标题： 与包络代数中Cartan子代数的交换子相关的超可积系统 显示英文标题

标题： Superintegrable systems associated to commutants of Cartan subalgebras in enveloping algebras

摘要： 我们提供了复半单李代数非例外型的Cartan子代数的中心化子在它们的泛包络代数中所张成元素的分类和显式公式，并证明这些元素生成多项式（泊松）代数。 对于秩为三的单李代数的情况，给出了精确的结构，并展示了相应多项式代数之间的包含关系。 我们发展了从中心化子子代数及其对称代数的生成元构造代数超可积系统的思路。 特别是，我们明确给出了对应于$\mathfrak{so}(6,\mathbb{C})$的超可积系统的例子。 显示英文摘要

摘要： We provide a classification and explicit formulas for the elements that span the centralizer of Cartan subalgebras of complex semisimple Lie algebras of non-exceptional type in their universal enveloping algebra, and show that these generate polynomial (Poisson) algebras. The precise structure for the case of rank three simple Lie algebras is provided, and the inclusion relations between the corresponding polynomial algebras illustrated. We develop the ideas of constructing algebraic superintegrable systems and their integrals from the generators of the centralizer subalgebras and their symmetry algebra. In particular, we present explicitly one example the superintegrable system corresponding to the Cartan centralizer of $\mathfrak{so}(6,\mathbb{C})$.