On the Iwasawa decomposition of a perplectic matrix

2020 ◽  
pp. 1-16
Author(s):  
Ralph John de la Cruz ◽  
Edgar Reyes
Keyword(s):  
Iwasawa Decomposition

Characteristic functions of semigroups in semi-simple Lie groups

2019 ◽  
Vol 31 (4) ◽  
pp. 815-842
Author(s):  
Luiz A. B. San Martin ◽  
Laercio J. Santos
Keyword(s):  
Laplace Transform ◽  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Convex Cone ◽  
Convex Set ◽  
Flag Manifold ◽  
Riemannian Metric ◽  
Iwasawa Decomposition ◽  
Characteristic Functions

Abstract Let G be a noncompact semi-simple Lie group with Iwasawa decomposition {G=KAN} . For a semigroup {S\subset G} with nonempty interior we find a domain of convergence of the Helgason–Laplace transform {I_{S}(\lambda,u)=\int_{S}e^{\lambda(\mathsf{a}(g,u))}\,dg} , where dg is the Haar measure of G, {u\in K} , {\lambda\in\mathfrak{a}^{\ast}} , {\mathfrak{a}} is the Lie algebra of A and {gu=ke^{\mathsf{a}(g,u)}n\in KAN} . The domain is given in terms of a flag manifold of G written {\mathbb{F}_{\Theta(S)}} called the flag type of S, where {\Theta(S)} is a subset of the simple system of roots. It is proved that {I_{S}(\lambda,u)<\infty} if λ belongs to a convex cone defined from {\Theta(S)} and {u\in\pi^{-1}(\mathcal{D}_{\Theta(S)}(S))} , where {\mathcal{D}_{\Theta(S)}(S)\subset\mathbb{F}_{\Theta(S)}} is a B-convex set and {\pi:K\rightarrow\mathbb{F}_{\Theta(S)}} is the natural projection. We prove differentiability of {I_{S}(\lambda,u)} and apply the results to construct of a Riemannian metric in {\mathcal{D}_{\Theta(S)}(S)} invariant by the group {S\cap S^{-1}} of units of S.

Involutive automorphisms and Iwasawa decomposition of some hyperbolic Kac–Moody algebras

1999 ◽  
Vol 40 (1) ◽  
pp. 501-510 ◽  
Author(s):  
K. C. Pati ◽  
D. Parashar ◽  
R. S. Kaushal
Keyword(s):  
Iwasawa Decomposition

su(2) -expansion of the Lorentz algebra so(3,1 )

2013 ◽  
Vol 91 (8) ◽  
pp. 589-598 ◽  
Author(s):  
Rutwig Campoamor-Stursberg ◽  
Hubert de Guise ◽  
Marc de Montigny
Keyword(s):  
Coherent State ◽  
Unitary Representations ◽  
Iwasawa Decomposition ◽  
Matrix Formalism ◽  
Matrix Elements ◽  
Infinite Dimensional ◽  
Lorentz Algebra ◽  
Finite Dimensional

We exploit the Iwasawa decomposition to construct coherent state representations of [Formula: see text], the Lorentz algebra in 3 + 1 dimensions, expanded on representations of the maximal compact subalgebra [Formula: see text]. Examples of matrix elements computation for finite dimensional and infinite-dimensional unitary representations are given. We also discuss different base vectors and the equivalence between these different choices. The use of the [Formula: see text]-matrix formalism to truncate the representation or to enforce unitarity is discussed.

scholarly journals Nonlinear Iwasawa decomposition of control flows

2007 ◽  
Vol 18 (2/3, June) ◽  
pp. 339-354 ◽  
Author(s):  
Paulo Ruffino ◽  
Fritz Colonius
Keyword(s):  
Iwasawa Decomposition
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