Auxiliary matrices for the six-vertex model at qN = 1 and a geometric interpretation of its symmetries

Korff, C. (2003) Auxiliary matrices for the six-vertex model at qN = 1 and a geometric interpretation of its symmetries. Journal of Physics A: Mathematical and General, 36(19), pp. 5229-5266. (doi: 10.1088/0305-4470/36/19/305)

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The construction of auxiliary matrices for the six-vertex model at a root of unity is investigated from a quantum group theoretic point of view. Employing the concept of intertwiners associated with the quantum loop algebra Uq(sl2) at qN = 1, a three-parameter family of auxiliary matrices is constructed. The elements of this family satisfy a functional relation with the transfer matrix allowing one to solve the eigenvalue problem of the model and to derive the Bethe ansatz equations. This functional relation is obtained from the decomposition of a tensor product of evaluation representations and involves auxiliary matrices with different parameters. Because of this dependence on additional parameters, the auxiliary matrices break in general the finite symmetries of the six-vertex model, such as spin-reversal or spin-conservation. More importantly, they also lift the extra degeneracies of the transfer matrix due to the loop symmetry present at rational coupling values. The extra parameters in the auxiliary matrices are shown to be directly related to the elements in the enlarged centre Z of the algebra Uq(sl2) at qN = 1. This connection provides a geometric interpretation of the enhanced symmetry of the six-vertex model at rational coupling. The parameters labelling the auxiliary matrices can be interpreted as coordinates on a hypersurface Spec Z ⊂ C4 which remains invariant under the action of an infinite-dimensional group G of analytic transformations, called the quantum coadjoint action.

Item Type:Articles
Glasgow Author(s) Enlighten ID:Korff, Professor Christian
Authors: Korff, C.
Subjects:Q Science > QA Mathematics
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of Physics A: Mathematical and General
Publisher:Institute of Physics Publishing Ltd.

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