Title: Subalgebras of Lie algebras. Example of sl(3,R) revisited Show Chinese title

Title: 李代数的子代数 重新审视 sl(3,R) 的例子

Abstract: Revisiting the results by Winternitz [Symmetry in physics, CRM Proc. Lecture Notes 34, American Mathematical Society, Providence, RI, 2004, pp. 215-227], we thoroughly refine his classification of Lie subalgebras of the real order-three special linear Lie algebra and thus present the correct version of this classification for the first time. A similar classification over the complex numbers is also carried out. We follow the general approach by Patera, Winternitz and Zassenhaus but in addition enhance it and rigorously prove its theoretical basis for the required specific cases of classifying subalgebras of real or complex finite-dimensional Lie algebras. As a byproduct, we first construct complete lists of inequivalent subalgebras of the rank-two affine Lie algebra over both the real and complex fields. Show Chinese abstract

Abstract: 重新审视Winternitz [对称性在物理中的应用，CRM研讨会讲稿系列34，美国数学学会，罗德岛普罗维登斯，2004年，第215-227页] 的结果，我们彻底改进了他对实数三阶特殊线性李代数的李子代数分类，从而首次提出了这一分类的正确版本。对复数的情况也进行了类似的分类。我们遵循Patera、Winternitz和Zassenhaus的一般方法，但除此之外还对其进行了增强，并严格证明了其理论基础，以适用于对实数或复数有限维李代数的子代数进行分类所需的特定情况。作为副产品，我们首次构建了在实数和复数域上二维仿射李代数的所有非等价子代数的完整列表。