Volterra层次的量子化
Quantisations of the Volterra hierarchy
摘要：
在本文中，我们探讨了一个最近出现的方法来量化的问题的基础上的概念，量化的理想。我们明确证明，nonabelian Volterra连同整个层次的对称性承认变形量化。我们表明，所有奇数度对称的Volterra层次承认也非变形量化。讨论了周期Volterra族的量子化问题，包括量子化代数的量子哈密顿量和中心元，并证明了所得到的量子系统的超可积性.证明了周期为$3$的Volterra系统存在双量子结构，这可以看作是其经典双哈密顿结构的量子变形.
英文摘要：
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 admit 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.
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