior if p>2, and that such a behavior is not possible in the cases 1<p≤2. These results are compared with the corresponding situation for Euclidean spaces and symmetric spaces of compact type, where such a behavior is not possible.

scholarly journals Equivariant harmonic maps depend real analytically on the representation

2021 ◽  
Author(s):  
Ivo Slegers
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Harmonic Maps ◽  
Main Tool ◽  
Target Space ◽  
Fixed Target ◽  
Compact Type ◽  
Proper Action ◽  
Real Analytic

AbstractWe consider harmonic maps into symmetric spaces of non-compact type that are equivariant for representations that induce a free and proper action on the symmetric space. We show that under suitable non-degeneracy conditions such equivariant harmonic maps depend in a real analytic fashion on the representation they are associated to. The main tool in the proof is the construction of a family of deformation maps which are used to transform equivariant harmonic maps into maps mapping into a fixed target space so that a real analytic version of the results in [4] can be applied.

scholarly journals An Anosov action on the bundle of Weyl chambers

1985 ◽  
Vol 5 (4) ◽  
pp. 587-593 ◽  
Author(s):  
Hans-Christoph Im Hof
Keyword(s):  
Symmetric Space ◽  
Geodesic Flow ◽  
Riemannian Symmetric Space ◽  
Compact Type ◽  
Rank One

AbstractWe introduce an Anosov action on the bundle of Weyl chambers of a riemannian symmetric space of non-compact type, which for rank one spaces coincides with the geodesic flow.

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