
SIGMA 20 (2024), 012, 23 pages arXiv:2305.00529
https://doi.org/10.3842/SIGMA.2024.012
Contribution to the Special Issue on Symmetry, Invariants, and their Applications in honor of Peter J. Olver
$\mathfrak{gl}(3)$ Polynomial Integrable System: Different Faces of the 3Body/${\mathcal A}_2$ Elliptic Calogero Model
Alexander V. Turbiner, Juan Carlos Lopez Vieyra and Miguel A. GuadarramaAyala
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70543, 04510 Ciudad de México, Mexico
Received July 26, 2023, in final form January 22, 2024; Published online February 03, 2024
Abstract
It is shown that the $\mathfrak{gl}(3)$ polynomial integrable system, introduced by SokolovTurbiner in [J. Phys. A 48 (2015), 155201, 15 pages, arXiv:1409.7439], is equivalent to the $\mathfrak{gl}(3)$ quantum EulerArnold top in a constant magnetic field. Their Hamiltonian as well as their thirdorder 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 nonlinear elements of the universal enveloping algebra of the 5dimensional 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 finitedifference operators on uniform or exponential lattices. We discovered the existence of two 2parametric 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 socalled 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 abovementioned 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 3body elliptic Calogero model, if $(\hat{p},\hat{q})$ are written in the standard coordinatemomentum representation. If $(\hat{p},\hat{q})$ are represented by finitedifference/discrete operators on uniform or exponential lattice, the Hamiltonian and the integral of the 3body elliptic Calogero model become the isospectral, finitedifference operators on uniformuniform or exponentialexponential lattices (or mixed) with polynomial coefficients. If $(\hat{p},\hat{q})$ are written in complex $(z, \bar{z})$ variables the Hamiltonian corresponds to a complexification of the 3body elliptic Calogero model on ${\mathbb C^2}$.
Key words: elliptic Calogero model; integrable systems; 3body systems.
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References
 Chryssomalakos C., Turbiner A.V., Canonical commutation relation preserving maps, J. Phys. A 34 (2001), 1047510485, arXiv:mathph/0104004.
 Lopez Vieyra J.C., Turbiner A.V.,Wolfes model aka $G_2/I_6$rational integrable model: $\mathfrak{g}^{(2)}$, $\mathfrak{g}^{(3)}$ hidden algebras and quartic polynomial algebra of integrals, arXiv:2310.20481.
 Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313404.
 Oshima T., Completely integrable systems associated with classical root systems, SIGMA 3 (2007), 061, 50 pages, arXiv:mathph/0502028.
 Smirnov Yu.F., Turbiner A.V., Lie algebraic discretization of differential equations, Modern Phys. Lett. A 10 (1995), 17951802, arXiv:functan/9501001.
 Sokolov V.V., Turbiner A.V., Quasiexactsolvability of the $A_2/G_2$ elliptic model: algebraic forms, $\mathfrak{sl}(3)/\mathfrak{g}^{(2)}$ hidden algebra, polynomial eigenfunctions, J. Phys. A 48 (2015), 155201, 15 pages, Corrigendum, J. Phys. A 48 (2015), 359501, 2 pages, arXiv:1409.7439.
 Tremblay F., Turbiner A.V., Winternitz P., An infinite family of solvable and integrable quantum systems on a plane, J. Phys. A 42 (2009), 242001, 10 pages, arXiv:0904.0738.
 Turbiner A.V., Lamé equation, $\mathfrak{sl}(2)$ algebra and isospectral deformations, J. Phys. A 22 (1989), L1L3.
 Turbiner A.V., Liealgebras and linear operators with invariant subspaces, in Lie Algebras, Cohomology, and New Applications to Quantum Mechanics (Springfield, MO, 1992),Contemp. Math., Vol. 160, American Mathematical Society, Providence, RI, 1994, 263310, arXiv:functan/9301001.
 Turbiner A.V., Different faces of harmonic oscillator, in SIDE III  Symmetries and Integrability of Difference Equations (Sabaudia, 1998), CRM Proc. Lecture Notes, Vol. 25, American Mathematical Society, Providence, RI, 2000, 407414, arXiv:mathph/9905006.
 Turbiner A.V., The Heun operator as a Hamiltonian, J. Phys. A 49 (2016), 26LT01, 8 pages, arXiv:1603.02053.
 Turbiner A.V., Miller Jr. W., EscobarRuiz M.A., From twodimensional (superintegrable) quantum dynamics to (superintegrable) threebody dynamics, J. Phys. A 54 (2021), 015204, 10 pages, arXiv:1912.05726.
 Turbiner A.V., Vasilevski N., Polyanalytic functions and representation theory, Complex Anal. Oper. Theory 15 (2021), 110, 24 pages, arXiv:2103.12771.
 Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1996.

