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Diagonalization of the \(XXZ\) Hamiltonian by vertex operators. (English) Zbl 0769.17020

The \(XXZ\) Hamiltonian is an operator \(H\) on the doubly infinite tensor product \(\bigotimes^ \infty_{k=-\infty}\mathbb{C}^ 2\) of copies of \(\mathbb{C}^ 2\). Viewing \(\mathbb{C}^ 2\) as the natural 2-dimensional representation of the quantum affine algebra \(U_ q(\widehat{sl}_ 2)\), it turns out that \(H\) commutes with the action of \(U_ q(\widehat{sl}_ 2)\). The authors analyze this representation of \(U_ q(\widehat{sl}_ 2)\) by crystal basis techniques, and use their results to diagonalize \(H\).

MSC:

17B81 Applications of Lie (super)algebras to physics, etc.
82B23 Exactly solvable models; Bethe ansatz
81R50 Quantum groups and related algebraic methods applied to problems in quantum theory
17B37 Quantum groups (quantized enveloping algebras) and related deformations
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