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Integrable quantum chain and the representation of the quantum group SU\(_ q\)(2). (English) Zbl 0745.17021

Summary: We discuss the \(XXZ\) spin chain with certain boundary terms and the representation of the quantum group \(SU_ q(2)\). It is shown that Bethe ansatz states are highest-weight states of \(SU_ q(2)\). With a generic \(q\) we construct the irreducible representations of the quantum group. For \(q\) being a root of unity, we show that there are new Bethe ansatz states, which coincide with null states of \(SU_ q(2)\). By taking certain limits we can derive the state \(\mid b\rangle\), which is necessary for constructing the indecomposable but reducible representations of \(SU_ q(2)\) and for the completeness of the state space. In this case the Hamiltonian may not be completely diagonalized.

MSC:

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