Zhukovsky-Volterra top and quantisation ideals

TL;DR

This paper explores the quantisation of the Zhukovsky-Volterra top, revealing two distinct quantisations linked to different algebraic structures, and discusses their relation to existing quantisation methods.

Contribution

It introduces a four-parametric family of compatible Poisson brackets and identifies two new quantisations of the Zhukovsky-Volterra top, expanding understanding of its algebraic structures.

Findings

Discovered a four-parametric pencil of compatible Poisson brackets.

Identified two distinct quantisations: one related to $so(3)$ and another to inhomogeneous quadratic Poisson brackets.

Established relationships between new quantisations and existing methods by Sklyanin and Levin-Olshanetsky-Zotov.

Abstract

In this letter, we revisit the quantisation problem for a fundamental model of classical mechanics - the Zhukovsky-Volterra top. We have discovered a four-parametric pencil of compatible Poisson brackets, comprising two quadratic and two linear Poisson brackets. Using the quantisation ideal method, we have identified two distinct quantisations of the Zhukovsky-Volterra top. The first type corresponds to the universal enveloping algebras of $so(3)$, leading to Lie-Poisson brackets in the classical limit. The second type can be regarded as a quantisation of the four-parametric inhomogeneous quadratic Poisson pencil. We discuss the relationships between the quantisations obtained in our paper, Sklyanin's quantisation of the Euler top, and Levin-Olshanetsky-Zotov's quantisation of the Zhukovsky-Volterra top.