現代数学の系譜工学物理雑談古典ガロア理論も読む78

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む78 (1002ﾚｽ)
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649: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/09(土) 06:49:24.66 ID:aIAMZK1h

>>607
>モノドロミーで、GL(n, C)が出てくるよ（＾＾

(参考)
https://ja.wikipedia.org/wiki/%E3%83%A2%E3%83%8E%E3%83%89%E3%83%AD%E3%83%9F%E3%83%BC
モノドロミー
脚注
2 ^ V.P. Kostov (2004), “The Deligne?Simpson problem ? a survey”, J. Algebra 281 (1): 83?108, doi:10.1016/j.jalgebra.2004.07.013, MR2091962 and the references therein.
V.P. Kostov / Journal of Algebra 281 (2004) 83?108
The Deligne?Simpson problem?a survey
Vladimir Petrov Kostov
Universite de Nice?Sophia Antipolis, Laboratoire de Mathematiques,
Parc Valrose, 06108 Nice cedex 2, France
Received 18 September 2002
Available online 3 September 2004
(抜粋)
1. Introduction
1.1. Regular and Fuchsian linear systems on Riemann’s sphere
The problem which is the subject of this paper admits a purely algebraic formulation.
Yet its importance lies in the analytic theory of systems of linear differential equations, this
is why we start by considering the linear system of ordinary differential equations defined
on Riemann’s sphere:
dX/dt = A(t)X. (1)
Here the n × n-matrix A is meromorphic on CP1, with poles at a1, . . . ,ap+1; the dependent
variables X form an n×n-matrix.Without loss of generality we assume that∞is not
among the poles aj and not a pole of the 1-form A(t) dt . In modern literature (see, e.g.,
[18]) the terminology of meromorphic connections and sections is often preferred to the
one of meromorphic linear systems and their solutions and there is a 1?1-correspondence
between the two languages.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1571400076/649

650: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/11/09(土) 06:50:09.05 ID:aIAMZK1h

>>649
つづき

This transformation preserves regularity but, in general, it does not preserve being Fuchsian.
The only invariant under the group of linear transformations (5) is the monodromy
group of the system.
Set Σ := CP1\{a1, . . . ,ap+1}. To define the monodromy group one has to fix a base
point a0 ∈ Σ and a matrix B ∈ GL(n,C). The monodromy group is defined only up to
conjugacy due to the freedom to choose a0 and B.

https://en.wikipedia.org/wiki/Monodromy
Notes
2 V. P. Kostov (2004), "The Deligne?Simpson problem ? a survey", J. Algebra, 281 (1): 83?108, arXiv:math/0206298, doi:10.1016/j.jalgebra.2004.07.013, MR 2091962 and the references therein.
（上記と５章が微妙に違うな）
https://arxiv.org/abs/math/0206298
The Deligne-Simpson problem -- a survey
Vladimir Petrov Kostov
(Submitted on 27 Jun 2002)
The Deligne-Simpson problem (DSP) (resp. the weak DSP) is formulated like this:
{\em give necessary and sufficient conditions for the choice of the conjugacy classes Cj⊂GL(n,C) or cj⊂gl(n,C) so that there exist irreducible (resp. with trivial centralizer) (p+1)-tuples of matrices Mj∈Cj or Aj∈cj satisfying the equality M1...Mp+1=I or A1+...+Ap+1=0}.
The matrices Mj and Aj are interpreted as monodromy operators of regular linear systems and as matrices-residua of Fuchsian ones on Riemann's sphere.
The present paper offers a survey of the results known up to now concerning the DSP.
https://arxiv.org/pdf/math/0206298.pdf
以上

http://rio2016.5ch.net/test/read.cgi/math/1571400076/650

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