Quasi-polynomials and the Bethe Ansatz

@article{Mukhin2006QuasipolynomialsAT,
  title={Quasi-polynomials and the Bethe Ansatz},
  author={Evgeny Mukhin and Alexander N. Varchenko},
  journal={arXiv: Quantum Algebra},
  year={2006}
}
We study solutions of the Bethe Ansatz equation related to the trigonometric Gaudin model associated to a simple Lie algebra g and a tensor product of irreducible finite-dimensional representations. Having one solution, we describe a construction of new solutions. The collection of all solutions obtained from a given one is called a population. We show that the Weyl group of g acts on the points of a population freely and transitively (under certain conditions). To a solution of the Bethe… 
On the Gaudin model of type G2
  • Kang Lu, E. Mukhin
  • Mathematics, Physics
    Communications in Contemporary Mathematics
  • 2019
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References

SHOWING 1-10 OF 31 REFERENCES
Discrete Miura Opers and Solutions of the Bethe Ansatz Equations
Solutions of the Bethe ansatz equations associated to the XXX model of a simple Lie algebra come in families called the populations. We prove that a population is isomorphic to the flag variety of
Opers on the projective line, flag manifolds and Bethe Ansatz
We consider the problem of diagonalization of the hamiltonians of the Gaudin model, which is a quantum chain model associated to a simple Lie algebra. The hamiltonians of this model act on the tensor
Integral representation of solutions of the elliptic knizhnik-zamolodchikov-bernard equations
We give an integral representation of solutions of the elliptic Knizhnik-Zamolodchikov-Bernard equations for arbitrary simple Lie algebras. If the level is a positive integer, we obtain formulas for
Solutions to the XXX type Bethe ansatz equations and flag varieties
We consider a version of the AN Bethe equation of XXX type and introduce a reporduction procedure constructing new solutions of this equation from a given one. The set of all solutions obtained from
Three formulas for eigenfunctions of integrable Schrödinger operators
We give three formulas for meromorphic eigenfunctions (scatteringstates) of Sutherland‘sintegrable N-body Schrödinger operators and their generalizations.The first is an explicit computation of the
Bethe eigenvectors of higher transfer matrices
We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra . The models are defined on a tensor product of irreducible -modules. For each model, there exist N
Critical Points of the Product of Powers of Linear Functions and Families of Bases of Singular Vectors
The quasiclassical asymptotics of the Knizhnik-Zamolodchikov equation with values in the tensor product of sl(2)- representations are considered. The first term of asymptotics is an eigenvector of a
Norm of a Bethe vector and the Hessian of the master function
We show that the norm of a Bethe vector in the $sl_{r+1}$ Gaudin model is equal to the Hessian of the corresponding master function at the corresponding critical point. In particular the Bethe
Solutions of Trigonometric KZ Equations satisfy Dynamical Difference Equations
The trigonometric KZ equations associated to a Lie algebra \g depend on a parameter \lambda in \h where \h is a Cartan subalgebra of \g. A system of dynamical difference equations with respect to
Populations of solutions of the XXX Bethe equations associated to Kac-Moody algebras
We consider the XXX Bethe equation associated with integral dominant weights of a Kac-Moody algebra and introduce a generating procedure constructing new solutions starting from a given one. The
...
1
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