𝖌𝔩(3) Polynomial Integrable System: Different Faces of the 3-Body/𝒜₂ Elliptic Calogero Model

Abstract

It is shown that the 𝖌𝔩(3) polynomial integrable system, introduced by Sokolov-Turbiner in [J. Phys. A 48 (2015), 155201, 15 pages, arXiv:1409.7439], is equivalent to the 𝖌𝔩(3) quantum Euler-Arnold top in a constant magnetic field. Their Hamiltonian and third-order integral can be rewritten in terms of the 𝖌𝔩(3) algebra generators. In turn, all these 𝖌𝔩(3) generators can be represented by the non-linear elements of the universal enveloping algebra of the 5-dimensional Heisenberg algebra 𝔥₅(𝑝^₁,₂, 𝑞^₁,₂, 𝐼); thus, the Hamiltonian and integral are two elements of the universal enveloping algebra 𝑈𝔥₅. In this paper, four different representations of the 𝔥₅ 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(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(𝖌𝔩(3)), which vanish once the representation of the 𝖌𝔩(3)-algebra generators is written in terms of the 𝔥₅(𝑝^₁,₂, 𝑞^₁,₂, 𝐼)-algebra generators. In this representation, all nine artifacts vanish, two of the above-mentioned elements of U(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 (𝑝^, 𝑞^) are written in the standard coordinate-momentum representation. If (𝑝^, 𝑞^) are represented by finite-difference/discrete operators on uniform or exponential lattices, 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, 𝓏¯) variables, the Hamiltonian corresponds to a complexification of the 3-body elliptic Calogero model on ℂ².