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As an immediate consequence of [[Birkhoff factorization|Birkhoff factorization]], [[#References|[a1]]], the group of differentiable invertible matrix loops <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b1202501.png" /> may be decomposed in a union of subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b1202502.png" />, labelled by unordered <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b1202503.png" />-tuples of integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b1202504.png" />. Each of these consists of all loops with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b1202505.png" /> as the set of partial indices. This decomposition is called a Birkhoff stratification. It reflects important properties of holomorphic vector bundles over the Riemann sphere [[#References|[a2]]], singular integral equations [[#References|[a3]]], and Riemann–Hilbert problems [[#References|[a4]]]. The structure of a Birkhoff stratification resembles those of Schubert decompositions of Grassmannians and Bruhat decompositions of complex Lie groups (cf. also [[Bruhat decomposition|Bruhat decomposition]]). The Birkhoff strata <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b1202506.png" /> are complex submanifolds of finite codimension in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b1202507.png" />. Codimension, homotopy type and cohomological fundamental class of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b1202508.png" /> are expressible in terms of the label <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b1202509.png" /> [[#References|[a5]]]. The adjacencies among the Birkhoff strata describe deformations of holomorphic vector bundles [[#References|[a5]]]. Birkhoff stratifications also exist for loop groups of compact Lie groups [[#References|[a6]]]. For the group of based loops <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b12025010.png" /> on a compact [[Lie group|Lie group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b12025011.png" />, the Birkhoff strata are contractible complex submanifolds labelled by the conjugacy classes of homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b12025012.png" /> [[#References|[a6]]]. Birkhoff stratification has a visual interpretation in the framework of [[Morse theory|Morse theory]] of the energy function on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b12025013.png" /> [[#References|[a6]]]. Certain geometric aspects of Birkhoff stratification may be described in terms of non-commutative differential geometry and Fredholm structures [[#References|[a7]]], [[#References|[a8]]]. In particular, the Birkhoff strata become Fredholm submanifolds of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b12025014.png" /> endowed with various Fredholm structures. Fredholm structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b12025015.png" /> arise from the natural Kähler structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b12025016.png" /> [[#References|[a7]]] and in the context of generalized Riemann–Hilbert problems with coefficients in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120250/b12025017.png" /> [[#References|[a8]]]. Curvatures and characteristic classes of Birkhoff strata may be computed in the spirit of non-commutative differential geometry, in terms of regularized traces of appropriate Toeplitz operators [[#References|[a7]]].
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As an immediate consequence of [[Birkhoff factorization|Birkhoff factorization]], [[#References|[a1]]], the group of differentiable invertible matrix loops $\operatorname{LGL} ( n , \mathbf{C} )$ may be decomposed in a union of subsets $B _ { \kappa }$, labelled by unordered $n$-tuples of integers $\kappa$. Each of these consists of all loops with $\kappa$ as the set of partial indices. This decomposition is called a Birkhoff stratification. It reflects important properties of holomorphic vector bundles over the Riemann sphere [[#References|[a2]]], singular integral equations [[#References|[a3]]], and Riemann–Hilbert problems [[#References|[a4]]]. The structure of a Birkhoff stratification resembles those of Schubert decompositions of Grassmannians and Bruhat decompositions of complex Lie groups (cf. also [[Bruhat decomposition|Bruhat decomposition]]). The Birkhoff strata $B _ { \kappa }$ are complex submanifolds of finite codimension in $\operatorname{LGL} ( n , \mathbf{C} )$. Codimension, homotopy type and cohomological fundamental class of $B _ { \kappa }$ are expressible in terms of the label $\kappa$ [[#References|[a5]]]. The adjacencies among the Birkhoff strata describe deformations of holomorphic vector bundles [[#References|[a5]]]. Birkhoff stratifications also exist for loop groups of compact Lie groups [[#References|[a6]]]. For the group of based loops $\Omega G$ on a compact [[Lie group|Lie group]] $G$, the Birkhoff strata are contractible complex submanifolds labelled by the conjugacy classes of homomorphisms $\mathcal{T} \rightarrow G$ [[#References|[a6]]]. Birkhoff stratification has a visual interpretation in the framework of [[Morse theory|Morse theory]] of the energy function on $\Omega G$ [[#References|[a6]]]. Certain geometric aspects of Birkhoff stratification may be described in terms of non-commutative differential geometry and Fredholm structures [[#References|[a7]]], [[#References|[a8]]]. In particular, the Birkhoff strata become Fredholm submanifolds of $\Omega G$ endowed with various Fredholm structures. Fredholm structures on $\Omega G$ arise from the natural Kähler structure on $\Omega G$ [[#References|[a7]]] and in the context of generalized Riemann–Hilbert problems with coefficients in $G$ [[#References|[a8]]]. Curvatures and characteristic classes of Birkhoff strata may be computed in the spirit of non-commutative differential geometry, in terms of regularized traces of appropriate Toeplitz operators [[#References|[a7]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.D. Birkhoff,  "Singular points of ordinary linear differential equations"  ''Trans. Amer. Math. Soc.'' , '''10'''  (1909)  pp. 436–470</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Grothendieck,  "Sur la classification des fibrés holomorphes sur la sphère de Riemann"  ''Amer. J. Math.'' , '''79'''  (1957)  pp. 121–138</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  B. Bojarski,  "On the stability of Hilbert problem for holomorphic vector"  ''Bull. Acad. Sci. Georgian SSR'' , '''21'''  (1958)  pp. 391–398</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  S. Disney,  "The exponents of loops on the complex general linear group"  ''Topology'' , '''12'''  (1973)  pp. 297–315</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  A. Pressley,  G. Segal,  "Loop groups" , Clarendon Press  (1986)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Freed,  "The geometry of loop groups"  ''J. Diff. Geom.'' , '''28'''  (1988)  pp. 223–276</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  G. Khimshiashvili,  "Lie groups and transmission problems on Riemann surfaces"  ''Contemp. Math.'' , '''131'''  (1992)  pp. 164–178</TD></TR></table>
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<table><tr><td valign="top">[a1]</td> <td valign="top">  G.D. Birkhoff,  "Singular points of ordinary linear differential equations"  ''Trans. Amer. Math. Soc.'' , '''10'''  (1909)  pp. 436–470</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  A. Grothendieck,  "Sur la classification des fibrés holomorphes sur la sphère de Riemann"  ''Amer. J. Math.'' , '''79'''  (1957)  pp. 121–138</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  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</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  B. Bojarski,  "On the stability of Hilbert problem for holomorphic vector"  ''Bull. Acad. Sci. Georgian SSR'' , '''21'''  (1958)  pp. 391–398</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  S. Disney,  "The exponents of loops on the complex general linear group"  ''Topology'' , '''12'''  (1973)  pp. 297–315</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  A. Pressley,  G. Segal,  "Loop groups" , Clarendon Press  (1986)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  D. Freed,  "The geometry of loop groups"  ''J. Diff. Geom.'' , '''28'''  (1988)  pp. 223–276</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  G. Khimshiashvili,  "Lie groups and transmission problems on Riemann surfaces"  ''Contemp. Math.'' , '''131'''  (1992)  pp. 164–178</td></tr></table>

Latest revision as of 17:01, 1 July 2020

As an immediate consequence of Birkhoff factorization, [a1], the group of differentiable invertible matrix loops $\operatorname{LGL} ( n , \mathbf{C} )$ may be decomposed in a union of subsets $B _ { \kappa }$, labelled by unordered $n$-tuples of integers $\kappa$. Each of these consists of all loops with $\kappa$ as the set of partial indices. This decomposition is called a Birkhoff stratification. It reflects important properties of holomorphic vector bundles over the Riemann sphere [a2], singular integral equations [a3], and Riemann–Hilbert problems [a4]. The structure of a Birkhoff stratification resembles those of Schubert decompositions of Grassmannians and Bruhat decompositions of complex Lie groups (cf. also Bruhat decomposition). The Birkhoff strata $B _ { \kappa }$ are complex submanifolds of finite codimension in $\operatorname{LGL} ( n , \mathbf{C} )$. Codimension, homotopy type and cohomological fundamental class of $B _ { \kappa }$ are expressible in terms of the label $\kappa$ [a5]. The adjacencies among the Birkhoff strata describe deformations of holomorphic vector bundles [a5]. Birkhoff stratifications also exist for loop groups of compact Lie groups [a6]. For the group of based loops $\Omega G$ on a compact Lie group $G$, the Birkhoff strata are contractible complex submanifolds labelled by the conjugacy classes of homomorphisms $\mathcal{T} \rightarrow G$ [a6]. Birkhoff stratification has a visual interpretation in the framework of Morse theory of the energy function on $\Omega G$ [a6]. Certain geometric aspects of Birkhoff stratification may be described in terms of non-commutative differential geometry and Fredholm structures [a7], [a8]. In particular, the Birkhoff strata become Fredholm submanifolds of $\Omega G$ endowed with various Fredholm structures. Fredholm structures on $\Omega G$ arise from the natural Kähler structure on $\Omega G$ [a7] and in the context of generalized Riemann–Hilbert problems with coefficients in $G$ [a8]. Curvatures and characteristic classes of Birkhoff strata may be computed in the spirit of non-commutative differential geometry, in terms of regularized traces of appropriate Toeplitz operators [a7].

References

[a1] G.D. Birkhoff, "Singular points of ordinary linear differential equations" Trans. Amer. Math. Soc. , 10 (1909) pp. 436–470
[a2] A. Grothendieck, "Sur la classification des fibrés holomorphes sur la sphère de Riemann" Amer. J. Math. , 79 (1957) pp. 121–138
[a3] 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
[a4] B. Bojarski, "On the stability of Hilbert problem for holomorphic vector" Bull. Acad. Sci. Georgian SSR , 21 (1958) pp. 391–398
[a5] S. Disney, "The exponents of loops on the complex general linear group" Topology , 12 (1973) pp. 297–315
[a6] A. Pressley, G. Segal, "Loop groups" , Clarendon Press (1986)
[a7] D. Freed, "The geometry of loop groups" J. Diff. Geom. , 28 (1988) pp. 223–276
[a8] G. Khimshiashvili, "Lie groups and transmission problems on Riemann surfaces" Contemp. Math. , 131 (1992) pp. 164–178
How to Cite This Entry:
Birkhoff stratification. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Birkhoff_stratification&oldid=12320
This article was adapted from an original article by G. Khimshiashvili (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article