Origin of quantum group and its application in integrable systems

1995 ◽  
Vol 5 (12) ◽  
pp. 2329-2344 ◽  
Author(s):  
Anjan Kundu
Keyword(s):  
Quantum Group ◽  
Integrable Systems

scholarly journals QUANTUM GROUP SYMMETRY OF INTEGRABLE SYSTEMS WITH OR WITHOUT BOUNDARY

2002 ◽  
Vol 17 (25) ◽  
pp. 3649-3661 ◽  
Author(s):  
E. RAGOUCY
Keyword(s):  
Quantum Group ◽  
Integrable Systems ◽  
Integrable Hierarchies ◽  
Group Symmetry ◽  
Integrals Of Motion ◽  
R Matrix ◽  
Explicit Expressions

We present a construction of integrable hierarchies without or with boundary, starting from a single R-matrix, or equivalently from a ZF algebra. We give explicit expressions for the Hamiltonians and the integrals of motion of the hierarchy in term of the ZF algebra. In the case without boundary, the integrals of motion form a quantum group, while in the case with boundary they form a Hopf coideal subalgebra of the quantum group.

scholarly journals Set-theoretic Yang–Baxter & reflection equations and quantum group symmetries

2021 ◽  
Vol 111 (4) ◽  
Author(s):  
Anastasia Doikou ◽  
Agata Smoktunowicz
Keyword(s):  
Quantum Group ◽  
Transfer Matrix ◽  
Integrable Systems ◽  
Hecke Algebra ◽  
Special Choice ◽  
Quantum Integrable Systems ◽  
Double Row ◽  
Reflection Equations ◽  
Group Symmetries ◽  
Reflection Algebra

AbstractConnections between set-theoretic Yang–Baxter and reflection equations and quantum integrable systems are investigated. We show that set-theoretic R-matrices are expressed as twists of known solutions. We then focus on reflection and twisted algebras and we derive the associated defining algebra relations for R-matrices being Baxterized solutions of the A-type Hecke algebra $${\mathcal {H}}_N(q=1)$$ H N ( q = 1 ) . We show in the case of the reflection algebra that there exists a “boundary” finite sub-algebra for some special choice of “boundary” elements of the B-type Hecke algebra $${\mathcal {B}}_N(q=1, Q)$$ B N ( q = 1 , Q ) . We also show the key proposition that the associated double row transfer matrix is essentially expressed in terms of the elements of the B-type Hecke algebra. This is one of the fundamental results of this investigation together with the proof of the duality between the boundary finite subalgebra and the B-type Hecke algebra. These are universal statements that largely generalize previous relevant findings and also allow the investigation of the symmetries of the double row transfer matrix.

scholarly journals Universal R-matrix and functional relations

2014 ◽  
Vol 26 (06) ◽  
pp. 1430005 ◽  
Author(s):  
Herman Boos ◽  
Frank Göhmann ◽  
Andreas Klümper ◽  
Khazret S. Nirov ◽  
Alexander V. Razumov
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Integrable Systems ◽  
Vertex Model ◽  
Quantum Space ◽  
R Matrix ◽  
Functional Relations

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.

On the Heisenberg Supermagnet Model in (2+1)-Dimensions

2017 ◽  
Vol 72 (4) ◽  
pp. 331-337 ◽  
Author(s):  
Zhao-Wen Yan
Keyword(s):  
Integrable System ◽  
Integrable Systems ◽  
Bäcklund Transformations ◽  
Quadratic Constraints ◽  
Backlund Transformations

AbstractThe Heisenberg supermagnet model is an important supersymmetric integrable system in (1+1)-dimensions. We construct two types of the (2+1)-dimensional integrable Heisenberg supermagnet models with the quadratic constraints and investigate the integrability of the systems. In terms of the gage transformation, we derive their gage equivalent counterparts. Furthermore, we also construct new solutions of the supersymmetric integrable systems by means of the Bäcklund transformations.

scholarly journals Separation of variables in AdS/CFT: functional approach for the fishnet CFT

2021 ◽  
Vol 2021 (6) ◽  
Author(s):  
Andrea Cavaglià ◽  
Nikolay Gromov ◽  
Fedor Levkovich-Maslyuk
Keyword(s):  
Integrable Systems ◽  
Density Operator ◽  
Numerical Evaluation ◽  
Lagrangian Density ◽  
Separation Of Variables ◽  
Functional Approach ◽  
Quantum Integrable Systems ◽  
Arbitrary State ◽  
Nontrivial Coupling ◽  
The One

Abstract The major simplification in a number of quantum integrable systems is the existence of special coordinates in which the eigenstates take a factorised form. Despite many years of studies, the basis realising the separation of variables (SoV) remains unknown in $$ \mathcal{N} $$ N = 4 SYM and similar models, even though it is widely believed they are integrable. In this paper we initiate the SoV approach for observables with nontrivial coupling dependence in a close cousin of $$ \mathcal{N} $$ N = 4 SYM — the fishnet 4D CFT. We develop the functional SoV formalism in this theory, which allows us to compute non-perturbatively some nontrivial observables in a form suitable for numerical evaluation. We present some applications of these methods. In particular, we discuss the possible SoV structure of the one-point correlators in presence of a defect, and write down a SoV-type expression for diagonal OPE coefficients involving an arbitrary state and the Lagrangian density operator. We believe that many of the findings of this paper can be applied in the $$ \mathcal{N} $$ N = 4 SYM case, as we speculate in the last part of the article.

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