On decomposition of generalized Riemannian symmetric spaces and complete lifts of Clifford translations

2021 ◽  
Author(s):  
Gholam Hossein Arab ◽  
Megerdich Toomanian
Keyword(s):  
Symmetric Spaces ◽  
Riemannian Symmetric Spaces

scholarly journals Wave propagation on Riemannian symmetric spaces

1992 ◽  
Vol 107 (2) ◽  
pp. 270-278 ◽  
Author(s):  
G. 'Olafsson ◽  
H. Schlichtkrull
Keyword(s):  
Wave Propagation ◽  
Symmetric Spaces ◽  
Riemannian Symmetric Spaces

scholarly journals Harmonic and minimal unit vector fields on Riemannian symmetric spaces

2003 ◽  
Vol 47 (4) ◽  
pp. 1273-1286 ◽  
Author(s):  
Jürgen Berndt ◽  
Lieven Vanhecke ◽  
László Verhóczki
Keyword(s):  
Symmetric Spaces ◽  
Vector Fields ◽  
Unit Vector ◽  
Unit Vector Fields ◽  
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.

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