scholarly journals The classification of simply-connected contact sub-riemannian symmetric spaces

1999 ◽  
Vol 188 (1) ◽  
pp. 65-82 ◽  
Author(s):  
Pierre Bieliavsky ◽  
Elisha Falbel ◽  
Claudio Gorodski
Keyword(s):  
Symmetric Spaces ◽  
Riemannian Symmetric Spaces ◽  
Simply Connected

Symmetry classes

2018 ◽  
Author(s):  
Martin R. Zirnbauer
Keyword(s):  
Symmetric Spaces ◽  
Fock Space ◽  
Minimal Extension ◽  
Dirac Fermions ◽  
Symmetry Classes ◽  
Organizational Principle ◽  
Large Families ◽  
Riemannian Symmetric Spaces ◽  
Modern Perspective

This article examines the notion of ‘symmetry class’, which expresses the relevance of symmetries as an organizational principle. In his 1962 paper The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, Dyson introduced the prime classification of random matrix ensembles based on a quantum mechanical setting with symmetries. He described three types of independent irreducible ensembles: complex Hermitian, real symmetric, and quaternion self-dual. This article first reviews Dyson’s threefold way from a modern perspective before considering a minimal extension of his setting to incorporate the physics of chiral Dirac fermions and disordered superconductors. In this minimally extended setting, Hilbert space is replaced by Fock space equipped with the anti-unitary operation of particle-hole conjugation, and symmetry classes are in one-to-one correspondence with the large families of Riemannian symmetric spaces.

Classification of five-dimensional naturally reductive spaces

1985 ◽  
Vol 97 (3) ◽  
pp. 445-463 ◽  
Author(s):  
Oldřich Kowalski ◽  
Lieven Vanhecke
Keyword(s):  
General Theory ◽  
Symmetric Spaces ◽  
Homogeneous Spaces ◽  
Natural Generalization ◽  
Riemannian Symmetric Spaces ◽  
Reductive Homogeneous Spaces ◽  
Naturally Reductive ◽  
Volume Preserving ◽  
Number Of Authors

Naturally reductive homogeneous spaces have been studied by a number of authors as a natural generalization of Riemannian symmetric spaces. A general theory with many examples was well-developed by D'Atri and Ziller[3]. D'Atri and Nickerson have proved that all naturally reductive spaces are spaces with volume-preserving local geodesic symmetries (see [1] and [2]).

The Structure of Semisimple Symmetric Spaces

1979 ◽  
Vol 31 (1) ◽  
pp. 157-180 ◽  
Author(s):  
W. Rossmann
Keyword(s):  
Symmetric Spaces ◽  
Automorphism Groups ◽  
General Structure ◽  
Point Group ◽  
Open Subgroup ◽  
Structure Theory ◽  
Semisimple Lie Groups ◽  
Riemannian Symmetric Spaces

A semisimple symmetric space can be defined as a homogeneous space G/H, where G is a semisimple Lie group, H an open subgroup of the fixed point group of an involutive automorphism of G. These spaces can also be characterized as the affine symmetric spaces or pseudo-Riemannian symmetric spaces or symmetric spaces in the sense of Loos [4] with semisimple automorphism groups [3, 4]. The connected semisimple symmetric spaces are all known: they have been classified by Berger [2] on the basis of Cartan's classification of the Riemannian symmetric spaces. However, the list of these spaces is much too long to make a detailed case by case study feasible. In order to do analysis on semisimple symmetric spaces, for example, one needs a general structure theory, just as in the case of Riemannian symmetric spaces and semisimple Lie groups.

Classification of flat connected quandles

2016 ◽  
Vol 25 (13) ◽  
pp. 1650071 ◽  
Author(s):  
Mahender Singh
Keyword(s):  
Abelian Group ◽  
Symmetric Spaces ◽  
Binary Operation ◽  
Divisible Group ◽  
Riemannian Symmetric Spaces

Let [Formula: see text] be an additive abelian group. Then the binary operation [Formula: see text] gives a quandle structure on [Formula: see text], denoted by [Formula: see text], and called the Takasaki quandle of [Formula: see text]. Viewing quandles as generalization of Riemannian symmetric spaces, Ishihara and Tamaru [Flat connected finite quandles, to appear in Proc. Amer. Math. Soc. (2016)] introduced flat quandles, and classified quandles which are finite, flat and connected. In this note, we classify all flat connected quandles. More precisely, we prove that a quandle [Formula: see text] is flat and connected if and only if [Formula: see text], where [Formula: see text] is a 2-divisible group.

scholarly journals Harmonic morphisms from solvable Lie groups

2009 ◽  
Vol 147 (2) ◽  
pp. 389-408 ◽  
Author(s):  
SIGMUNDUR GUDMUNDSSON ◽  
MARTIN SVENSSON
Keyword(s):  
Lie Groups ◽  
Symmetric Spaces ◽  
Global Solutions ◽  
Riemannian Symmetric Space ◽  
Continuous Family ◽  
Harmonic Morphisms ◽  
Solvable Lie Groups ◽  
3 Dimensional ◽  
Riemannian Symmetric Spaces ◽  
Simply Connected

AbstractIn this paper we introduce two new methods for constructing harmonic morphisms from solvable Lie groups. The first method yields global solutions from any simply connected nilpotent Lie group and from any Riemannian symmetric space of non-compact type and rank r ≥ 3. The second method provides us with global solutions from any Damek–Ricci space and many non-compact Riemannian symmetric spaces. We then give a continuous family of 3-dimensional solvable Lie groups not admitting any complex-valued harmonic morphisms, not even locally.

The Focal Locus of a Riemannian 4-Symmetric Space

1988 ◽  
Vol 31 (2) ◽  
pp. 175-181
Author(s):  
J. Alfredo Jimenez
Keyword(s):  
Symmetric Space ◽  
Normal Bundle ◽  
Symmetric Spaces ◽  
Fibre Bundle ◽  
Exponential Map ◽  
Focal Points ◽  
Cut Points ◽  
Covering Map ◽  
Simply Connected

Abstractcompact Riemannian 4-symmetric space M can be regarded as a fibre bundle over a Riemannian 2-symmetric space with totally geodesic fibres isometric to a 2-symmetric space. Here the result of R. Crittenden for conjugate and cut points in a 2-symmetric space is extended to the focal points of the fibres of M. Also the restriction of the exponential map of M up to the first focal locus in the normal bundle of a fibre is proved to yield a covering map onto its image. It is shown that for the noncompact dual M*, the fibres have no focal points and hence the exponential map of M* restricted to the normal bundle of a fibre is a covering map. The classification of the compact simply connected 4-symmetric spaces G/L with G classical simple provides a large class of examples of these fibrations.

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