nLab Birkhoff decomposition

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.

References

Related $n$Lab entries: loop group, Riemann-Hilbert problem, Wiener-Hopf decomposition

• wikipedia Birkhoff factorization, Birkhoff–Grothendieck theorem
• eom: Birkhoff factorization
• George David 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.
• Andrew Pressley, Graeme Segal, Loop groups Oxford University Press (1988)
• 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)