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Bethe Eigenvectors of Higher Transfer Matrices

Evgeny Mukhin

Published Jan 1, 2006• 88 citations
OpenAlex

Abstract

We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra gl N. The models are defined on a tensor product M1 ⊗... ⊗ Mn of irreducible gl N-modules. For each model, there exist N oneparameter families of commuting operators on M1 ⊗... ⊗ Mn, called the transfer matrices. We show that the Bethe vectors for these models, given by the algebraic nested Bethe ansatz, are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues.

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We consider the XXX-type and Gaudin quantum integrable models associated with the Lie algebra gl N. The models are defined on a tensor product M1 ⊗... ⊗ Mn of irreducible gl N-modules. For each model, there exist N oneparameter families of commuting operators on M1 ⊗... ⊗ Mn, called the transfer matrices. We show that the Bethe vectors for these models, given by the algebraic nested Bethe ansatz, are eigenvectors of higher transfer matrices and compute the corresponding eigenvalues.
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To each representation of the elliptic quantum group Eτ,η(sl2)E_{\tau,\eta}(sl_2)Eτ,η​(sl2​) is associated a family of commuting transfer matrices. We give common eigenvectors by a version of the algebraic Bethe ansatz method. Special cases of this construction give eigenvectors for IRF models, for the eight-vertex model and for the two-body Ruijsenaars operator. The latter is a qqq-deformation of Hermite's solution of the Lam\'e equation.