Lesson plan
Birkhoff factorization
In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by , is a generalization of the LU decomposition (i.e. Gauss elimination) to loop groups. The factorization of an invertible matrix M\in\mathrm{GL}_n(\mathbb{C}[z,z^{-1}]) with coefficients that are Laurent polynomials in z is given by a product M=M^{+}M^{0}M^{-} , where M^{+} has entries that are polynomials in z , M^{0}=\mathrm{diag}(z^{k_1}, z^{k_2},...,z^{k_n}) is diagonal with k_i\in\mathbb{Z} for 1\leq i\leq n and k_1\geq k_2\geq ...\geq k_n , and M^{-} has entries that are polynomials in z^{-1} . For a generic matrix we have M^{0}=\mathrm{id} . Birkhoff factorization implies the Birkhoff–Grothendieck theorem of that vector bundles over the projective line are sums of line bundles. There are several variations where the general linear group is replaced by some other reductive algebraic group, due to . 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.
Article summary
In mathematics, Birkhoff factorization or Birkhoff decomposition, introduced by , is a generalization of the LU decomposition (i.e. Gauss elimination) to loop groups. The factorization of an invertible matrix M\in\mathrm{GL}_n(\mathbb{C}[z,z^{-1}]) with coefficients that are Laurent polynomials in z is given by a product M=M^{+}M^{0}M^{-} , where M^{+} has entries that are polynomials in z , M^{0}=\mathrm{diag}(z^{k_1}, z^{k_2},...,z^{k_n}) is diagonal with k_i\in\mathbb{Z} for 1\leq i\leq n and k_1\geq k_2\geq ...\geq k_n , and M^{-} has entries that are polynomials in z^{-1} . For a generic matrix we have M^{0}=\mathrm{id} . Birkhoff factorization implies the Birkhoff–Grothendieck theorem of that vector bundles over the projective line are sums of line bundles. There are several variations where the general linear group is replaced by some other reductive algebraic group, due to . 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.
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Discussion questions
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