scholarly journals On the Classification of Reflective Submanifolds of Riemannian Symmetric Spaces

1974 ◽  
Vol 24 (4) ◽  
pp. 327-339 ◽  
Author(s):  
Dominic Leung
Keyword(s):  
Symmetric Spaces ◽  
Riemannian Symmetric Spaces

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]).

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

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 On some recent progress in the classification of $$(P\hbox { and }Q)$$-polynomial association schemes

2019 ◽  
Author(s):  
Alexander L. Gavrilyuk ◽  
Jack H. Koolen
Keyword(s):  
Symmetric Spaces ◽  
Recent Progress ◽  
Association Schemes ◽  
Compact Symmetric Spaces ◽  
Expository Paper

AbstractThe problem of classification of $$(P\hbox { and }Q)$$(PandQ)-polynomial association schemes, as a finite analogue of E. Cartan’s classification of compact symmetric spaces, was posed in the monograph “Association schemes” by E. Bannai and T. Ito in the early 1980s. In this expository paper, we report on some recent results towards its solution.

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