标题： 非线性全纯超对称，Dolan-Grady关系和Onsager代数 显示英文标题

标题： Nonlinear holomorphic supersymmetry, Dolan-Grady relations and Onsager algebra

摘要： 最近，我们注意到高阶$n\in\N, n>1$的非线性全纯超对称性（$n$-HSUSY）具有代数起源。 我们证明了Onsager代数是$n$-HSUSY 的基础，并在后者的背景下研究了前者的结构。 发现了一组新的无限相互对易的电荷，这些电荷不同于Dolan-Grady集中的电荷，它们包含Onsager代数生成元的二次项。 这使我们能够找到$n$-HSUSY 超代数的一般形式，并明确确定$n=2,3,4,5,6$的情况。 对于通过收缩Dolan-Grady关系生成的Onsager代数的新收缩形式，得到了类似的结果。 作为应用，明确了已知的具有$n$-HSUSY 的一维和二维系统的代数结构，并提出了将该构造推广到非线性伪超对称性的方法。 这种推广应用于一些可积自旋模型，并借此得到一组出现在$PT$对称量子力学中的准精确可解系统。 显示英文摘要

摘要： Recently, it was noticed by us that the nonlinear holomorphic supersymmetry of order $n\in\N, n>1$, ($n$-HSUSY) has an algebraic origin. We show that the Onsager algebra underlies $n$-HSUSY and investigate the structure of the former in the context of the latter. A new infinite set of mutually commuting charges is found which, unlike those from the Dolan-Grady set, include the terms quadratic in the Onsager algebra generators. This allows us to find the general form of the superalgebra of $n$-HSUSY and fix it explicitly for the cases of $n=2,3,4,5,6$. The similar results are obtained for a new, contracted form of the Onsager algebra generated via the contracted Dolan-Grady relations. As an application, the algebraic structure of the known 1D and 2D systems with $n$-HSUSY is clarified and a generalization of the construction to the case of nonlinear pseudo-supersymmetry is proposed. Such a generalization is discussed in application to some integrable spin models and with its help we obtain a family of quasi-exactly solvable systems appearing in the $PT$-symmetric quantum mechanics.