Communications in Mathematical Physics

, Volume 260, Issue 1, pp 17–44 | Cite as

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

  • Konstantin G. Boreskov
  • Alexander V. Turbiner
  • Juan Carlos Lopez Vieyra


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.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Differential Operator 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2005

Authors and Affiliations

  • Konstantin G. Boreskov
    • 1
  • Alexander V. Turbiner
    • 2
  • Juan Carlos Lopez Vieyra
    • 2
  1. 1.Institute for Theoretical and Experimental PhysicsMoscowRussia
  2. 2.Instituto de Ciencias Nucleares, UNAMMexico D.FMexico

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