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arXiv · hep-th/0112158

Nonlinear holomorphic supersymmetry, Dolan-Grady relations and Onsager algebra

Sergey M. Klishevich·Mikhail S. Plyushchay
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Abstract

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.

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Sergey M. Klishevich, Mikhail S. Plyushchay. 2002-02-15. Nonlinear holomorphic supersymmetry, Dolan-Grady relations and Onsager algebra. https://doi.org/10.1016/s0550-3213(02)00071-8

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