$\mathfrak{gl}(3)$ Polynomial Integrable System: Different Faces of the 3-Body/${\mathcal A}_2$ Elliptic Calogero Model
Alexander V. Turbiner, Juan Carlos Lopez Vieyra, Miguel Ayala
TL;DR
The paper builds a $\frak{gl}(3)$ polynomial integrable system associated with the ${\cal A}_2$ elliptic (3‑body) Calogero model by realizing the Hamiltonian and a cubic integral as elements of $U_{\frak{gl}(3)}$ and, crucially, as nonlinear elements of the 5‑dimensional Heisenberg algebra $\frak{h}_5$ inside $U_{\frak{h}_5}$. This reveals a commuting pair $h$ and $k$ in the $\frak{gl}(3)$ algebra that vanish their commutator in the $U_{\frak{h}_5}$ realization, thereby reproducing the integrable top in various representations; the authors construct four distinct families of isospectral polynomial systems on different two‑dimensional spaces: differential, uniform and exponential lattices, and complex space $\mathbb{C}^2$. A detailed theory of “artifacts” $A_1,...,A_9$ shows they vanish in the $U_{\frak{h}_5}$ realization, while the explicit commutator in the abstract algebra decomposes into these artifacts; the work extends to $G_2$ elliptic 3‑body dynamics, where a higher‑degree commuting operator exists and yields finite‑dimensional invariant subspaces for certain $\nu$. Overall, the paper unifies continuous and discrete realizations of an integrable system through a unifying $U_{\frak{h}_5}$ / $\frak{gl}(3)$ framework, highlighting hidden algebra structures and offering new exactly solvable perspectives for two‑dimensional Calogero‑type models.
Abstract
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.