Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras
Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras
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泛包络代数中子代数交换子的代数（超）可积性
DOI：
10.1088/1751-8121/acb576
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发表时间：
2022
期刊：
影响因子：
--
通讯作者：
Yao
中科院分区：
文献类型：
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作者：
R. Campoamor;D. Latini;I. Marquette;Yao
从一个基于泛包络代数中的子代数的交换子的纯代数过程出发，提出了生成多项式对称代数的代数Hamilton算子和运动常数的概念.本文详细讨论了特殊线性李代数sl（n）的情形，得到了交换子关于Cartan子代数的显式基，并计算了多项式代数的阶.进一步表明，通过sl（n）的适当实现，这提供了与（n−1）维球面Sn−1上的一般超可积模型和相关的Racah代数R（n）的明确联系。特别是，我们明确地表明，如何在两个球和三个球和相关的对称代数的模型可以得到从二次和三次多项式代数所产生的交换子定义在sl（3）和sl（4）的包络代数，分别。该构造是在经典（或泊松-李）上下文中进行的，其中Berezin括号取代了交换子。
Starting from a purely algebraic procedure based on the commutant of a subalgebra in the universal enveloping algebra of a given Lie algebra, the notion of algebraic Hamiltonians and the constants of the motion generating a polynomial symmetry algebra is proposed. The case of the special linear Lie algebra sl(n) is discussed in detail, where an explicit basis for the commutant with respect to the Cartan subalgebra is obtained, and the order of the polynomial algebra is computed. It is further shown that, with an appropriate realization of sl(n) , this provides an explicit connection with the generic superintegrable model on the (n−1) -dimensional sphere Sn−1 and the related Racah algebra R(n). In particular, we show explicitly how the models on the two-sphere and three-sphere and the associated symmetry algebras can be obtained from the quadratic and cubic polynomial algebras generated by the commutants defined in the enveloping algebra of sl(3) and sl(4) , respectively. The construction is performed in the classical (or Poisson-Lie) context, where the Berezin bracket replaces the commutator.