3,031 research outputs found
A note on Iwasawa-type decomposition
We study the Iwasawa-type decomposition of an open subset of SL(n,C) as
SU(p,q)AN. We show that the dressing action of SU(p,q) is globally defined on
the space of admissible elements in AN. We also show that the space of
admissible elements is a multiplicative subset of AN. We establish a geometric
criterion: the symmetrization of an admissible element maps the positive cone
in C^n into itself.Comment: 10 page
Iwasawa nilpotency degree of non compact symmetric cosets in N-extended Supergravity
We analyze the polynomial part of the Iwasawa realization of the coset
representative of non compact symmetric Riemannian spaces. We start by studying
the role of Kostant's principal SU(2)_P subalgebra of simple Lie algebras, and
how it determines the structure of the nilpotent subalgebras. This allows us to
compute the maximal degree of the polynomials for all faithful representations
of Lie algebras. In particular the metric coefficients are related to the
scalar kinetic terms while the representation of electric and magnetic charges
is related to the coupling of scalars to vector field strengths as they appear
in the Lagrangian. We consider symmetric scalar manifolds in N-extended
supergravity in various space-time dimensions, elucidating various relations
with the underlying Jordan algebras and normed Hurwitz algebras. For magic
supergravity theories, our results are consistent with the Tits-Satake
projection of symmetric spaces and the nilpotency degree turns out to depend
only on the space-time dimension of the theory. These results should be helpful
within a deeper investigation of the corresponding supergravity theory, e.g. in
studying ultraviolet properties of maximal supergravity in various dimensions.Comment: 40 page
Dressing preserving the fundamental group
In this note we consider the relationship between the dressing action and the
holonomy representation in the context of constant mean curvature surfaces. We
characterize dressing elements that preserve the topology of a surface and
discuss dressing by simple factors as a means of adding bubbles to a class of
non finite type cylinders.Comment: 36 pages, 1 figur
Generalized DPW method and an application to isometric immersions of space forms
Let be a complex Lie group and denote the group of maps from
the unit circle into , of a suitable class. A differentiable
map from a manifold into , is said to be of \emph{connection
order } if the Fourier expansion in the loop parameter of the
-family of Maurer-Cartan forms for , namely F_\lambda^{-1}
\dd F_\lambda, is of the form . Most
integrable systems in geometry are associated to such a map. Roughly speaking,
the DPW method used a Birkhoff type splitting to reduce a harmonic map into a
symmetric space, which can be represented by a certain order map,
into a pair of simpler maps of order and respectively.
Conversely, one could construct such a harmonic map from any pair of
and maps. This allowed a Weierstrass type description
of harmonic maps into symmetric spaces. We extend this method to show that, for
a large class of loop groups, a connection order map, for ,
splits uniquely into a pair of and maps. As an
application, we show that constant non-zero curvature submanifolds with flat
normal bundle of a sphere or hyperbolic space split into pairs of flat
submanifolds, reducing the problem (at least locally) to the flat case. To
extend the DPW method sufficiently to handle this problem requires a more
general Iwasawa type splitting of the loop group, which we prove always holds
at least locally.Comment: Some typographical correction
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