Quantisations of the Volterra hierarchy
Quantisations of the Volterra hierarchy
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DOI：
10.1007/s11005-022-01588-1
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发表时间：
2022-04
影响因子：
1.2
通讯作者：
Sylvain Carpentier;A. Mikhailov;Jing Ping Wang
中科院分区：
文献类型：
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作者：
Sylvain Carpentier;A. Mikhailov;Jing Ping Wang
在这篇文章中，我们探索了一种最近出现的基于量子化理想概念的量子化问题的方法。我们明确地证明了非阿贝尔Volterra以及它的整个对称族都允许形变量子化。我们证明了Volterra族的所有奇次对称也允许非形变量子化。我们讨论了周期Volterra族的量子化问题，包括它们的量子哈密顿量，量子化代数的中心元，并证明了所得到的量子系统的超可积性。我们证明了周期为3的Volterra系统具有双量子结构，这可以看作是其经典双哈密顿结构的量子形变。
In this paper, we explore a recently emerged approach to the problem of quantisation based on the notion of quantisation ideals. We explicitly prove that the nonabelian Volterra together with the whole hierarchy of its symmetries admits a deformation quantisation. We show that all odd-degree symmetries of the Volterra hierarchy admit also a non-deformation quantisation. We discuss the quantisation problem for periodic Volterra hierarchy including their quantum Hamiltonians, central elements of the quantised algebras, and demonstrate super-integrability of the quantum systems obtained. We show that the Volterra system with period 3 admits a bi-quantum structure, which can be regarded as a quantum deformation of its classical bi-Hamiltonian structure.