Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

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 3-Body/${\mathcal A}_2$ Elliptic Calogero Model

Alexander V. Turbiner, Juan Carlos Lopez Vieyra and Miguel A. Guadarrama-Ayala
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México, Apartado Postal 70-543, 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 Sokolov-Turbiner in [J. Phys. A 48 (2015), 155201, 15 pages, 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. If $(\hat{p},\hat{q})$ are represented by finite-difference/discrete 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 $(\hat{p},\hat{q})$ are written in complex $(z, \bar{z})$ variables the Hamiltonian corresponds to a complexification of the 3-body elliptic Calogero model on ${\mathbb C^2}$.

Key words: elliptic Calogero model; integrable systems; 3-body systems.

pdf (502 kb)   tex (27 kb)  

References

1. Chryssomalakos C., Turbiner A.V., Canonical commutation relation preserving maps, J. Phys. A 34 (2001), 10475-10485, arXiv:math-ph/0104004.
2. 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.
3. Olshanetsky M.A., Perelomov A.M., Quantum integrable systems related to Lie algebras, Phys. Rep. 94 (1983), 313-404.
4. Oshima T., Completely integrable systems associated with classical root systems, SIGMA 3 (2007), 061, 50 pages, arXiv:math-ph/0502028.
5. Smirnov Yu.F., Turbiner A.V., Lie algebraic discretization of differential equations, Modern Phys. Lett. A 10 (1995), 1795-1802, arXiv:funct-an/9501001.
6. Sokolov V.V., Turbiner A.V., Quasi-exact-solvability 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.
7. 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.
8. Turbiner A.V., Lamé equation, $\mathfrak{sl}(2)$ algebra and isospectral deformations, J. Phys. A 22 (1989), L1-L3.
9. Turbiner A.V., Lie-algebras 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, 263-310, arXiv:funct-an/9301001.
10. 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, 407-414, arXiv:math-ph/9905006.
11. Turbiner A.V., The Heun operator as a Hamiltonian, J. Phys. A 49 (2016), 26LT01, 8 pages, arXiv:1603.02053.
12. Turbiner A.V., Miller Jr. W., Escobar-Ruiz M.A., From two-dimensional (super-integrable) quantum dynamics to (super-integrable) three-body dynamics, J. Phys. A 54 (2021), 015204, 10 pages, arXiv:1912.05726.
13. Turbiner A.V., Vasilevski N., Poly-analytic functions and representation theory, Complex Anal. Oper. Theory 15 (2021), 110, 24 pages, arXiv:2103.12771.
14. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Math. Lib., Cambridge University Press, Cambridge, 1996.