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

SIGMA 17 (2021), 069, 21 pages      arXiv:2107.08376      https://doi.org/10.3842/SIGMA.2021.069

Separation of Variables, Quasi-Trigonometric $r$-Matrices and Generalized Gaudin Models

Taras Skrypnyk
Bogolyubov Institute for Theoretical Physics, 14-b Metrolohichna Str., Kyiv, 03680, Ukraine

Received March 29, 2021, in final form July 07, 2021; Published online July 18, 2021

Abstract
We construct two new one-parametric families of separated variables for the classical Lax-integrable Hamiltonian systems governed by a one-parametric family of non-skew-symmetric, non-dynamical $\mathfrak{gl}(2)\otimes \mathfrak{gl}(2)$-valued quasi-trigonometric classical $r$-matrices. We show that for all but one classical $r$-matrices in the considered one-parametric families the corresponding curves of separation differ from the standard spectral curve of the initial Lax matrix. The proposed scheme is illustrated by an example of separation of variables for $N=2$ quasi-trigonometric Gaudin models in an external magnetic field.

Key words: integrable systems; separation of variables; classical $r$-matrices.

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References

  1. Adams M.R., Harnad J., Hurtubise J., Darboux coordinates and Liouville-Arnold integration in loop algebras, Comm. Math. Phys. 155 (1993), 385-413, arXiv:hep-th/9210089.
  2. Adams M.R., Harnad J., Hurtubise J., Darboux coordinates on coadjoint orbits of Lie algebras, Lett. Math. Phys. 40 (1997), 41-57, arXiv:solv-int/9603001.
  3. Al'ber S.I., Investigation of equations of Korteweg-de Vries type by the method of recurrence relations, J. London Math. Soc. 19 (1979), 467-480.
  4. Avan J., Talon M., Rational and trigonometric constant nonantisymmetric $R$-matrices, Phys. Lett. B 241 (1990), 77-82.
  5. Babelon O., Viallet C.-M., Hamiltonian structures and Lax equations, Phys. Lett. B 237 (1990), 411-416.
  6. Belavin A.A., Drinfel'd V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl. 16 (1982), 159-180.
  7. Cavaglià A., Gromov N., Levkovich-Maslyuk F., Separation of variables and scalar products at any rank, J. High Energy Phys. 2019 (2019), no. 9, 052, 28 pages, arXiv:1907.03788.
  8. Diener P., Dubrovin B., Algebraic-geometrical Darboux coordinates in $R$-matrix formalism, Preprint 88/94/FM, SISSA, 1994, available at http://preprints.sissa.it/xmlui/bitstream/handle/1963/3655/convert_SCAN-9409240.pdf?sequence=2&isAllowed=y.
  9. Dubrovin B., Krichever I., Novikov S., Integrable systems. I, in Dynamical Systems, IV, Encyclopaedia Math. Sci., Vol. 4, Springer, Berlin, 2001, 177-332.
  10. Dubrovin B., Skrypnyk T., Separation of variables for linear Lax algebras and classical $r$-matrices, J. Math. Phys. 59 (2018), 091405, 39 pages.
  11. Falqui G., Magri F., Pedroni M., Bihamiltonian geometry and separation of variables for Toda lattices, J. Nonlinear Math. Phys. 8 (2001), 118-127, arXiv:nlin.SI/0002008.
  12. Flaschka H., McLaughlin D.W., Canonically conjugate variables for the Korteweg-de Vries equation and the Toda lattice with periodic boundary conditions, Progr. Theoret. Phys. 55 (1976), 438-456.
  13. Freidel L., Maillet J.M., Quadratic algebras and integrable systems, Phys. Lett. B 262 (1991), 278-284.
  14. Kalnins E., Kuznetsov V., Miller Jr. W., Separation of variables and the XXZ Gaudin magnet, arXiv:hep-th/9412190.
  15. Maillet J.M., Niccoli G., On quantum separation of variables, J. Math. Phys. 59 (2018), 091417, 47 pages, arXiv:1807.11572.
  16. Ryan P., Volin D., Separated variables and wave functions for rational $\mathfrak{gl}(N)$ spin chains in the companion twist frame, J. Math. Phys. 60 (2019), 032701, 23 pages, arXiv:1810.10996.
  17. Sklyanin E.K., On complete integrability of the Landau-Lifshitz equation, Preprint LOMI, E-3-79, 1979.
  18. Sklyanin E.K., Separation of variables in the Gaudin model, J. Sov. Math. 47 (1989), 2473-2488.
  19. Sklyanin E.K., Separation of variables - new trends, Progr. Theoret. Phys. Suppl. 118 (1995), 35-60, arXiv:solv-int/9504001.
  20. Skrypnik T., Quasigraded Lie algebras, the Kostant-Adler scheme, and integrable hierarchies, Theoret. and Math. Phys. 142 (2005), 329-345.
  21. Skrypnyk T., Integrable quantum spin chains, non-skew symmetric $r$-matrices and quasigraded Lie algebras, J. Geom. Phys. 57 (2006), 53-67.
  22. Skrypnyk T., Generalized Gaudin systems in a magnetic field and non-skew-symmetric $r$-matrices, J. Phys. A: Math. Theor. 40 (2007), 13337-13352.
  23. Skrypnyk T., Lie algebras with triangular decompositions, non-skew-symmetric classical $r$-matrices and Gaudin-type integrable systems, J. Geom. Phys. 60 (2010), 491-500.
  24. Skrypnyk T., Infinite-dimensional Lie algebras, classical $r$-matrices, and Lax operators: two approaches, J. Math. Phys. 54 (2013), 103507, 32 pages.
  25. Skrypnyk T., ''Many-poled'' $r$-matrix Lie algebras, Lax operators, and integrable systems, J. Math. Phys. 55 (2014), 083507, 19 pages.
  26. Skrypnyk T., Reductions in finite-dimensional integrable systems and special points of classical $r$-matrices, J. Math. Phys. 57 (2016), 123504, 38 pages.
  27. Skrypnyk T., Classical $r$-matrices, ''elliptic'' BCS and Gaudin-type hamiltonians and spectral problem, Nuclear Phys. B 941 (2019), 225-248.
  28. Skrypnyk T., Separation of variables, Lax-integrable systems and $\mathfrak{gl}(2)\otimes \mathfrak{gl}(2)$-valued classical $r$-matrices, J. Geom. Phys. 155 (2020), 103733, 17 pages.
  29. Skrypnyk T., Symmetric separation of variables for trigonometric integrable models, Nuclear Phys. B 957 (2020), 115101, 16 pages.
  30. Veselov A.P., Novikov S.P., On Poisson brackets compatible with algebraic geometry and Korteweg-de Vries dynamics on the set of finite-zone potentials, Sov. Math. Dokl. 26 (1982), 357-362.

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