Riemann-Birkhoff factorization/decomposition is a realization of a holomorphic matrix function of a circle as a product of a matrix holomorphic on a neighborhood of closed disk and a function of a matrix holomorphic on a neighborhood of an exterior of the disk including infinity and the circle itself. This decomposition is in the essence of Riemann-Hilbert problem. The interpretation in terms of loop groups is related to Bruhat decomposition.

There is an algebraic Birkhoff decomposition discovered in the study of Connes-Kreimer Hopf algebraic approach to renormalization in QFT.

G. D. Birkhoff, The generalized Riemann problem for linear differential equations and the allied problems for linear difference and q-difference equations, Proc. Amer. Acad. Arts and Sci. 49 (1913), 531-568.

G. Segal, A. Pressley, Loop groups, Oxford University Press

I. Z. Gohberg, M. G. Krein, Systems of integral equations on a half-line with kernels depending on the difference of the arguments, Transl. Amer. Math. Soc. 14 (1960) pp. 217–284

K. F. Clancey, I. Z. Gohberg, Factorization of matrix functions and singular integral operators, Birkhäuser (1981)

Created on January 4, 2014 at 10:59:15.
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