Title: Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrödinger algebras Show Chinese title

Title: 二次代数作为Schrödinger代数包络代数中代数哈密顿量的对易子

Abstract: We discuss a procedure to determine finite sets $\mathcal{M}$ within the commutant of an algebraic Hamiltonian in the enveloping algebra of a Lie algebra $\mathfrak{g}$ such that their generators define a quadratic algebra. Although independent from any realization of Lie algebras by differential operators, the method is partially based on an analytical approach, and uses the coadjoint representation of the Lie algebra $\mathfrak{g}$. The procedure, valid for non-semisimple algebras, is tested for the centrally extended Schr\"odinger algebras $\widehat{S}(n)$ for various different choices of algebraic Hamiltonian. For the so-called extended Cartan solvable case, it is shown how the existence of minimal quadratic algebras can be inferred without explicitly manipulating the enveloping algebra. Show Chinese abstract

Abstract: 我们讨论一种程序，用于确定代数哈密顿量在李代数 $\mathfrak{g}$的包络代数的换位子中的有限集合 $\mathcal{M}$，使得它们的生成元定义一个二次代数。 尽管与任何由微分算子实现的李代数无关，该方法部分基于一种分析方法，并利用李代数 $\mathfrak{g}$的共伴表示。 该程序适用于非半单代数，并针对中心扩张的薛定谔代数 $\widehat{S}(n)$进行了测试，对于不同的代数哈密顿量选择进行了验证。 对于所谓的扩展卡当可解情况，展示了如何在不显式操作包络代数的情况下推断出最小二次代数的存在性。