Birkhoff decompositions and Iwasawa decompositions for loop groups
type: Item
instance of: scholarly article
title: Birkhoff decompositions and Iwasawa decompositions for loop groups
zbMATH Open document ID: 0996.22013
author: Vladimir Balan
author: Josef F. Dorfmeister
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2002-11-04T00:00:00Z
review text: Recently, loop groups have been used to investigate families of geometric objects, e.g., the study of harmonic maps from surfaces to compact Lie groups, and later, the study of surfaces with constant mean curvature or constant Gauss curvature, Willmore surfaces and affine spheres. The PDW study of the last two cases involves homogeneous spaces with noncompact transitive group. Within this framework, it is useful to establish, for general Lie groups and general homogeneous spaces, the basic properties for loop groups. The most salient features of loop groups used in this paper are certainly the generalized Birkhoff and Iwasawa decompositions for loop groups. In the present paper, the authors establish these generalized decomposition theorems for loop groups associated with arbitrary Lie groups and illustrate the results by several typical examples. A concise introduction to the topic (Section 1) is followed in Section 2 by the basic preliminaries regarding subsequent decompositions of connected real analytic Lie groups \(G\) which admit a faithful finite-dimensional continuous representation. Section 3 is devoted to loop groups: the Banach space structure, the Wiener topology of \(\Lambda G^C\) (where \(G^C\) is the complexified group of \(G)\), and the subgroups \(\Lambda^+G^C\) and \(\Lambda^-G^C\) are described. In Section 4, after technical preliminary results which provide the Birkhoff splitting of the loop subgroups attached to the reductive, solvable, nilradical and abelian subgroups of \(G^C\), a theorem is derived which states the Birkhoff decomposition of \(\Lambda G^C\). Section 5 shows that the ``big cell'' (the main canonic coset) of the splitting is open and dense in \(\Lambda G^C\). In Section 6 the Iwasawa decompositions are proved for the loop groups of the canonically attached reductive, solvable, nilradical and abelian subgroups of \(G^C\) what leads to the main result: the Iwasawa decomposition of \(G^C\). Note that the main proof includes the Iwa…
reviewed by: Constantin Udrişte
Mathematics Subject Classification ID: 22E67
Mathematics Subject Classification ID: 22E15
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 1749368
zbMATH Keywords: Wiener algebra
zbMATH Keywords: reductive subgroup
zbMATH Keywords: maximal torus
zbMATH Keywords: loop groups
zbMATH Keywords: compact Lie groups
zbMATH Keywords: Birkhoff and Iwasawa decompositions
zbMATH Keywords: tangent group
zbMATH Keywords: infinite-dimensional Lie groups
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1113247803
OpenAlex ID: W2025440365
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DOI: 10.2748/TMJ/1113247803
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description: scientific article; zbMATH DE number 1749368