The argument shift method and the Gaudin model View Full Text


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Article Info

DATE

2006-07

AUTHORS

L. G. Rybnikov

ABSTRACT

We construct a family of maximal commutative subalgebras in the tensor product of n copies of the universal enveloping algebra U (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{g}$$ \end{document}) of a semisimple Lie algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{g}$$ \end{document}. This family is parameterized by finite sequences µ, z1, ..., zn, where µ ∈ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{g}$$ \end{document}* and zi ∈ 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(\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{g}$$ \end{document}) 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\mathfrak{g}$$ \end{document}-modules and the Gaudin model. More... »

PAGES

188-199

References to SciGraph publications

  • 2001-09-01. Jet schemes of locally complete intersection canonical singularities in INVENTIONES MATHEMATICAE
  • 1996-05. Bethe subalgebras in twisted Yangians in COMMUNICATIONS IN MATHEMATICAL PHYSICS
  • 1994-12. Gaudin model, Bethe Ansatz and critical level in COMMUNICATIONS IN MATHEMATICAL PHYSICS
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    URI

    http://scigraph.springernature.com/pub.10.1007/s10688-006-0030-3

    DOI

    http://dx.doi.org/10.1007/s10688-006-0030-3

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