Birkhoff decompositions and Iwasawa decompositions for loop groups
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 Iwasawa decomposition of loop groups of the semisimple part of the reductive subgroup, basically described in the Appendix of the paper, and recently developed by \textit{Peter Kellersch} [Eine Verallgemeinerung der Iwasawa Zerlegung in Loop Gruppen, Dissertation, TU Munich, 1999]. Section 7 shows that the ``big cell'' \(\widetilde P_G= \Lambda G\cdot \Lambda^+G^C\) of the Iwasawa splitting is open in \(\Lambda^+ G^C\), states a sufficient condition for \(\widetilde P_G\) being dense in \(\Lambda G^C\), and proves that the big cell covers the whole loop group \(\Lambda G^C\) iff the semisimple part of \(G\) is compact. Sections 8 and 9 illustrate by examples the Birkhoff and Iwasawa decompositions, respectively, and consider for \(G\) the cases of nilpotent \(2\times 2\) and \(3\times 3\) matrices, \(SL(2,\mathbb{C})\) and the tangent group of a finite-dimensional Lie group. All in all, the present paper represents a significant step in understanding the main features of loop groups within the field of infinite-dimensional Lie groups and that the two developed general loop group decompositions provide notable further applications in analysis and geometry.
scientific article; zbMATH DE number 1749368
scientific article