Abel群的一些分解定理的推广(Ⅲ)

秦鑫, 刘合国

数学学报 ›› 2019, Vol. 62 ›› Issue (3) : 361-372.

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数学学报 ›› 2019, Vol. 62 ›› Issue (3) : 361-372. DOI: 10.12386/A2019sxxb0034
论文

Abel群的一些分解定理的推广(Ⅲ)

    秦鑫1, 刘合国2
作者信息 +

Some Extensions of Decomposition Theorems in Abelian Groups (Ⅲ)

    Xin QIN1, He Guo LIU2
Author information +
文章历史 +

摘要

从主理想整环上有界模分解的Prüfer-Baer-Baer定理出发,研究(无限维)向量空间的代数的线性变换的几个基本问题,得到了如下结果:
V是域F上的(无限维)向量空间,AV上的一个代数的线性变换,则有
(1)若任何与A可交换的线性变换均与线性变换B可交换,则B=fA),其中fF上的多项式.进而线性变换B也是代数的.
(2)V中存在一组基,使A在这组基下的矩阵是有理标准型(经典标准型)矩阵.当F是代数闭域时,经典标准型矩阵即为若当标准型矩阵.
(3)当F是代数闭域时,A存在相应的Jordan-Chevalley分解.进一步,该结论在完全域上仍成立.
这些研究推广了有限维向量空间上线性变换的相关结果.

Abstract

On the basis of the Prüfer-Baer theorem of the bounded module over a principal ideal domain, this paper study several basic problems about the algebraic linear transformation of some vector space (infinite dimensional). Let V be a vector space (infinite dimensional) over a field F, A be an algebraic linear transformation of V:
(1) Suppose any linear transformation commuting with A commutes also with a linear transformation B, then B=f(A), where f is a polynomial over F.
(2) There exists a basis for V such that the matrix of A relative to this basis has the rational canonical form (classical canonical form). Moreover the classical canonical form becomes the Jordan canonical form when F is algebraic closed.
(3) There exists the Jordan-Chevalley decomposition of A when F is algebraically closed.
This result prevails for the perfect field in general. These results extend some theorems of finite dimensional vector spaces to infinite dimensional vector spaces.

关键词

Prü / fer-Baer-Baer定理 / 向量空间 / 双重中心化子定理 / Jordan-Chevalley分解

Key words

Prüfer-Baer theorem / vector space / double centralizer theorem / Jordan-Chevalley decomposition

引用本文

导出引用
秦鑫, 刘合国. Abel群的一些分解定理的推广(Ⅲ). 数学学报, 2019, 62(3): 361-372 https://doi.org/10.12386/A2019sxxb0034
Xin QIN, He Guo LIU. Some Extensions of Decomposition Theorems in Abelian Groups (Ⅲ). Acta Mathematica Sinica, Chinese Series, 2019, 62(3): 361-372 https://doi.org/10.12386/A2019sxxb0034

参考文献

[1] Greub W. H., Linear Algebra, Springer-Verlag, New York, 1975.
[2] Humphreys J. E., Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, 1972.
[3] Humphreys J. E., Linear Algebraic Groups, Springer-Verlag, New York, 1981.
[4] Jacobson N., Lectures in Abstract Algebra:Ⅱ, Linear Algebra, Springer-Verlag, New York, 1953.
[5] Liu H. G., Luo X. L., Qin X., et al., Some Extensions of Decomposition Theorems In Abelian Groups (I), Acta Math. Sin., Chin. Ser., 2017, 60(6):1065-1074.
[6] Prasolov V. V., Problems and Theorems in Linear Algebra, American Mathematical Soc., 1994.
[7] Suprunenko D. A., Tyshkevich R. I., Commutative Matrices, Academic Press, New York, London, 1968.
[8] Waterhouse W. C., Introduction to Affine Group Schemes, Springer-Verlag, New York, 1979.

基金

国家自然科学基金资助项目(11771129);湖北省高等学校优秀中青年科技创新团队计划(T201601)及湖北省新世纪高层次人才工程专项基金

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