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Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case

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Abstract

Solvability of the rational quantum integrable systems related to exceptional root spaces G2,F4 is re-examined and for E6,7,8 is established in the framework of a unified approach. It is shown that Hamiltonians take algebraic form being written in certain Weyl-invariant variables. It is demonstrated that for each Hamiltonian the finite-dimensional invariant subspaces are made from polynomials and they form an infinite flag. A notion of minimal flag is introduced and minimal flag for each Hamiltonian is found. Corresponding eigenvalues are calculated explicitly while the eigenfunctions can be computed by pure linear algebra means for arbitrary values of the coupling constants. The Hamiltonian of each model can be expressed in the algebraic form as a second degree polynomial in the generators of some infinite-dimensional but finitely-generated Lie algebra of differential operators, taken in a finite-dimensional representation.

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Authors and Affiliations

  1. Institute for Theoretical and Experimental Physics, Moscow, 117259, Russia

    Konstantin G. Boreskov

  2. Instituto de Ciencias Nucleares, UNAM, A.P. 70-543, 04510, Mexico D.F, Mexico

    Alexander V. Turbiner & Juan Carlos Lopez Vieyra

Authors
  1. Konstantin G. Boreskov
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  2. Alexander V. Turbiner
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  3. Juan Carlos Lopez Vieyra
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Correspondence to Konstantin G. Boreskov.

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Communicated by N.A. Nekrasov

Alexander V. Turbiner: On leave of absence from the Institute for Theoretical and Experimental Physics, Moscow 117259, Russia.

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Boreskov, K., Turbiner, A. & Vieyra, J. Solvability of the Hamiltonians Related to Exceptional Root Spaces: Rational Case. Commun. Math. Phys. 260, 17–44 (2005). https://doi.org/10.1007/s00220-005-1401-y

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  • DOI: https://doi.org/10.1007/s00220-005-1401-y

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