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QUANTUM GROUPS AND FUNCTIONAL RELATIONS FOR HIGHER RANK

Hermann Boos, Khazret S. Nirov, Alexander, V. Razumov

Published Aug 16, 2016• 25 citations
OpenAlex

Abstract

A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described.

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A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described.
A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described.
A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group Uq(L(sl2))\mathrm U_q(\mathcal L(\mathfrak{sl}_2))Uq​(L(sl2​)) is given. The full proof of the functional relations in the form independent of the representation of the quantum group on the quantum space is presented. The case of the general gradation and general twisting is treated. The specialization of the universal functional relations to the case when the quantum space is the state space of a discrete spin chain is described. This is a degression of the corresponding consideration for the case of the quantum group Uq(L(sl3))\mathrm U_q(\mathcal L(\mathfrak{sl}_3))Uq​(L(sl3​)) with an extensions to the higher spin case.
By the universal integrability objects we mean certain monodromy-type and transfer- type operators, where the representation in the auxiliary space is properly fixed, while the representation in the quantum space is not. This notion is actually determined by the structure of the universal R-matrix. We call functional relations between such universal integrability objects, and so, being independent of the representation in the quantum space, the universal functional relations. We present a short review of the universal functional relations for the quantum integrable systems associated with the quantum groups of loop Lie algebras.
The authors discuss the XXZ spin chain with certain boundary terms and the representation of quantum group SU q (2). It is shown that Bethe ansatz states are highest-weight states of SU q (2). With a generic q they construct the irreducible representations of the quantum group. For q being a root of unity, they show that there are new Bethe ansatz states, which coincide with null states of SU q (2). By taking certain limits the authors can derive the state mod b), 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.