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

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

scholarly journals Proof of the radical conjecture for homogeneous Kähler manifolds

1989 ◽  
Vol 114 ◽  
pp. 77-122 ◽  
Author(s):  
Josef Dorfmeister
Keyword(s):  
Lie Algebra ◽  
Kähler Manifold ◽  
Transitive Group ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Fiber Space ◽  
Kahler Manifold ◽  
Solvable Ideal ◽  
Simply Connected ◽  
Bounded Homogeneous Domain

In 1967 Gindikin and Vinberg stated the Fundamental Conjecture for homogeneous Kähler manifolds. It (roughly) states that every homogeneous Kähler manifold is a fiber space over a bounded homogeneous domain for which the fibers are a product of a flat with a simply connected compact homogeneous Kähler manifold. This conjecture has been proven in a number of cases (see [6] for a recent survey). In particular, it holds if the homogeneous Kähler manifold admits a reductive or an arbitrary solvable transitive group of automorphisms [5]. It is thus tempting to think about the general case. It is natural to expect that lack of knowledge about the radical of a transitive group G of automorphisms of a homogeneous Kähler manifold M is the main obstruction to a proof of the Fundamental Conjecture for M. Thus it is of importance to consider the Kähler algebra generated by the radical of the Lie algebra of G. Computations in this context suggest that one rather considers Kähler algebras generated by an arbitrary solvable ideal.

scholarly journals New representation of the non-symmetric homogeneous bounded domains inℂ4andℂ5

1992 ◽  
Vol 15 (4) ◽  
pp. 741-752
Author(s):  
Gr. Tsagas ◽  
G. Dimou
Keyword(s):  
Lie Algebra ◽  
Kähler Manifolds ◽  
Bounded Domains ◽  
Kahler Manifolds ◽  
Riemann Manifold ◽  
Solvable Lie Algebra

This paper deals with the corresponding solvable Lie algebra to each of non-symmetric homogeneous bounded domains inℂ4andℂ5by special set of matrices. Some interesting properties of Kähler manifolds are found. The theory ofs-structure on a complete Riemann manifold is also studied.

scholarly journals Hamiltonian 2-forms in Kähler geometry, IV Weakly Bochner-flat Kähler manifolds

2008 ◽  
Vol 16 (1) ◽  
pp. 91-126 ◽  
Author(s):  
Vestislav Apostolov ◽  
Vestislav Apostolov Apostolov ◽  
Vestislav Apostolov Apostolov ◽  
David M. J. Calderbank ◽  
David M.J. Calderbank ◽  
...  
Keyword(s):  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Kahler Geometry

scholarly journals Construction of projective special Kähler manifolds

2021 ◽  
Author(s):  
Mauro Mantegazza
Keyword(s):  
Lie Groups ◽  
Symmetric Tensor ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Kähler Structure ◽  
Special Kähler Manifolds

AbstractIn this paper, we present an intrinsic characterisation of projective special Kähler manifolds in terms of a symmetric tensor satisfying certain differential and algebraic conditions. We show that this tensor vanishes precisely when the structure is locally isomorphic to a standard projective special Kähler structure on $$\mathrm {SU}(n,1)/\mathrm {S}(\mathrm {U}(n)\mathrm {U}(1))$$ SU ( n , 1 ) / S ( U ( n ) U ( 1 ) ) . We use this characterisation to classify 4-dimensional projective special Kähler Lie groups.

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