Wolfes model aka $G_2/I_6$-rational integrable model: $g^{(2)}, g^{(3)}$ hidden algebras and quartic polynomial algebra of integrals

TL;DR

This paper analyzes the Wolfes 3-body integrable model, revealing its algebraic structure involving hidden $g^{(2)}$ and $g^{(3)}$ algebras, and identifies a quartic polynomial algebra of integrals.

Contribution

It introduces a minimal algebraic representation of the model's integrals using hidden algebra generators and uncovers a quartic polynomial algebra structure.

Findings

Hamiltonian and integrals expressed as algebraic differential operators

Identified quartic polynomial algebra of integrals embedded in $g^{(3)}$

Compared with the cubic algebra of the Calogero model

Abstract

One-dimensional 3-body Wolfes model with 2- and 3-body interactions also known as $G_2/I_6$-rational integrable model of the Hamiltonian reduction is exactly-solvable and superintegrable. Its Hamiltonian $H$ and two integrals ${\cal I}_{1}, {\cal I}_{2}$, which can be written as algebraic differential operators in two variables (with polynomial coefficients) of the 2nd and 6th orders, respectively, are represented as non-linear combinations of $g^{(2)}$ or $g^{(3)}$ (hidden) algebra generators in a minimal manner. By using a specially designed MAPLE-18 code to deal with algebraic operators it is found that $(H, {\cal I}_1, {\cal I}_2, {\cal I}_{12} \equiv [{\cal I}_1, {\cal I}_2])$ are the four generating elements of the {\it quartic} polynomial algebra of integrals. This algebra is embedded into the universal enveloping algebra $g^{(3)}$. In turn, 3-body/$A_2$-rational Calogero model is characterized by cubic polynomial algebra of integrals, it is mentioned briefly.