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

SIGMA 21 (2025), 105, 32 pages      arXiv:2211.00451      https://doi.org/10.3842/SIGMA.2025.105

Quantum Groups, Discrete Magnus Expansion, Pre-Lie and Tridendriform Algebras

Anastasia Doikou ab
a) Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, UK
b) Maxwell Institute for Mathematical Sciences, Edinburgh EH8 9BT, UK

Received July 09, 2024, in final form December 01, 2025; Published online December 11, 2025

Abstract
We review the discrete evolution problem and the corresponding solution as a discrete Dyson series in order to rigorously derive a generalized discrete version of the Magnus expansion. We also systematically derive the discrete analogue of the pre-Lie Magnus expansion and express the elements of the discrete Dyson series in terms of a tridendriform algebra binary operation. In the generic discrete case, extra significant terms that are absent in the continuous or the linear discrete case appear in both Dyson and Magnus expansions. Based on the rigorous discrete derivation key links between quantum algebras, tridendriform and pre-Lie algebras are then established. This is achieved by examining tensor realizations of quantum groups, such as the Yangian. We show that these realizations can be expressed in terms of tridendriform and pre-Lie algebras. The continuous limit as expected provides the corresponding non-local charges of the Yangian as members of the pre-Lie Magnus expansion.

Key words: Yangians; integrability; discrete evolution problem; Magnus expansion; pre-Lie algebras; (tri)dendriform algebras.

pdf (624 kb)   tex (45 kb)  

References

  1. Ablowitz M.J., Kaup D.J., Newell A.C., Segur H., Nonlinear-evolution equations of physical significance, Phys. Rev. Lett. 31 (1973), 125-127.
  2. Agrachev A.A., Gamkrelidze R.V., Chronological algebras and non-stationary vector fields, J. Sov. Math. 17 (1981), 1650-1675.
  3. Aguiar M., Infinitesimal Hopf algebras, in New Trends in Hopf Algebra Theory (La Falda, 1999), Contemp. Math., Vol. 267, American Mathematical Society, Providence, RI, 2000, 1-29.
  4. Aguiar M., Infinitesimal bialgebras, pre-Lie and dendriform algebras, in Hopf Algebras, Lecture Notes in Pure and Appl. Math., Vol. 237, Dekker, New York, 2004, 1-33, arXiv:math.QA/0211074.
  5. Avan J., Doikou A., Sfetsos K., Systematic classical continuum limits of integrable spin chains and emerging novel dualities, Nuclear Phys. B 840 (2010), 469-490, arXiv:1005.4605.
  6. Bai C., Introduction to pre-Lie algebras, in Algebra and Applications 1: Non-Associative Algebras and Categories, John Wiley & Sons, 2021, 245-273.
  7. Bai C., Guo L., Sheng Y., Tang R., Post-groups, (Lie-)Butcher groups and the Yang-Baxter equation, Math. Ann. 388 (2024), 3127-3167, arXiv:2306.11196.
  8. Bandiera R., Schaetz F., Eulerian idempotent, pre-Lie logarithm and combinatorics of trees, arXiv:1702.08907.
  9. Baxter G., An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math. 10 (1960), 731-742.
  10. Baxter R.J., Exactly solved models in statistical mechanics, Academic Press, London, 1982.
  11. Blanes S., Casas F., Oteo J.A., Ros J., The Magnus expansion and some of its applications, Phys. Rep. 470 (2009), 151-238, arXiv:0810.5488.
  12. Burum D.P., Magnus expansion generator, Phys. Rev. B 24 (1981), 3684-3692.
  13. Cedó F., Jespers E., Okniński J., Braces and the Yang-Baxter equation, Comm. Math. Phys. 327 (2014), 101-116, arXiv:1205.3587.
  14. Chapoton F., A rooted-trees $q$-series lifting a one-parameter family of Lie idempotents, Algebra Number Theory 3 (2009), 611-636, arXiv:0807.1830.
  15. Chapoton F., Livernet M., Pre-Lie algebras and the rooted trees operad, Int. Math. Res. Not. 2001 (2001), 395-408, arXiv:math.QA/0002069.
  16. Chari V., Pressley A., A guide to quantum groups, Cambridge University Press, Cambridge, 1995.
  17. Cherednik I.V., Factorizing particles on a half line, and root systems, Theoret. and Math. Phys. 61 (1984), 977-983.
  18. Doikou A., Set-theoretic Yang-Baxter equation, braces and Drinfeld twists, J. Phys. A 54 (2021), 415201, 21 pages, arXiv:2102.13591.
  19. Doikou A., Yangians as pre-Lie and tridendriform algebras, in Geometric Methods in Physics XL, Trends Math., Birkhäuser, Cham, 2024, 233-250.
  20. Doikou A., Ghionis A., Vlaar B., Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation, Lett. Math. Phys. 112 (2022), 78, 29 pages, arXiv:2203.03400.
  21. Doikou A., Smoktunowicz A., Set-theoretic Yang-Baxter & reflection equations and quantum group symmetries, Lett. Math. Phys. 111 (2021), 105, 40 pages, arXiv:2003.08317.
  22. Doikou A., Smoktunowicz A., From braces to Hecke algebras and quantum groups, J. Algebra Appl. 22 (2023), 2350179, 28 pages, arXiv:1912.03091.
  23. Drinfeld V.G., Hopf algebras and the quantum Yang-Baxter equation, Sov. Dokl. Math. 32 (1985), 256-258.
  24. Drinfeld V.G., Quantum groups, in Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986), American Mathematical Society, Providence, RI, 1987, 798-820.
  25. Drinfeld V.G., A new realization of Yangians and of quantum affine algebras, Sov. Dokl. Math. 36 (1988), 212-216.
  26. Drinfeld V.G., On some unsolved problems in quantum group theory, in Quantum Groups (Leningrad, 1990), Lecture Notes in Math., Vol. 1510, Springer, Berlin, 1992, 1-8.
  27. Drinfeld V.G., Sokolov V.V., Equations of Korteweg-de Vries type, and simple Lie algebras, Sov. Math. Dokl. 23 (1981), 457-462.
  28. Dyson F.J., The radiation theories of Tomonaga, Schwinger, and Feynman, Phys. Rev. 75 (1949), 486-502.
  29. Ebrahimi-Fard K., Loday-type algebras and the Rota-Baxter relation, Lett. Math. Phys. 61 (2002), 139-147, arXiv:math-ph/0207043.
  30. Ebrahimi-Fard K., Guo L., Rota-Baxter algebras and dendriform algebras, J. Pure Appl. Algebra 212 (2008), 320-339, arXiv:math.RA/0503647.
  31. Ebrahimi-Fard K., Lundervold A., Munthe-Kaas H.Z., On the Lie enveloping algebra of a post-Lie algebra, J. Lie Theory 25 (2015), 1139-1165, arXiv:1410.6350.
  32. Ebrahimi-Fard K., Malham S.J.A., Patras F., Wiese A., Flows and stochastic Taylor series in Itô calculus, J. Phys. A 48 (2015), 495202, 17 pages, arXiv:1504.07226.
  33. Ebrahimi-Fard K., Manchon D., A Magnus- and Fer-type formula in dendriform algebras, Found. Comput. Math. 9 (2009), 295-316, arXiv:0707.0607.
  34. Ebrahimi-Fard K., Manchon D., The tridendriform structure of a discrete magnus expansion, Discrete Contin. Dyn. Syst. 34 (2014), 1021-1040, arXiv:1306.6439.
  35. Ebrahimi-Fard K., Mencattini I., Munthe-Kaas H., Post-Lie algebras and factorization theorems, J. Geom. Phys. 119 (2017), 19-33, arXiv:1701.07786.
  36. Ebrahimi-Fard K., Mencattini I., Quesney A., What is the Magnus expansion?, J. Comput. Dyn. 12 (2025), 115-159, arXiv:2312.16674.
  37. Ebrahimi-Fard K., Patras F., The pre-Lie structure of the time-ordered exponential, Lett. Math. Phys. 104 (2014), 1281-1302, arXiv:1305.3856.
  38. Ebrahimi-Fard K., Patras F., From iterated integrals and chronological calculus to Hopf and Rota-Baxter algebras, in Algebra and Applications 2: Combinatorial Algebra and Hopf Algebras, ISTE, London, 2021, 55-118, arXiv:1911.08766.
  39. Faddeev L.D., Takhtajan L.A., Hamiltonian methods in the theory of solitons, Springer Ser. Sov. Math., Springer, Berlin, 1987.
  40. Gerstenhaber M., The cohomology structure of an associative ring, Ann. of Math. 78 (1963), 267-288.
  41. Golubchik I.Z., Sokolov V.V., Generalized operator Yang-Baxter equations, integrable ODEs and nonassociative algebras, J. Nonlinear Math. Phys. 7 (2000), 184-197, arXiv:nlin.SI/0003034.
  42. Guarnieri L., Vendramin L., Skew braces and the Yang-Baxter equation, Math. Comp. 86 (2017), 2519-2534, arXiv:1511.03171.
  43. Hudson R., Sticky shuffle product Hopf algebras and their stochastic representations, in New Trends in Stochastic Analysis and Related Topics, Interdiscip. Math. Sci., Vol. 12, World Scientific Publishing, Hackensack, NJ, 2012, 165-181.
  44. Jimbo M., A $q$-difference analogue of $U({\mathfrak g})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69.
  45. Lax P.D., Integrals of nonlinear equations of evolution and solitary waves, Comm. Pure Appl. Math. 21 (1968), 467-490.
  46. Loday J.-L., Dialgebras, in Dialgebras and Related Operads, Lecture Notes in Math., Vol. 1763, Springer, Berlin, 2001, 7-66.
  47. Loday J.-L., On the algebra of quasi-shuffles, Manuscripta Math. 123 (2007), 79-93, arXiv:math.QA/0506498.
  48. Loday J.-L., Ronco M., Trialgebras and families of polytopes, in Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic $K$-theory, Contemp. Math., Vol. 346, American Mathematical Society, Providence, RI, 2004, 369-398, arXiv:math.AT/0205043.
  49. MacKay N.J., Lattice quantization of Yangian charges, Phys. Lett. B 349 (1995), 94-98, arXiv:hep-th/9501079.
  50. Magnus W., On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math. 7 (1954), 649-673.
  51. Majid S., Foundations of quantum group theory, Cambridge University Press, Cambridge, 1995.
  52. Manchon D., A short survey on pre-Lie algebras, in Noncommutative Geometry and Physics: Renormalisation, Motives, Index Theory, ESI Lect. Math. Phys., European Mathematical Society, Zürich, 2011, 89-102.
  53. Matveev V.B., Salle M.A., Darboux transformations and solitons, Springer Ser. Nonlinear Dynam., Springer, Berlin, 1991.
  54. Molev A., Nazarov M., Olshanski G., Yangians and classical Lie algebras, Russian Math. Surveys 51 (1996), 205-282, arXiv:hep-th/9409025.
  55. Pei J., Bai C., Guo L., Rota-Baxter operators on ${\rm sl}(2,{\mathbb C})$ and solutions of the classical Yang-Baxter equation, J. Math. Phys. 55 (2014), 021701, 17 pages, arXiv:1311.0612.
  56. Reshetikhin N.Yu., Takhtajan L.A., Faddeev L.D., Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225.
  57. Rota G.-C., Baxter algebras and combinatorial identities. I, Bull. Amer. Math. Soc. 75 (1969), 325-329.
  58. Rota G.-C., Baxter algebras and combinatorial identities. II, Bull. Amer. Math. Soc. 75 (1969), 330-334.
  59. Rump W., Braces, radical rings, and the quantum Yang-Baxter equation, J. Algebra 307 (2007), 153-170.
  60. Rump W., The brace of a classical group, Note Mat. 34 (2014), 115-144.
  61. Salzman W.R., New criterion for convergence of exponential perturbation theory in the Schrödinger representation, Phys. Rev. A 36 (1987), 5074-5076.
  62. Semenov-Tyan-Shanskii M.A., What is a classical $r$-matrix?, Funct. Anal. Appl. 17 (1983), 259-272.
  63. Sklyanin E.K., Method of the inverse scattering problem and the nonlinear quantum Schrödinger equation, Sov. Phys. Dokl. 24 (1979), 107-109.
  64. Sklyanin E.K., On complete integrability of the Landau-Lifshitz equation, Preprint E-3-79, Leningrad Branch of Steklov Mathematical Institute, 1979.
  65. Sklyanin E.K., Boundary conditions for integrable equations, Funct. Anal. Appl. 21 (1987), 164-166.
  66. Sklyanin E.K., Boundary conditions for integrable quantum systems, J. Phys. A 21 (1988), 2375-2389.
  67. Smoktunowicz A., A new formula for Lazard's correspondence for finite braces and pre-Lie algebras, J. Algebra 594 (2022), 202-229, arXiv:2011.07611.
  68. Smoktunowicz A., Algebraic approach to Rump's results on relations between braces and pre-Lie algebras, J. Algebra Appl. 21 (2022), 2250054, 13 pages, arXiv:2007.09403.
  69. Vinberg E.B., The theory of homogeneous convex cones, Trans. Mosc. Math. Soc. 12 (1963), 340-403.
  70. Yang C.N., Some exact results for the many-body problem in one dimension with repulsive delta-function interaction, Phys. Rev. Lett. 19 (1967), 1312-1315.
  71. Zakharov V.E., Shabat A.B., Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media, J. Exp. Theor. Phys. 34 (1970), 62-69.
  72. Zhang Y., Bai C., Guo L., Totally compatible associative and Lie dialgebras, tridendriform algebras and PostLie algebras, Sci. China Math. 57 (2014), 259-273, arXiv:1510.03698.

Previous article  Next article  Contents of Volume 21 (2025)