scholarly journals On the classification of the coadjoint orbits of the Sobolev Bott–Virasoro group

2009 ◽  
Vol 256 (9) ◽  
pp. 2815-2841 ◽  
Author(s):  
François Gay-Balmaz
Keyword(s):  
Coadjoint Orbits ◽  
Virasoro Group

scholarly journals GEODESIC FLOW ON EXTENDED BOTT–VIRASORO GROUP AND GENERALIZED TWO-COMPONENT PEAKON TYPE DUAL SYSTEMS

2008 ◽  
Vol 20 (10) ◽  
pp. 1191-1208 ◽  
Author(s):  
PARTHA GUHA
Keyword(s):  
Evolution Equations ◽  
Boussinesq System ◽  
Coadjoint Orbits ◽  
Dual Systems ◽  
Hamiltonian Flows ◽  
Lie Algebraic Structure ◽  
Two Component ◽  
Integrable Evolution ◽  
Virasoro Group ◽  
Systematic Derivation

This paper discusses an algorithmic way of constructing integrable evolution equations based on Lie algebraic structure. We derive, in a pedagogical style, a large class of two-component peakon type dual systems from their two-component soliton equations counter part. We study the essential aspects of Hamiltonian flows on coadjoint orbits of the centrally extended semidirect product group [Formula: see text] to give a systematic derivation of the dual counter parts of various two-component of integrable systems, viz., the dispersive water wave equation, the Kaup–Boussinesq system and the Broer–Kaup system, using moment of inertia operators method and the (frozen) Lie–Poisson structure. This paper essentially gives Lie algebraic explanation of Olver–Rosenau's paper [31].

scholarly journals Gravitational edge modes, coadjoint orbits, and hydrodynamics

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
William Donnelly ◽  
Laurent Freidel ◽  
Seyed Faroogh Moosavian ◽  
Antony J. Speranza
Keyword(s):  
General Relativity ◽  
Boundary Surface ◽  
Infinite Family ◽  
Symmetry Algebra ◽  
Complete Classification ◽  
Coadjoint Orbits ◽  
Poincaré Group ◽  
Poincare Group ◽  
Dimensional Group

Abstract The phase space of general relativity in a finite subregion is characterized by edge modes localized at the codimension-2 boundary, transforming under an infinite-dimensional group of symmetries. The quantization of this symmetry algebra is conjectured to be an important aspect of quantum gravity. As a step towards quantization, we derive a complete classification of the positive-area coadjoint orbits of this group for boundaries that are topologically a 2-sphere. This classification parallels Wigner’s famous classification of representations of the Poincaré group since both groups have the structure of a semidirect product. We find that the total area is a Casimir of the algebra, analogous to mass in the Poincaré group. A further infinite family of Casimirs can be constructed from the curvature of the normal bundle of the boundary surface. These arise as invariants of the little group, which is the group of area-preserving diffeomorphisms, and are the analogues of spin. Additionally, we show that the symmetry group of hydrodynamics appears as a reduction of the corner symmetries of general relativity. Coadjoint orbits of both groups are classified by the same set of invariants, and, in the case of the hydrodynamical group, the invariants are interpreted as the generalized enstrophies of the fluid.

scholarly journals Classification of Positive Energy Representations of the Virasoro Group

2014 ◽  
Vol 2015 (18) ◽  
pp. 8620-8656 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Hadi Salmasian
Keyword(s):  
Positive Energy ◽  
Virasoro Group

Foliations Formed by Generic Coadjoint Orbits of a Class of Real Seven-Dimensional Solvable Lie Groups

2021 ◽  
Vol 61 ◽  
pp. 79-104
Author(s):  
Tuyen Nguyen ◽  
◽  
Vu Le
Keyword(s):  
Lie Groups ◽  
Lie Algebras ◽  
Topological Classification ◽  
Coadjoint Orbits ◽  
Coadjoint Representation ◽  
Nilpotent Lie Algebra ◽  
The Family ◽  
Measurable Foliation ◽  
Simply Connected

In this paper, we consider exponential, connected and simply connected Lie groups which are corresponding to seven-dimensional Lie algebras such that their nilradical is a five-dimensional nilpotent Lie algebra $\mathfrak{g}_{5,2}$ given in Table~\ref{tab1}. In particular, we give a description of the geometry of the generic orbits in the coadjoint representation of some considered Lie groups. We prove that, for each considered group, the family of the generic coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.

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