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Affine Algebra
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S Pakuliak – One of the best experts on this subject based on the ideXlab platform.

Bethe Vectors of Quantum Integrable Models with GL(3) Trigonometric $R$Matrix
Symmetry Integrability and Geometry : Methods and Applications, 2013CoAuthors: S. Belliard, S Pakuliak, Eric Ragoucy, Nikita A. SlavnovAbstract:We study quantum integrable models with GL(3) trigonometric $R$matrix and solvable by the nested Algebraic Bethe ansatz. Using the presentation of the universal Bethe vectors in terms of projections of products of the currents of the quantum Affine Algebra $U_q(\hat{\mathfrak{gl}}_3)$ onto intersections of different types of Borel subAlgebras, we prove that the set of the nested Bethe vectors is closed under the action of the elements of the monodromy matrix.

Bethe Ansatz and Bethe Vectors Scalar Products
Symmetry Integrability and Geometry: Methods and Applications, 2010CoAuthors: S. Belliard, S Pakuliak, Eric RagoucyAbstract:An integral presentation for the scalar products of nested Bethe vectors for the quantum integrable models associated with the quantum Affine Algebra $U_q(\hat{\mathfrak{gl}}_3)$ is given. This result is obtained in the framework of the universal Bethe ansatz, using presentation of the universal Bethe vectors in terms of the total currents of a “new” realization of the quantum Affine Algebra $U_q(\hat{\mathfrak{gl}}_3)$.

a computation of universal weight function for quantum Affine Algebra u_q widehat mathfrak gl _n
Journal of Mathematics of Kyoto University, 2008CoAuthors: S M Khoroshkin, S PakuliakAbstract:We compute weight functions (offshell Bethe vectors) in any representation with a weight singular vector of the quantum Affine Algebra $U_q(\widehat{\mathfrak{gl}}_N)$ applying the method of projections of Drinfeld currents developed in {EKP}.
S M Khoroshkin – One of the best experts on this subject based on the ideXlab platform.

a computation of universal weight function for quantum Affine Algebra u_q widehat mathfrak gl _n
Journal of Mathematics of Kyoto University, 2008CoAuthors: S M Khoroshkin, S PakuliakAbstract:We compute weight functions (offshell Bethe vectors) in any representation with a weight singular vector of the quantum Affine Algebra $U_q(\widehat{\mathfrak{gl}}_N)$ applying the method of projections of Drinfeld currents developed in {EKP}.

Weight functions and Drinfeld currents
Communications in Mathematical Physics, 2007CoAuthors: Benjamin Enriquez, S M Khoroshkin, S PakuliakAbstract:A universal weight function for a quantum Affine Algebra is a family of functions with values in a quotient of its Borel subAlgebra, satisfying certain coAlgebraic properties. In representations of the quantum Affine Algebra it gives offshell Bethe vectors and is used in the construction of solutions of the qKZ equations. We construct a universal weight function for each untwisted quantum Affine Algebra, using projections onto the intersection of Borel subAlgebras of different types, and study its functional properties.

a computation of an universal weight function for the quantum Affine Algebra u_q hat mathfrak gl _n
arXiv: Quantum Algebra, 2007CoAuthors: S M Khoroshkin, S PakuliakAbstract:We compute an universal weight function (offshell Bethe vectors) in any representation with a weight singular vector of the quantum Affine Algebra $U_q(\hat{\mathfrak{gl}}_N)$ applying the method of projections of Drinfeld currents developed in arXiv:math/0610398.
Honglian Zhang – One of the best experts on this subject based on the ideXlab platform.

Twoparameter quantum Affine Algebra of type G2(1), Drinfeld realization and vertex representation
Journal of Mathematical Physics, 2015CoAuthors: Yun Gao, Honglian ZhangAbstract:In this paper, we define the twoparameter quantum Affine Algebra for type G2(1) and give the (r, s)Drinfeld realization of Ur,s(G2(1)), as well as establish and prove its Drinfeld isomorphism. We construct and verify explicitly the levelone vertex representation of twoparameter quantum Affine Algebra Ur,s(G2(1)), which also supports an evidence in nontwisted type G2(1) for the uniform defining approach via the twoparameter τinvariant generating functions proposed in Hu and Zhang [Generating functions with τinvariance and vertex representations of twoparameter quantum Affine Algebras Ur,s(g): Simply laced cases eprint arXiv:1401.4925]

two parameter quantum Affine Algebra u_ r s widehat frak sl _n drinfel d realization and quantum Affine lyndon basis
arXiv: Quantum Algebra, 2008CoAuthors: Marc Rosso, Honglian ZhangAbstract:We further define twoparameter quantum Affine Algebra $U_{r,s}(\widehat{\frak {sl}_n})$ $(n>2)$ after the work on the finite cases (see [BW1], [BGH1], [HS] & [BH]), which turns out to be a Drinfel’d double. Of importance for the quantum {\it Affine} cases is that we can work out the compatible twoparameter version of the Drinfel’d realization as a quantum affinization of $U_{r,s}(\frak{sl}_n)$ and establish the Drinfel’d isomorphism Theorem in the twoparameter setting, via developing a new combinatorial approach (quantum calculation) to the quantum {\it Affine} Lyndon basis we present (with an explicit valid algorithm based on the use of Drinfel’d generators).

two parameter quantum Affine Algebra u_ r s widehat mathfrak sl _n drinfel d realization and quantum Affine lyndon basis
Communications in Mathematical Physics, 2008CoAuthors: Marc Rosso, Honglian ZhangAbstract:We further define twoparameter quantum Affine Algebra \(U_{r,s}(\widehat{\mathfrak {sl}_n})\) (n > 2) after the work on the finite cases (see [BW1,BGH1,HS,BH]), which turns out to be a Drinfel’d double. Of importance for the quantum Affine cases is that we can work out the compatible twoparameter version of the Drinfel’d realization as a quantum affinization of \(U_{r,s}({\mathfrak{sl}}_n)\) and establish the Drinfel’d Isomorphism Theorem in the twoparameter setting, via developing a new combinatorial approach (quantum calculation) to the quantum Affine Lyndon basis we present (with an explicit valid algorithm based on the use of Drinfel’d generators).
Jonathan Beck – One of the best experts on this subject based on the ideXlab platform.

Crystal Structure of Level Zero Extremal Weight Modules
Letters in Mathematical Physics, 2002CoAuthors: Jonathan BeckAbstract:We consider the crystal structure of the level zero extremal weight modules V(λ) using the crystal base of the quantum Affine Algebra constructed in Duke Math. J.99 (1999), 455–487. This approach yields an explicit form for extremal weight vectors in the U− part of each connected component of the crystal, which are given as Schur functions in the imaginary root vectors. We show the map \(\Phi _\lambda\) induces a correspondence between the global crystal base of V(λ) and elements \(s_{c_0 } \left( {z^{ – 1} } \right)G\left( b \right),b \in B_0 \left( {U_q \left[ {{z_{i,k}}^{{ \pm 1}} } \right]u\prime } \right)\).

Crystal structure of level zero extremal weight modules
arXiv: Quantum Algebra, 2002CoAuthors: Jonathan BeckAbstract:We consider the crystal structure of the level zero extremal weight modules $V(\lambda)$ using the crystal base of the quantum Affine Algebra constructed by Beck, Chari and Pressley. This approach yields an explicit form for the U^ extremal weight vectors in each connected component of the crystal of $V(\lambda)$, which are given as Schur functions in the imaginary root vectors. We use this fact to demonstrate Kashiwara’s conjectures regarding the crystal structure of $V(\lambda)$.

Braid group action and quantum Affine Algebras
Communications in Mathematical Physics, 1994CoAuthors: Jonathan BeckAbstract:We lift the lattice of translations in the extended Affine Weyl group to a braid group action on the quantum Affine Algebra. This action fixes the Heisenberg subAlgebra pointwise. Looplike generators of the Algebra are obtained which satisfy the relations of Drinfel’d’s new realization. Coproduct formulas are given and a PBW type basis is constructed.
Kentaroh Yoshida – One of the best experts on this subject based on the ideXlab platform.

a deformation of quantum Affine Algebra in squashed wess zumino novikov witten models
Journal of Mathematical Physics, 2014CoAuthors: Io Kawaguchi, Kentaroh YoshidaAbstract:We proceed to study infinitedimensional symmetries in twodimensional squashed WessZuminoNovikovWitten models at the classical level. The target space is given by squashed S3 and the isometry is SU(2)L × U(1)R. It is known that SU(2)L is enhanced to a couple of Yangians. We reveal here that an infinitedimensional extension of U(1)R is a deformation of quantum Affine Algebra, where a new deformation parameter is provided with the coefficient of the WessZumino term. Then we consider the relation between the deformed quantum Affine Algebra and the pair of Yangians from the viewpoint of the leftright duality of monodromy matrices. The integrable structure is also discussed by computing the r/smatrices that satisfy the extended classical YangBaxter equation. Finally, two degenerate limits are discussed.

a deformation of quantum Affine Algebra in squashed wznw models
arXiv: High Energy Physics – Theory, 2013CoAuthors: Io Kawaguchi, Kentaroh YoshidaAbstract:We proceed to study infinitedimensional symmetries in twodimensional squashed WessZuminoNovikovWitten (WZNW) models at the classical level. The target space is given by squashed S^3 and the isometry is SU(2)_L x U(1)_R. It is known that SU(2)_L is enhanced to a couple of Yangians. We reveal here that an infinitedimensional extension of U(1)_R is a deformation of quantum Affine Algebra, where a new deformation parameter is provided with the coefficient of the WessZumino term. Then we consider the relation between the deformed quantum Affine Algebra and the pair of Yangians from the viewpoint of the leftright duality of monodromy matrices. The integrable structure is also discussed by computing the r/smatrices that satisfy the extended classical YangBaxter equation. Finally two degenerate limits are discussed.

on the classical equivalence of monodromy matrices in squashed sigma model
Journal of High Energy Physics, 2012CoAuthors: Io Kawaguchi, Takuya Matsumoto, Kentaroh YoshidaAbstract:We proceed to study the hybrid integrable structure in twodimensional nonlinear sigma models with target space threedimensional squashed spheres. A quan tum Affine Algebra and a pair of Yangian Algebras are realized in the sigma models and, according to them, there are two descriptions to describe the classical dynamics 1) the trigonometric description and 2) the rational description, respectively. For every description, a Lax pair is constructed and the associated monodromy matrix is also constructed. In this paper we show the gaugeequivalence of the monodromy matrices in the trigonometric and rational description under a certain relation between spectral parameters and the rescalings of sl(2) generators.