论文标题
$ su（3）$的多项式代数和两个领域的通用模型
Polynomial algebras from $su(3)$ and the generic model on the two sphere
论文作者
论文摘要
多年来，已经引入了基于Lie代数的可促进系统的构建。但是，这些方法取决于明确的实现，例如作为基础代数的差异操作员。构建其相关对称代数的情况也是如此，通常采用有限生成的二次代数的形式。这些代数通常显示依赖中心元素的结构常数，尤其是哈密顿元素。在本文中，我们开发了一种新的方法，重新审查了2个速率上通用可促进系统的情况，该系统众所周知对称代数为RACAH代数$ r（3）$。这样的模型与共同扁平空间及其12个等效类别上的59美元$ 2D $可促进系统有关。我们证明，在$ su（3）$ su（3）$的代数中使用2,3和4的进一步多项式，一个人只能基于$ su（3）$ lie代数的抽象换向关系而产生代数，而无需明确的限制。这种结构依赖于最大的阿贝尔亚军，也称为MASA，它是纸箱发电机及其换向物。我们获得了一个新的6维立方代数，其中结构常数是整数数字，该数字从四分之一的代数中降低，结构常数取决于Cartan Generator和Casimir不变。我们还使用$ su（3）$的二次和立方casimir不变式提出了对称代数的其他形式。仅当使用显式实现时，它才能减少为已知的二次RACAH代数$ r（3）$。该代数结构描述了两个球体上通用共聚系统的对称性。我们还对另一个6维立方代数的收缩将对应于Smorodinsky-Winternitz模型的对称代数。
Construction of superintegrable systems based on Lie algebras have been introduced over the years. However, these approaches depend on explicit realisations, for instance as a differential operators, of the underlying Lie algebra. This is also the case for the construction of their related symmetry algebra which take usually the form of a finitely generated quadratic algebra. These algebras often display structure constants which depend on the central elements and in particular on the Hamiltonian. In this paper, we develop a new approach reexamining the case of the generic superintegrable systems on the 2-sphere for which a symmetry algebra is known to be the Racah algebra $R(3)$. Such a model is related to the 59 $2D$ superintegrable systems on conformally flat spaces and their 12 equivalence classes. We demonstrate that using further polynomials of degree 2,3 and 4 in the enveloping algebra of $su(3)$ one can generate an algebra based only on abstract commutation relations of $su(3)$ Lie algebra without explicit constraints on the representations or realisations. This construction relies on the maximal Abelian subalgebra, also called MASA, which are the Cartan generators and their commutant. We obtain a new 6-dimensional cubic algebra where the structure constant are integer numbers which reduce from a quartic algebra for which the structure constant depend on the Cartan generator and the Casimir invariant. We also present other form of the symmetry algebra using the quadratic and cubic Casimir invariants of $su(3)$. It reduces as the known quadratic Racah algebra $R(3)$ only when using an explicit realization. This algebraic structure describe the symmetry of the generic superintegrable systems on the 2 sphere. We also present a contraction to another 6-dimensional cubic algebra which would corresponding to the symmetry algebra of a Smorodinsky-Winternitz model.
