The notion of hidden symmetry algebra used in the context of exactly solvable systems is re-examined from the purely algebraic way, analyzing subspaces of commuting polynomials that generate finite-dimensional quadratic algebras. By construction, these algebras do not depend on the choice of realizations by vector fields of the underlying Lie algebra, allowing to propose a procedure to analyze polynomial algebras as those subspaces in an enveloping algebra that commute with a given algebraic Hamiltonian.

中文翻译：

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隐藏对称代数与超可积系统二次代数的构造

在完全可解系统的上下文中使用的隐藏对称代数的概念从纯代数方式重新检查，分析生成有限维二次代数的交换多项式的子空间。通过构造，这些代数不依赖于底层李代数的向量场实现的选择，允许提出一个程序来分析多项式代数作为包络代数中与给定代数哈密顿量交换的那些子空间。