# Affine Algebra - Explore the Science & Experts | ideXlab

## Affine Algebra

The Experts below are selected from a list of 25371 Experts worldwide ranked by ideXlab platform

### 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, 2013
Co-Authors: S. Belliard, S Pakuliak, Eric Ragoucy, Nikita A. Slavnov
Abstract:

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, 2010
Co-Authors: S. Belliard, S Pakuliak, Eric Ragoucy
Abstract:

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, 2008
Co-Authors: S M Khoroshkin, S Pakuliak
Abstract:

We compute weight functions (off-shell 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, 2008
Co-Authors: S M Khoroshkin, S Pakuliak
Abstract:

We compute weight functions (off-shell 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, 2007
Co-Authors: Benjamin Enriquez, S M Khoroshkin, S Pakuliak
Abstract:

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 off-shell 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, 2007
Co-Authors: S M Khoroshkin, S Pakuliak
Abstract:

We compute an universal weight function (off-shell 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.

• ##### Two-parameter quantum Affine Algebra of type G2(1), Drinfeld realization and vertex representation
Journal of Mathematical Physics, 2015
Co-Authors: Yun Gao, Honglian Zhang
Abstract:

In this paper, we define the two-parameter 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 level-one vertex representation of two-parameter quantum Affine Algebra Ur,s(G2(1)), which also supports an evidence in nontwisted type G2(1) for the uniform defining approach via the two-parameter τ-invariant generating functions proposed in Hu and Zhang [Generating functions with τ-invariance and vertex representations of two-parameter quantum Affine Algebras Ur,s(g): Simply laced cases e-print 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, 2008
Co-Authors: Marc Rosso, Honglian Zhang
Abstract:

We further define two-parameter 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 two-parameter 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 two-parameter 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, 2008
Co-Authors: Marc Rosso, Honglian Zhang
Abstract:

We further define two-parameter 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 two-parameter 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 two-parameter 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, 2002
Co-Authors: Jonathan Beck
Abstract:

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, 2002
Co-Authors: Jonathan Beck
Abstract:

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, 1994
Co-Authors: Jonathan Beck
Abstract:

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. Loop-like 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, 2014
Co-Authors: Io Kawaguchi, Kentaroh Yoshida
Abstract:

We proceed to study infinite-dimensional symmetries in two-dimensional squashed Wess-Zumino-Novikov-Witten 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 infinite-dimensional extension of U(1)R is a deformation of quantum Affine Algebra, where a new deformation parameter is provided with the coefficient of the Wess-Zumino term. Then we consider the relation between the deformed quantum Affine Algebra and the pair of Yangians from the viewpoint of the left-right duality of monodromy matrices. The integrable structure is also discussed by computing the r/s-matrices that satisfy the extended classical Yang-Baxter equation. Finally, two degenerate limits are discussed.

• ##### a deformation of quantum Affine Algebra in squashed wznw models
arXiv: High Energy Physics – Theory, 2013
Co-Authors: Io Kawaguchi, Kentaroh Yoshida
Abstract:

We proceed to study infinite-dimensional symmetries in two-dimensional squashed Wess-Zumino-Novikov-Witten (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 infinite-dimensional extension of U(1)_R is a deformation of quantum Affine Algebra, where a new deformation parameter is provided with the coefficient of the Wess-Zumino term. Then we consider the relation between the deformed quantum Affine Algebra and the pair of Yangians from the viewpoint of the left-right duality of monodromy matrices. The integrable structure is also discussed by computing the r/s-matrices that satisfy the extended classical Yang-Baxter equation. Finally two degenerate limits are discussed.

• ##### on the classical equivalence of monodromy matrices in squashed sigma model
Journal of High Energy Physics, 2012
Co-Authors: Io Kawaguchi, Takuya Matsumoto, Kentaroh Yoshida
Abstract:

We proceed to study the hybrid integrable structure in two-dimensional non-linear sigma models with target space three-dimensional 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 gauge-equivalence of the monodromy matrices in the trigonometric and rational description under a certain relation between spectral parameters and the rescalings of sl(2) generators.