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arXiv:math/0612086 · Symmetry Integrability and Geometry Methods and Applications

Construction of the Bethe State for the E τ,η (so 3 ) Elliptic Quantum Group

N. Manojlović

Published Jan 5, 2007• 2 citations
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Abstract

Elliptic quantum groups can be associated to solutions of the star-triangle relation of statistical mechanics.In this paper, we consider the particular case of the E τ,η (so 3 ) elliptic quantum group.In the context of algebraic Bethe ansatz, we construct the corresponding Bethe creation operator for the transfer matrix defined in an arbitrary representation of E τ,η (so 3 ).

DOI
10.3842/sigma.2007.004

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Elliptic quantum groups can be associated to solutions of the star-triangle relation of statistical mechanics.In this paper, we consider the particular case of the E τ,η (so 3 ) elliptic quantum group.In the context of algebraic Bethe ansatz, we construct the corresponding Bethe creation operator for the transfer matrix defined in an arbitrary representation of E τ,η (so 3 ).
Algebraic Bethe ansatz for the elliptic quantum group Eτ,η(sl2)E_{\tau,\eta}(sl_2)Eτ,η​(sl2​)
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.