Quantisations of the Volterra hierarchy

TL;DR

This paper investigates the quantisation of the Volterra hierarchy using the concept of quantisation ideals, demonstrating deformation quantisation for the hierarchy and non-deformation quantisation for odd symmetries, with applications to periodic systems and super-integrability.

Contribution

It provides the first explicit proof of deformation quantisation for the nonabelian Volterra hierarchy and explores quantum structures in periodic cases, including bi-quantum structures.

Findings

Deformation quantisation of the Volterra hierarchy and its symmetries.

Existence of non-deformation quantisation for odd symmetries.

Super-integrability of quantum periodic Volterra systems.

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 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.