Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1312.2484 · Journal of Physics A Mathematical and Theoretical

Quantum groups and functional relations for higher rank

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

Published Jun 18, 2014• 22 citations
arXivHTMLPDF

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.

DOI
10.1088/1751-8113/47/27/275201

Related papers

QUANTUM GROUPS AND FUNCTIONAL RELATIONS FOR HIGHER RANK

Hermann Boos, Khazret S. Nirov, Alexander +1 more

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.

•Aug 16, 2016•25 citations

Quantum groups and functional relations for lower rank

Khazret S. Nirov, A. V. Razumov

A detailed construction of the universal integrability objects related to the integrable systems associated with the quantum group $\mathrm U_q(\mathcal L(\mathfrak{sl}_2))$ 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 $\mathrm U_q(\mathcal L(\mathfrak{sl}_3))$ with an extensions to the higher spin case.

arXiv:1412.7342•Dec 23, 2014•0 citations

Universal R-matrix and functional relations

Herman E. Boos, Frank Göhmann, Andreas Klümper +2 more

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.

arXiv:1205.1631•Reviews in Mathematical Physics•Jun 1, 2014•32 citations

Quantum affine algebras and universal functional relations

Khazret S. Nirov, A. V. Razumov

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.

arXiv:1512.04308•Journal of Physics Conference Series•Jan 25, 2016•1 citation

Quantum groups and functional relations for arbitrary rank

A. V. Razumov

The quantum integrable systems associated with the quantum loop algebras Uq(L(sll+1)) are considered. The factorized form of the transfer operators related to the infinite dimensional evaluation representations is found and the determinant form of the transfer operators related to the finite dimensional evaluation representations is obtained. The master TQ- and TT-relations are derived. The operatorial T- and Q-systems are found. The nested Bethe equations are obtained.

arXiv:2104.12603•Nuclear Physics B•Aug 9, 2021•5 citations

Integrable quantum chain and the representation of the quantum group SUq(2)

Bo-Yu Hou, Kangjie Shi, Xhong-Xia Yang +1 more

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.

Journal of Physics A Mathematical and General•Aug 21, 1991•12 citations
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.