SL(n,C)中的一些特殊可解子群及应用（英文）

Some special solvable subgroups of SL(n,C) and their application

Fuchs方程在许多物理问题中有着广泛而重要的应用,所以判定给定的Fuchs方程的可积性及解的性质在理论与应用中都有意义。根据Khovanskiy定理,Fuchs方程的可积性判定问题可转化为对其单值群的计算并判断其可解性,但由于这方面理论及计算的发展尚不完善。到目前为止,对任意给定的Fuchs方程,并不存在行之有效的方法求出单值群以及判断其可解性。给出了SL(n;C)中的几类特殊可解子群,并应用于Fuchs系统.由Fuchs方程的单值群的可解性与其可积性的关系,得出结论,若Fuchs系统解的Riemann曲面是二维有界闭流形上除去有限个极点的曲面,则其单值群必然是有限生成的线性群。特别若生成元满足本文所列之条件,则单值群必可解,从而Fuchs方程可积。

It is well known that Fuchsian equations have widespread and important application in mathematical physics problems. Therefore the research on the judgment of integrability for some given Fuchsian equations has significance both in theory and application. By Khovanskiy theorem, the problem on judging the integrability of Fuchsian equations can be changed into deducing the corresponding monodromy groups and checking their solvability. But as the theory is still imperfect, up to now, no effective method has been introduced to resolve this problem for a certain Fuchsian equation. In this paper, the author gives several classes of special solvable subgroups in SL（n,C）, and their application for Fuchsian systems. By the relation between the solvability of monodromy group and the integrability of Fuchsian equations, the conclusion is if solution of Riemann surface of Fuchsian system is a surface of two dimensional bounded closed mainfold with getting rid of poles, where the number of poles is finite, then monodromy group of this system must be finite and linear. Especially, if generated elements satisfy the condition of the theorems in the paper, the monodromy group must be solvable. Thus the system is integral in quadratures.