Birkhoff factorization

In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by Birkhoff (1909), is the factorization of an invertible matrix M with coefficients that are Laurent polynomials in z into a product M = M⁺M⁰M⁻, where M⁺ has entries that are polynomials in z, M⁰ is diagonal, and M⁻ has entries that are polynomials in z⁻¹. There are several variations where the general linear group is replaced by some other reductive algebraic group, due to Grothendieck (1957).

Birkhoff factorization implies the Birkhoff–Grothendieck theorem of Grothendieck (1957) that vector bundles over the projective line are sums of line bundles.

Birkhoff factorization follows from the Bruhat decomposition for affine Kac-Moody groups (or loop groups), and conversely the Bruhat decomposition for the affine general linear group follows from Birkhoff factorization together with the Bruhat decomposition for the ordinary general linear group.

References

• Birkhoff, George David (1909), "Singular points of ordinary linear differential equations", Transactions of the American Mathematical Society 10 (4): 436–470, doi:10.2307/1988594, ISSN 0002-9947, JFM 40.0352.02, JSTOR 1988594
• Grothendieck, Alexander (1957), "Sur la classification des fibrés holomorphes sur la sphère de Riemann", American Journal of Mathematics 79: 121–138, doi:10.2307/2372388, ISSN 0002-9327, JSTOR 2372388, MR 0087176
• Khimshiashvili, G. (2001), "b/b120240", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
• Pressley, Andrew; Segal, Graeme (1986), Loop groups, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, ISBN 978-0-19-853535-5, MR 900587