2004/07/22 by Sergei Chmutov, S. Chmutov, Chmutov, S. +3
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Nonlinear Waves and Solitons #Representation Theory (math.RT) #math-ph #math.MP #math.RT
paper · pdf · doi:10.48550/arxiv.math/0407367
the final version, to appear in the IMRN
openalex publication_date 2004/07/22 · arxiv created 2005/05/25 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The note deals with the Gaudin model associated with the tensor product of n irreducible finite-dimensional slN+1-modules marked by distinct complex numbers z1,..., zn. The Bethe Ansatz is a method to construct common eigenvectors of the Gaudin hamiltonians by means of chosen singular vectors in the factors and zj's. These vectors are called Bethe vectors. The question if the Bethe vectors are non-zero vectors is open. By the moment, the only way to verify that was based on a relation to critical points of the master function of the Gaudin model, and non-triviality of a Bethe vector was proved only in the case when the corresponding critical point is non-degenerate ([ScV], [MV1]). However degenerate critical points do appear in the Gaudin model (see Section12 of [ReV]). We believe that the Bethe vectors never vanish, and suggest an approach that does not depend on non-degeneracy of the corresponding critical point. The idea is for a Bethe vector to choose a suitable subspace in the weight space and to check that the projection of the Bethe vector to this subspace is non-zero. We apply this approach to verify non-triviality of Bethe vectors in new examples.