Birkhoff decompositions and Iwasawa decompositions for loop groups

Abstract

Representations of arbitrary real or complex invertible matrices as products of matrices of special type have been used for many purposes. The matrix form of the Gram-Schmidt orthonormalization procedure and the Gauss elimination process are instances of such matrix factorizations. For arbitrary, finite-dimensional, semisimple Lie groups, the corresponding matrix factorizations are known as Iwasawa decomposition and Bruhat decomposition. The work of Matsuki and Rossmann has generalized the Iwasawa decomposition for the finite-dimensional, semisimple Lie groups. In infinite dimensions, for affine loop groups/Kac-Moody groups, the Bruhat decomposition has an, also classical, competitor, the Birkhoff decomposition. Both decompositions (in infinite dimensions), the Iwasawa decomposition and the Birkhoff decomposition, have had important applications to analysis, e.g., to the Riemann-Hilbert problem, and to geometry, like to the construction of harmonic maps from Riemann surfaces to compact symmetric spaces and compact Lie groups. The Matsuki/Rossmann decomposition has been generalized only very recently to untwisted affine loop groups by Kellersch and facilitates the discussion of harmonic maps from Riemann surfaces to semisimple symmetric spaces. In the present paper we extend the decompositions of Kellersch and Birkhoff for untwisted affine loop groups to general Lie groups. These generalized decompositions have already been used in the discussion of harmonic maps from Riemann surfaces to arbitrary loop groups.