Actions of compact Lie groups on homogeneous spaces

1985 ◽  
Vol 189 (4) ◽  
pp. 475-486 ◽  
Author(s):  
Volker Hauschild
Keyword(s):  
Lie Groups ◽  
Homogeneous Spaces ◽  
Compact Lie Groups

scholarly journals Singular fibres of the Gelfand–Cetlin system on MANI( n )*

2018 ◽  
Vol 376 (2131) ◽  
pp. 20170423 ◽  
Author(s):  
D. Bouloc ◽  
E. Miranda ◽  
N.T. Zung
Keyword(s):  
Lie Groups ◽  
Homogeneous Spaces ◽  
Lagrangian Submanifolds ◽  
Unitary Groups ◽  
Direct Products ◽  
Compact Lie Groups ◽  
Lie Groupoids ◽  
Finite Dimensional ◽  
Adjoint Orbits ◽  
Free Actions

In this paper, we show that every singular fibre of the Gelfand–Cetlin system on co-adjoint orbits of unitary groups is a smooth isotropic submanifold which is diffeomorphic to a two-stage quotient of a compact Lie group by free actions of two other compact Lie groups. In many cases, these singular fibres can be shown to be homogeneous spaces or even diffeomorphic to compact Lie groups. We also give a combinatorial formula for computing the dimensions of all singular fibres, and give a detailed description of these singular fibres in many cases, including the so-called (multi-)diamond singularities. These (multi-)diamond singular fibres are degenerate for the Gelfand–Cetlin system, but they are Lagrangian submanifolds diffeomorphic to direct products of special unitary groups and tori. Our methods of study are based on different ideas involving complex ellipsoids, Lie groupoids and also general ideas coming from the theory of singularities of integrable Hamiltonian systems. This article is part of the theme issue ‘Finite dimensional integrable systems: new trends and methods’.

scholarly journals Connes-Landi spheres are homogeneous spaces

2019 ◽  
Vol 53 (supl) ◽  
pp. 257-271
Author(s):  
Mitsuru Wilson
Keyword(s):  
Fixed Point ◽  
Lie Groups ◽  
Quantum Groups ◽  
Homogeneous Spaces ◽  
Compact Lie Groups ◽  
Recent Developments ◽  
Compact Quantum Groups

In this paper, we review some recent developments of compact quantum groups that arise as θ-deformations of compact Lie groups of rank at least two. A θ-deformation is merely a 2-cocycle deformation using an action of a torus of dimension higher than 2. Using the formula (Lemma 5.3) developed in [11], we derive the noncommutative 7-sphere in the sense of Connes and Landi [3] as the fixed-point subalgebra.

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