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Funktsional. Anal. i Prilozhen., 2006, Volume 40, Issue 3, Pages 30–43 (Mi faa741)  

This article is cited in 30 scientific papers (total in 30 papers)

The Argument Shift Method and the Gaudin Model

L. G. Rybnikovab

a Independent University of Moscow
b M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics

Abstract: We construct a family of maximal commutative subalgebras in the tensor product of $n$ copies of the universal enveloping algebra $U(\mathfrak{g})$ of a semisimple Lie algebra $\mathfrak{g}$. This family is parameterized by finite sequences $\mu$, $z_1,…,z_n$, where $\mu\in\mathfrak{g}^*$ and $z_i\in\mathbb{C}$. The construction presented here generalizes the famous construction of the higher Gaudin Hamiltonians due to Feigin, Frenkel, and Reshetikhin. For $n=1$, the corresponding commutative subalgebras in the Poisson algebra $S(\mathfrak{g})$ were obtained by Mishchenko and Fomenko with the help of the argument shift method. For commutative algebras of our family, we establish a connection between their representations in the tensor products of finite-dimensional $\mathfrak{g}$-modules and the Gaudin model.

Keywords: Gaudin model, argument shift method, Mishchenko–Fomenko subalgebra, affine Kac–Moody algebra, critical level

DOI: https://doi.org/10.4213/faa741

Full text: PDF file (267 kB)
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English version:
Functional Analysis and Its Applications, 2006, 40:3, 188–199

Bibliographic databases:

UDC: 512.813.4
Received: 09.04.2005

Citation: L. G. Rybnikov, “The Argument Shift Method and the Gaudin Model”, Funktsional. Anal. i Prilozhen., 40:3 (2006), 30–43; Funct. Anal. Appl., 40:3 (2006), 188–199

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Skrypnyk T., “Generalized Gaudin systems in a magnetic field and non-skew-symmetric $r$-matrices”, J. Phys. A, 40:44 (2007), 13337–13352  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    2. Panyushev D.I., Yakimova O.S., “The argument shift method and maximal commutative subalgebras of Poisson algebras”, Math. Res. Lett., 15:2-3 (2008), 239–249  crossref  mathscinet  zmath  isi  elib  scopus
    3. Chervov A., Falqui G., “Manin matrices and Talalaev's formula”, J. Phys. A, 41:19 (2008), 194006, 28 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    4. Feigin B., Frenkel E., Rybnikov L., “On the endomorphisms of Weyl modules over affine Kac-Moody algebras at the critical level”, Lett. Math. Phys., 88:1-3 (2009), 163–173  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    5. Alexander Chervov, Gregorio Falqui, Leonid Rybnikov, “Limits of Gaudin Systems: Classical and Quantum Cases”, SIGMA, 5 (2009), 029, 17 pp.  mathnet  crossref  mathscinet  zmath
    6. Chervov A., Falqui G., Rybnikov L., “Limits of Gaudin algebras, quantization of bending flows, Jucys-Murphy elements and Gelfand-Tsetlin bases”, Lett. Math. Phys., 91:2 (2010), 129–150  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    7. Feigin B., Frenkel E., Toledano Laredo V., “Gaudin models with irregular singularities”, Adv. Math., 223:3 (2010), 873–948  crossref  mathscinet  zmath  isi  elib  scopus
    8. Feigin B., Frenkel E., Rybnikov L., “Opers with irregular singularity and spectra of the shift of argument subalgebra”, Duke Math. J., 155:2 (2010), 337–363  crossref  mathscinet  zmath  isi  elib  scopus
    9. Skrypnyk T., “Generalized Gaudin systems in an external magnetic field and reflection equation algebras”, J. Stat. Mech. Theory Exp., 2010, no. 6, P06028, 12 pp.  crossref  mathscinet  isi  elib  scopus
    10. Feigin B., Finkelberg M., Frenkel I., Rybnikov L., “Gelfand-Tsetlin algebras and cohomology rings of Laumon spaces”, Selecta Math. (N.S.), 17:2 (2011), 337–361  crossref  mathscinet  zmath  isi  scopus
    11. Feigin B. Frenkel E., “Quantization of Soliton Systems and Langlands Duality”, Exploring New Structures and Natural Constructions in Mathematical Physics, Advanced Studies in Pure Mathematics, 61, ed. Hasegawa K. Hayashi T. Hosono S. Yamada Y., Math Soc Japan, 2011, 185–274  mathscinet  zmath  isi
    12. Gaiotto D., Witten E., “Knot Invariants From Four-Dimensional Gauge Theory”, Adv. Theor. Math. Phys., 16:3 (2012), 935–1086  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    13. Molev A.I., “Feigin-Frenkel Center in Types B, C and D”, Invent. Math., 191:1 (2013), 1–34  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. Panyushev D.I., Yakimova O.S., “On Maximal Commutative Subalgebras of Poisson Algebras Associated With Involutions of Semisimple Lie Algebras”, Bull. Sci. Math., 138:6 (2014), 705–720  crossref  mathscinet  zmath  isi  scopus
    15. Molev A.I. Mukhin E.E., “Invariants of the Vacuum Module Associated With the Lie Superalgebra Gl(1 Vertical Bar 1)”, J. Phys. A-Math. Theor., 48:31, SI (2015), 314001  crossref  mathscinet  zmath  isi  elib  scopus
    16. Futorny V., Molev A., “Quantization of the shift of argument subalgebras in type A”, Adv. Math., 285 (2015), 1358–1375  crossref  mathscinet  zmath  isi  elib  scopus
    17. B. L. Feigin, “Integrable systems, shuffle algebras, and Bethe equations”, Trans. Moscow Math. Soc., 77 (2016), 203–246  mathnet  crossref  elib
    18. Frappat L., Jing N., Molev A., Ragoucy E., “Higher Sugawara Operators for the Quantum Affine Algebras of Type A”, Commun. Math. Phys., 345:2 (2016), 631–657  crossref  mathscinet  zmath  isi  elib  scopus
    19. Molev A.I., Ragoucy E., Rozhkovskaya N., “Segal–Sugawara vectors for the Lie algebra of type G 2”, J. Algebra, 455 (2016), 386–401  crossref  mathscinet  zmath  isi  elib  scopus
    20. Bolsinov A.V. Izosimov A.M. Tsonev D.M., “Finite-dimensional integrable systems: A collection of research problems”, J. Geom. Phys., 115 (2017), 2–15  crossref  mathscinet  zmath  isi  scopus
    21. A. I. Molev, E. E. Mukhin, “Eigenvalues of Bethe vectors in the Gaudin model”, Theoret. and Math. Phys., 192:3 (2017), 1258–1281  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    22. Vicedo B., Young Ch., “Cyclotomic Gaudin Models With Irregular Singularities”, J. Geom. Phys., 121 (2017), 247–278  crossref  mathscinet  zmath  isi  scopus
    23. Trans. Moscow Math. Soc., 78 (2017), 217–234  mathnet  crossref  elib
    24. Moreau A., “Centralizers of Nilpotent Elements and Related Problems, a Survey”, Perspectives in Lie Theory, Springer Indam Series, 19, eds. Callegaro F., Carnovale G., Caselli F., DeConcini C., DeSole A., Springer International Publishing Ag, 2017, 331–346  crossref  mathscinet  zmath  isi  scopus
    25. Jing N., Kozic S., Molev A., Yang F., “Center of the Quantum Affine Vertex Algebra in Type a”, J. Algebra, 496 (2018), 138–186  crossref  mathscinet  zmath  isi  scopus
    26. Ilin A., Rybnikov L., “Degeneration of Bethe Subalgebras in the Yangian of Gl(N)”, Lett. Math. Phys., 108:4 (2018), 1083–1107  crossref  mathscinet  zmath  isi  scopus
    27. Moreau A., “A Remark on Mishchenko-Fomenko Algebras and Regular Sequences”, Sel. Math.-New Ser., 24:3 (2018), 2651–2657  crossref  mathscinet  zmath  isi  scopus
    28. Benoît Vicedo, Charles Young, “$({\mathfrak{gl}}_M, {\mathfrak{gl}}_N)$-Dualities in Gaudin Models with Irregular Singularities”, SIGMA, 14 (2018), 040, 28 pp.  mathnet  crossref
    29. Bolsinov A. Matveev V.S. Miranda E. Tabachnikov S., “Open Problems, Questions and Challenges in Finite-Dimensional Integrable Systems”, Philos. Trans. R. Soc. A-Math. Phys. Eng. Sci., 376:2131 (2018), 20170430  crossref  isi  scopus
    30. Lacroix S., Vicedo B., Young Ch., “Affine Gaudin Models and Hypergeometric Functions on Affine Opers”, Adv. Math., 350 (2019), 486–546  crossref  isi
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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