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
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.
Generalized m-parabolic K?hler manifolds are defined and holomorphically
projective mappings between such manifolds have been considered. Two
non-linear systems of PDE?s in covariant derivatives of the first and second
kind for the existence of such mappings are given. Also, relations between
five linearly independent curvature tensors of generalized m-parabolic K?hler
manifolds with respect to these mappings are examined.