Tohoku Mathematical Journal
2001
December
SECOND SERIES VOL. 53, NO. 4

Tohoku Math. J.
53 (2001), 593-615

Title BIRKHOFF DECOMPOSITION AND IWASAWA DECOMPOSITION FOR LOOP GROUPS

Author Vladimir Balan and Josef Dorfmeister

(Received February 9, 2000, revised December 25, 2000)
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 [2].

[2] V. Balan and J. Dorfmeister, The DPW method for harmonic maps from Riemann surfaces to general Lie groups, Balkan J. Geom. Appl. 5(2000), 7-37.

2000 Mathematics Subject Classification. Primary 22E67; Secondary 22E15, 22E25, 22E46.

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