On the symplectic structure of coadjoint orbits of (solvable) Lie groups and applications. I

Niels Vigand Pedersen

doi:10.1007/bf01456843

On the symplectic structure of coadjoint orbits of (solvable) Lie groups and applications. I

Mathematische Annalen ◽

10.1007/bf01456843 ◽

1988 ◽

Vol 281 (4) ◽

pp. 633-669 ◽

Cited By ~ 14

Author(s):

Niels Vigand Pedersen

Keyword(s):

Lie Groups ◽

Symplectic Structure ◽

Coadjoint Orbits ◽

Solvable Lie Groups

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Some groups with T1 primitive ideal spaces

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s144678870002259x ◽

1985 ◽

Vol 38 (1) ◽

pp. 55-64 ◽

Cited By ~ 2

Author(s):

A. L. Carey ◽

W. Moran

Keyword(s):

Normal Subgroup ◽

Lie Groups ◽

Lie Group ◽

Locally Compact Group ◽

Primitive Ideal ◽

Ideal Space ◽

Solvable Lie Groups ◽

Locally Compact ◽

Simply Connected ◽

Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

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The Darboux coordinate system and Holstein–Primakoff representations on Kähler manifolds

Canadian Journal of Physics ◽

10.1139/p06-081 ◽

2006 ◽

Vol 84 (10) ◽

pp. 891-904

Author(s):

J R Schmidt

Keyword(s):

Coordinate System ◽

Lie Groups ◽

Lie Algebra ◽

Poisson Structure ◽

Kähler Manifolds ◽

Coadjoint Orbits ◽

Kahler Manifolds ◽

Canonical Transformations ◽

Kahler Geometry

The Kahler geometry of minimal coadjoint orbits of classical Lie groups is exploited to construct Darboux coordinates, a symplectic two-form and a Lie–Poisson structure on the dual of the Lie algebra. Canonical transformations cast the generators of the dual into Dyson or Holstein–Primakoff representations.PACS Nos.: 02.20.Sv, 02.30.Ik, 02.40.Tt

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Lattices of some solvable Lie groups and actions of products of affine groups

Tohoku Mathematical Journal ◽

10.2748/tmj/1255700199 ◽

2009 ◽

Vol 61 (3) ◽

pp. 349-364 ◽

Cited By ~ 2

Author(s):

Nobuo Tsuchiya ◽

Aiko Yamakawa

Keyword(s):

Lie Groups ◽

Solvable Lie Groups ◽

Affine Groups

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Analysis of a Distinguished Laplacian on Solvable Lie Groups

Mathematische Nachrichten ◽

10.1002/mana.19931630115 ◽

1993 ◽

Vol 163 (1) ◽

pp. 151-162 ◽

Cited By ~ 6

Author(s):

Saverio Giulini ◽

Giancarlo Mauceri

Keyword(s):

Lie Groups ◽

Solvable Lie Groups

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Spectral multipliers on exponential growth solvable Lie groups

Mathematische Zeitschrift ◽

10.1007/pl00004662 ◽

1998 ◽

Vol 229 (3) ◽

pp. 435-441 ◽

Cited By ~ 5

Author(s):

Waldemar Hebisch

Keyword(s):

Lie Groups ◽

Exponential Growth ◽

Spectral Multipliers ◽

Solvable Lie Groups

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Harmonicity of vector fields on a class of Lorentzian solvable Lie groups

Advances in Geometry ◽

10.1515/advgeom-2017-0014 ◽

2018 ◽

Vol 18 (3) ◽

pp. 337-344 ◽

Cited By ~ 1

Author(s):

Ju Tan ◽

Shaoqiang Deng

Keyword(s):

Vector Field ◽

Lie Groups ◽

Lie Algebras ◽

Critical Points ◽

Vector Fields ◽

Energy Functional ◽

Smooth Vector ◽

Solvable Lie Groups ◽

Invariant Vector ◽

Invariant Vector Fields

AbstractIn this paper, we consider a special class of solvable Lie groups such that for any x, y in their Lie algebras, [x, y] is a linear combination of x and y. We investigate the harmonicity properties of invariant vector fields of this kind of Lorentzian Lie groups. It is shown that any invariant unit time-like vector field is spatially harmonic. Moreover, we determine all vector fields which are critical points of the energy functional restricted to the space of smooth vector fields.

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