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

SIGMA 20 (2024), 087, 26 pages      arXiv:2404.13688      https://doi.org/10.3842/SIGMA.2024.087

Algebraic Complete Integrability of the $a_4^{(2)}$ Toda Lattice

Bruce Lionnel Lietap Ndi ^a, Djagwa Dehainsala ^b and Joseph Dongho ^a
a) University of Maroua, Faculty of Sciences, Department of Mathematics Computer Sciences, P.O. Box 814, Maroua, Cameroon
^b) Department of Mathematics, Faculty of Exact and Applied Sciences, University of NDjamena, 1 route de Farcha, P.O. Box 1027, NDjamena, Chad

Received April 25, 2024, in final form September 25, 2024; Published online October 05, 2024

Abstract
The aim of this work is focused on the investigation of the algebraic complete integrability of the Toda lattice associated with the twisted affine Lie algebra $a_4^{(2)}$. First, we prove that the generic fiber of the momentum map for this system is an affine part of an abelian surface. Second, we show that the flows of integrable vector fields on this surface are linear. Finally, using the formal Laurent solutions of the system, we provide a detailed geometric description of these abelian surfaces and the divisor at infinity.

Key words: Toda lattice; integrable system; algebraic integrability; abelian surface.

pdf (493 kb)   tex (29 kb)  

References

1. Adler M., van Moerbeke P., Linearization of Hamiltonian systems, Jacobi varieties and representation theory, Adv. Math. 38 (1980), 318-379.
2. Adler M., van Moerbeke P., The complex geometry of the Kowalewski-Painlevé analysis, Invent. Math. 97 (1989), 3-51.
3. Adler M., van Moerbeke P., The Toda lattice, Dynkin diagrams, singularities and abelian varieties, Invent. Math. 103 (1991), 223-278.
4. Adler M., van Moerbeke P., Vanhaecke P., Algebraic integrability, Painlevé geometry and Lie algebras, Ergeb. Math. Grenzgeb. (3), Vol. 47, Springer, Berlin, 2004.
5. Bogoyavlensky O.I., On perturbations of the periodic Toda lattice, Comm. Math. Phys. 51 (1976), 201-209.
6. Dehainsala D., Sur l'intégrabilité algébrique des réseaux de Toda: cas particuliers des réseaux $c_2^{(1)}$ et $d_3^{(2)}$, Ph.D. Thesis, Université de Poitiers, 2008.
7. Dehainsala D., Algebraic complete integrability of the $\mathfrak c_2^{(1)}$ Toda lattice, J. Geom. Phys. 135 (2019), 80-97.
8. Flaschka H., The Toda lattice. II. Existence of integrals, Phys. Rev. B 9 (1974), 1924-1925.
9. Haine L., Geodesic flow on ${\rm SO}(4)$ and abelian surfaces, Math. Ann. 263 (1983), 435-472.
10. Hénon M., Integrals of the Toda lattice, Phys. Rev. B 9 (1974), 1921-1923.
11. Kowalevski S., Sur le problème de la rotation d'un corps solide autour d'un point fixe, in The Kowalevski Property (Leeds, 2000), CRM Proc. Lecture Notes, Vol. 32, American Mathematical Society, Providence, RI, 2002, 315-372.
12. Toda M., One-dimensional dual transformation, J. Phys. Soc. Japan 20 (1965), 2095A.
13. Toda M., One-dimensional dual transformation, Prog. Theor. Phys. Suppl. 36 (1966), 113-119.