$\mathfrak{gl}(3)$ Polynomial Integrable System: Different Faces of the 3-Body/${\mathcal A}_2$ Elliptic Calogero Model

It is shown that the $\mathfrak{gl}(3)$ polynomial integrable system, introduced by Sokolov-Turbiner in [arXiv:1409.7439], is equivalent to the $\mathfrak{gl}(3)$ quantum Euler-Arnold top in a constant magnetic field. Their Hamiltonian as well as their third-order integral can be rewritten in terms of $\mathfrak{gl}(3)$ algebra generators. In turn, all these $\mathfrak{gl}(3)$ generators can be represented by the non-linear elements of the universal enveloping algebra of the 5-dimensional Heisenberg algebra $\mathfrak{h}_5(\hat{p}_{1,2},\hat{q}_{1,2}, I)$, thus, the Hamiltonian and integral are two elements of the universal enveloping algebra $U_{\mathfrak{h}_5}$. In this paper, four different representations of the $\mathfrak{h}_5$ Heisenberg algebra are used: (I) by differential operators in two real (complex) variables, (II) by finite-difference operators on uniform or exponential lattices. We discovered the existence of two 2-parametric bilinear and trilinear elements (denoted $H$ and $I$, respectively) of the universal enveloping algebra $U(\mathfrak{gl}(3))$ such that their Lie bracket (commutator) can be written as a linear superposition of nine so-called artifacts - the special bilinear elements of $U(\mathfrak{gl}(3))$, which vanish once the representation of the $\mathfrak{gl}(3)$-algebra generators is written in terms of the $\mathfrak{h}_5(\hat{p}_{1,2},\hat{q}_{1,2},I)$-algebra generators. In this representation all nine artifacts vanish, two of the above-mentioned elements of $U(\mathfrak{gl}(3))$ (called the Hamiltonian $H$ and the integral $I$) commute(!); in particular, they become the Hamiltonian and the integral of the 3-body elliptic Calogero model, if $(\hat{p},\hat{q})$ are written in the standard coordinate-momentum representation.