Quantisations of the Volterra hierarchy

Abstract

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.

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Item Type:	Article
Authors/Creators:	Carpentier, S Mikhailov, AV https://orcid.org/0000-0003-4238-6995 Wang, JP
Copyright, Publisher and Additional Information:	© The Author(s) 2022. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Keywords:	The quantum Volterra equation; Quantum integrability; Super integable systems; Non-deformation quantisation; Quantised algebra
Dates:	Published: October 2022 Published (online): 18 September 2022 Accepted: 6 September 2022
Institution:	The University of Leeds
Academic Units:	The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Applied Mathematics (Leeds)
Depositing User:	Symplectic Publications
Date Deposited:	16 Sep 2022 14:13
Last Modified:	01 Feb 2023 02:00
Status:	Published
Publisher:	Springer
Identification Number:	10.1007/s11005-022-01588-1
Open Archives Initiative ID (OAI ID):	oai:eprints.whiterose.ac.uk:190998

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Quantisations of the Volterra hierarchy

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