$gl(3)$ polynomial integrable system: different faces of the 3-body/$A_2$ elliptic Calogero model

Abstract

It is shown that the $gl(3)$ polynomial integrable system, introduced by Sokolov-Turbiner, is equivalent to the $gl(3)$ quantum Euler-Arnold top in a constant magnetic field. Their Hamiltonian and Integral can be rewritten in terms of $gl(3)$ algebra generators. All these $gl(3)$ generators can be represented by the non-linear elements of the universal enveloping algebra of the Heisenberg algebra $h_5({p}_{1,2},{q}_{1,2}, I)$, thus, the Hamiltonian and Integral are two elements of the universal enveloping algebra $U_{h_5}$. In this paper four different representations of the $h_5$ Heisenberg algebra are used: by differential operators in two real (complex) variables and by finite-difference operators on uniform or exponential lattices. If $({p},{q})$ are represented by finite-difference operators on uniform or exponential lattice, the Hamiltonian and the Integral of the 3-body elliptic Calogero model become the isospectral, finite-difference operators on uniform-uniform or exponential-exponential lattices (or mixed) with polynomial coefficients. If $({p},{q})$ are written in complex $(z, \bar{z})$ variables the Hamiltonian corresponds to a complexification of the 3-body elliptic Calogero model on ${\bf C^2}$.Comment: 28 pages, considerable editing performe