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A bi-Hamiltonian nature of the Gaudin algebras

Jan 1, 2023
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  • Adv.Math. 412 (2023) 108805
  • Published: Jan 1, 2023
DOI:
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Abstract: (Elsevier Inc.)
Let q be a Lie algebra over a field k and p,p˜∈k[t] two different normalised polynomials of degree n⩾2. As vector spaces, both quotient Lie algebras q[t]/(p) and q[t]/(p˜) can be identified with W=q⋅1⊕qt¯⊕…⊕qt¯n−1. If deg⁡(p−p˜)⩽1, then the Lie brackets [,]p, [,]p˜ induced on W by p and p˜, respectively, are compatible. Making use of the Lenard–Magri scheme, we construct a subalgebra Z=Z(p,p˜)⊂S(W)q⋅1 such that {Z,Z}p={Z,Z}p˜=0. If tr.degS(q)q=indq and q has the codim–2 property, then tr.degZ takes the maximal possible value, which is n−12dim⁡q+n+12indq. If q=g is semisimple, then Z contains the Hamiltonians of a suitably chosen Gaudin model. Furthermore, if p and p˜ do not have common roots, then there is a Gaudin subalgebra C⊂U(g⊕n) such that Z=gr(C), up to a certain identification. In a non-reductive case, we obtain a completely integrable generalisation of Gaudin models. For a wide class of Lie algebras, which extends the reductive setting, Z(p,p+t) coincides with the image of the Poisson-commutative algebra Z(qˆ,t)=S(tq[t])q[t−1] under the quotient map ψp:S(q[t])→S(W), providing p(0)≠0.
  • 17B63
  • 17B65
  • 17B80
  • Integrable systems
  • Takiff Lie algebras
  • Symmetric invariants
  • [1]
    Труды Моск. Матем. Общества 78 (2) (2017) 261–281 =
    • T. Arakawa
      ,
    • A. Premet
      • Trans. Mosc. Math. Soc. 78 (2017) 217-234
  • [2]
    • A. Bolsinov
      • Acta Appl.Math. 24 (1991) 3, 253-274
  • [4]
    • J.-P. Dufour
      ,
    • N.T. Zung
      • Prog.Math. vol. 242 (2005)
  • [5]
    • B. Feigin
      ,
    • E. Frenkel
      • Int.J.Mod.Phys.A 7 (1992) Suppl. 1A, 197-215
  • [7]
    • I.M. Gelfand
      ,
    • I.S. Zakharevich
      • Selecta Math. 6 (2000) 131-183
  • [8]
    • F. Geoffriau
      • J.Algebra 171 (1995) 444-456
  • [9]
    • J. Greenstein
      ,
    • V. Mazorchuk
      • Algebr.Represent.Theory 20 (2017) 3, 675-694
  • [10]
    • A. Ilin
      ,
    • L. Rybnikov
  • [11]
    • B. Kostant
      • Am.J.Math. 85 (1963) 327-404
  • [12]
    • A. Molev
      ,
    • O. Yakimova
      • Represent.Theory 23 (2019) 350-378
  • [13]
    • D. Panyushev
      • Adv.Math. 213 (2007) 1, 380-404
  • [14]
    • D. Panyushev
      ,
    • A. Premet
      ,
    • O. Yakimova
      • J.Algebra 313 (2007) 343-391
  • [15]
    • D.I. Panyushev
      ,
    • O.S. Yakimova
      • Transform.Groups 25 (2020) 609-624
  • [16]
    • D.I. Panyushev
      ,
    • O.S. Yakimova
      • Math.Z. 295 (2020) 101-127
  • [17]
    • D. Panyushev
      ,
    • O. Yakimova
      • Int.Math.Res.Not. 2021 (2021) 23, 18367-18406
  • [18]
    • D. Panyushev
      ,
    • O. Yakimova
  • [19]
    • M. Raïs
      ,
    • P. Tauvel
      • J.Reine Angew.Math. 425 (1992) 123-140
  • [20]
    • L.G. Rybnikov
      • Rept.Math.Phys. 61 (2008) 247-252
  • [21]
    • T.A. Springer
  • [22]
    • S.J. Takiff
      • Trans.Am.Math.Soc. 160 (1971) 249-262
  • [23]
    Э.Б. Винберг, О некоторых коммутативных подалгебрах универсальной обертывающей алгебры, Изв. АН СССР. Сер. Матем. 54 (1) (1990) 3–25 (in Russian); English translation:
    • E.B. Vinberg
      • Math.USSR Izv. 36 (1991) 1-22
  • [24]
    Э.Б. Винберг, В.В. Горбацевич, А.Л. Онищик, Группы и алгебры Ли - 3, Соврем. пробл. математики. Фундам. направл., т. 41, ВИНИТИ, Москва, 1990 (in Russian); English translation:
    • V.V. Gorbatsevich
      ,
    • A.L. Onishchik
      ,
    • E.B. Vinberg
      • Encyclopaedia Math. Sci. vol. 41 (1994)
  • [25]
    • B.J. Wilson
      • J.Algebra 336 (2011) 1-27
  • 1-25 of 28
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