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.

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.

Scattering on Riemannian Symmetric Spaces and Huygens Principle

2018 ◽  
Vol 30 (08) ◽  
pp. 1840015
Author(s):  
Michael Semenov-Tian-Shansky
Keyword(s):  
Wave Equation ◽  
Scattering Theory ◽  
Symmetric Spaces ◽  
Evolution System ◽  
Semisimple Lie Groups ◽  
Curved Space ◽  
Zero Curvature ◽  
Famous Paper ◽  
Huygens Principle ◽  
Riemannian Symmetric Spaces

The famous paper by L. D. Faddeev and B. S. Pavlov (1972) on automorphic wave equation explored a highly romantic link between Scattering Theory (in the sense of Lax and Phillips) and Riemann hypothesis. An attempt to generalize this approach to general semisimple Lie groups leads to an interesting evolution system with multidimensional time explored by the author in 1976. In the present paper, we compare this system with a simpler one defined for zero curvature symmetric spaces and show that the Huygens principle for this system in the curved space holds if and only if it holds in the zero curvature limit.

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

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 Isomorphy classes of k-involutions of G2

2014 ◽  
Vol 13 (07) ◽  
pp. 1450039 ◽  
Author(s):  
John Hutchens
Keyword(s):  
Fixed Point ◽  
Symmetric Spaces ◽  
Point Group ◽  
Algebraic Groups ◽  
Type A ◽  
Rational Points ◽  
Combinatorial Classification ◽  
Generalized Symmetric Spaces ◽  
Appl Math

Isomorphy classes of k-involutions have been studied for their correspondence with symmetric k-varieties, also called generalized symmetric spaces. A symmetric k-variety of a k-group G is defined as Gk/Hk where θ : G → G is an automorphism of order 2 that is defined over k and Gk and Hk are the k-rational points of G and H = Gθ, the fixed point group of θ, respectively. This is a continuation of papers written by A. G. Helminck and collaborators [Involutions of SL (2, k), (n > 2), Acta Appl. Math.90(1–2) (2006) 91–119, Classification of involutions of SO (n; k; b), to appear, On the classification of k-involutions, Adv. Math.153(1) (1988) 1–117, Classification of involutions of SL (2, k), Comm. Algebra30(1) (2002) 193–203] expanding on his combinatorial classification over certain fields. Results have been achieved for groups of type A, B and D. Here we begin a series of papers doing the same for algebraic groups of exceptional type.

scholarly journals Assessment of health/functioning of older adults who consume psychoactive substances

2018 ◽  
Vol 71 (3) ◽  
pp. 942-950
Author(s):  
Vania Dias Cruz ◽  
Silvana Sidney Costa Santos ◽  
Jamila Geri Tomaschewski-Barlem ◽  
Bárbara Tarouco da Silva ◽  
Celmira Lange ◽  
...  
Keyword(s):  
Older Adults ◽  
Qualitative Case Study ◽  
International Classification Of Functioning ◽  
Psychoactive Substances ◽  
Systematic Observation ◽  
Health Functioning ◽  
Final Consideration ◽  
Theory Of Complexity

ABSTRACT Objective: To assess the health/functioning of the older adult who consumes psychoactive substances through the International Classification of Functioning, Disability and Health, considering the theory of complexity. Method: Qualitative case study, with 11 older adults, held between December 2015 and February 2016 in the state of Rio Grande do Sul, using interviews, documents and non-systematic observation. It was approved by the ethics committee. The analysis followed the propositions of the case study, using the complexity of Morin as theoretical basis. Results: We identified older adults who consider themselves healthy and show alterations - the alterations can be exacerbated by the use of psychoactive substances - of health/functioning expected according to the natural course of aging such as: systemic arterial hypertension; depressive symptoms; dizziness; tinnitus; harmed sleep/rest; and inadequate food and water consumption. Final consideration: The assessment of health/functioning of older adults who use psychoactive substances, guided by complex thinking, exceeds the accuracy limits to risk the understanding of the phenomena in its complexity.

scholarly journals Sparse Autoencoder-based Multi-head Deep Neural Networks for Machinery Fault Diagnostics with Detection of Novelties

2021 ◽  
Vol 34 (1) ◽  
Author(s):  
Zhe Yang ◽  
Dejan Gjorgjevikj ◽  
Jianyu Long ◽  
Yanyang Zi ◽  
Shaohui Zhang ◽  
...  
Keyword(s):  
Fault Diagnosis ◽  
Supervised Classification ◽  
Deep Neural Networks ◽  
Activation Function ◽  
Reconstruction Error ◽  
Fine Tuning ◽  
Fault Diagnostics ◽  
Sparse Autoencoder

AbstractSupervised fault diagnosis typically assumes that all the types of machinery failures are known. However, in practice unknown types of defect, i.e., novelties, may occur, whose detection is a challenging task. In this paper, a novel fault diagnostic method is developed for both diagnostics and detection of novelties. To this end, a sparse autoencoder-based multi-head Deep Neural Network (DNN) is presented to jointly learn a shared encoding representation for both unsupervised reconstruction and supervised classification of the monitoring data. The detection of novelties is based on the reconstruction error. Moreover, the computational burden is reduced by directly training the multi-head DNN with rectified linear unit activation function, instead of performing the pre-training and fine-tuning phases required for classical DNNs. The addressed method is applied to a benchmark bearing case study and to experimental data acquired from a delta 3D printer. The results show that its performance is satisfactory both in detection of novelties and fault diagnosis, outperforming other state-of-the-art methods. This research proposes a novel fault diagnostics method which can not only diagnose the known type of defect, but also detect unknown types of defects.

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