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arXiv:1205.1631 · Reviews in Mathematical Physics

Universal R-matrix and functional relations

Herman E. Boos, Frank Göhmann, Andreas Klümper, Khazret S. Nirov, A. V. Razumov

Published Jun 1, 2014• 32 citations
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Abstract

We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group [Formula: see text] related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.

DOI
10.1142/s0129055x14300052

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We collect and systematize general definitions and facts on the application of quantum groups to the construction of functional relations in the theory of integrable systems. As an example, we reconsider the case of the quantum group [Formula: see text] related to the six-vertex model. We prove the full set of the functional relations in the form independent of the representation of the quantum group in the quantum space and specialize them to the case of the six-vertex model.
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.
ntegrable models in statistical mechanics and quantum field theory constitute a rich research field at the crossroads of modern mathematics and theoretical physics. An important issue to understand is the space of local operators in the system and, ultimately, their correlation functions and form factors. This book is the first published monograph on this subject. It treats integrable lattice models, notably the six-vertex model and the XXZ Heisenberg spin chain. A pair of fermions is introduced and used to create a basis of the space of local operators, leading to the result that all correlation functions at finite distances are expressible in terms of two transcendental functions with rational coefficients. Step-by-step explanations are given for all materials necessary for this construction, ranging from algebraic Bethe ansatz, representations of quantum groups, and the Bazhanov-Lukyanov-Zamolodchikov construction in conformal field theory to Riemann surfaces and their Jacobians. Several examples and applications are given along with numerical results.
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.
Results obtained by us are overviewed from a general set up. The universal RRR-matrix is exploited to obtain various important relations and structures involved in quantum group algebra, which are used subsequently for generating different classes of quantum integrable systems through a systematic scheme. This recovers known models as well as discovers a series of new ones.