A non-symmetric Yang-Baxter algebra for the quantum nonlinear Schrödinger model

We study certain non-symmetric wavefunctions associated to the quantum nonlinear Schrödinger (QNLS) model, introduced by Komori and Hikami using representations of the degenerate affine Hecke algebra. In particular, they can be generated using a vertex operator formalism analogous to the recursion that defines the symmetric QNLS wavefunction in the quantum inverse scattering method. Furthermore, some of the commutation relations encoded in the Yang-Baxter equation are generalized to the non-symmetric case

Universal k-matrices for quantum Kac-Moody algebras

We define the notion of an \emph{almost cylindrical} bialgebra, which is roughly a quasitriangular bialgebra endowed with a universal solution of a {twisted} reflection equation, called a {twisted} universal k--matrix, yielding an action of cylindrical braid groups on tensor products of its representations. The definition is a nontrivial generalization of the notion of cylinder--braided bialgebras due to tom Dieck--H\"{a}ring-Oldenburg and Balagovi\'{c}--Kolb. Namely, the twisting involved in the reflection equation does not preserve the quasitriangular structure. Instead, it is only required to be an algebra automorphism, whose defect in being a morphism of quasitriangular bialgebras is controlled by a Drinfeld twist. We prove that examples of such new twisted universal k--matrices arise from quantum symmetric pairs of Kac--Moody type, whose controlling combinatorial datum is a pair of compatible generalized Satake diagrams. In finite type, this yields a refinement of the result obtained by Balagovi\'c--Kolb, producing a family of inequivalent solutions interpolating between the \emph{quasi}--k--matrix and the {\em full} universal k--matrix. This new framework is motivated by the study of solutions of the parameter--dependent reflection equation (spectral k--matrices) in the category of finite--dimensional representations of quantum affine algebras. Indeed, as an application, we prove that our construction leads to (formal) spectral k--matrices in evaluation representations of $U_qL{\mathfrak{sl}_2}$.Comment: 67 pages; made minor changes throughout the documen

Rational K-matrices for finite-dimensional representations of quantum affine algebras

Let $\mathfrak{g}$ be a complex simple Lie algebra. We prove that every finite-dimensional representation of the (untwisted) quantum affine algebra $U_qL\mathfrak{g}$ gives rise to a family of spectral K-matrices, namely solutions of Cherednik's generalized reflection equation, which depends upon the choice of a quantum affine symmetric pair $U_q\mathfrak{k}\subset U_qL\mathfrak{g}$. Moreover, we prove that every irreducible representation over $U_qL\mathfrak{g}$ remains generically irreducible under restriction to $U_q\mathfrak{k}$. From the latter result, we deduce that every obtained K-matrix can be normalized to a matrix-valued rational function in a multiplicative parameter, known in the study of quantum integrability as a trigonometric K-matrix. Finally, we show that our construction recovers many of the known solutions of the standard reflection equation and gives rise to a large class of new solutions.Comment: 37 page

Solutions of the U q (ŜI N ) reflection equations

We find the complete set of invertible solutions of the untwisted and twisted reflection equations for the Bazhanov-Jimbo R-matrix of type . We also show that all invertible solutions can be obtained by an appropriate affinization procedure from solutions of the constant untwisted and twisted reflection equations

Quasitriangular coideal subalgebras of Uq(g)U_q(\mathfrak{g}) in terms of generalized Satake diagrams

Let $\mathfrak{g}$ be a finite-dimensional semisimple complex Lie algebra and $\theta$ an involutive automorphism of $\mathfrak{g}$. It is well-known from works of Letzter, Kolb and Balagovi\'c that the fixed-point subalgebra $\mathfrak{k} = \mathfrak{g}^\theta$ has a quantum counterpart $B$, a coideal subalgebra of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{g})$ possessing a cylinder-twisted universal K-matrix $\mathcal{K}$. The objects $\theta$, $\mathfrak{k}$, $B$ and $\mathcal{K}$ can all be described in terms of a combinatorial datum, a Satake diagram. In the present work we extend this construction to generalized Satake diagrams, objects first considered by Heck. A generalized Satake diagram defines a semisimple automorphism of $\mathfrak{g}$ restricting to the standard Cartan subalgebra $\mathfrak{h}$ as an involution. We show that it naturally leads to a subalgebra $\mathfrak{k}\subset \mathfrak{g}$, not necessarily a fixed-point subalgebra, but still satisfying $\mathfrak{k} \cap \mathfrak{h} = \mathfrak{h}^\theta$. Such a subalgebra $\mathfrak{k}$ can be quantized to a coideal subalgebra of $U_q(\mathfrak{g})$ endowed with a cylinder-twisted universal K-matrix. We conjecture that all such coideal subalgebras of $U_q(\mathfrak{g})$ arise from generalized Satake diagrams in this way

Integral solutions to boundary quantum Knizhnik–Zamolodchikov equations

We construct integral representations of solutions to the boundary quantum Knizhnik–Zamolodchikov equations. These are difference equations taking values in tensor products of Verma modules of quantum affine , with the K-operators acting diagonally. The integrands in question are products of scalar-valued elliptic weight functions with vector-valued trigonometric weight functions (boundary Bethe vectors). These integrals give rise to a basis of solutions of the boundary qKZ equations over the field of quasi-constant meromorphic functions in weight subspaces of the tensor product

Boundary quantum Knizhnik-Zamolodchikov equations and fusion

In this paper we extend our previous results concerning Jackson integral solutions of the boundary quantum Knizhnik-Zamolodchikov equations with diagonal K-operators to higher-spin representations of quantum affine $\mathfrak{sl}_2$. First we give a systematic exposition of known results on $R$-operators acting in the tensor product of evaluation representations in Verma modules over quantum $\mathfrak{sl}_2$. We develop the corresponding fusion of $K$-operators, which we use to construct diagonal $K$-operators in these representations. We construct Jackson integral solutions of the associated boundary quantum Knizhnik-Zamolodchikov equations and explain how in the finite-dimensional case they can be obtained from our previous results by the fusion procedure

A non-symmetric Yang-Baxter Algebra for the Quantum Nonlinear Schr\"odinger Model

We study certain non-symmetric wavefunctions associated to the quantum nonlinear Schr\"odinger model, introduced by Komori and Hikami using Gutkin's propagation operator, which involves representations of the degenerate affine Hecke algebra. We highlight how these functions can be generated using a vertex-type operator formalism similar to the recursion defining the symmetric (Bethe) wavefunction in the quantum inverse scattering method. Furthermore, some of the commutation relations encoded in the Yang-Baxter equation for the relevant monodromy matrix are generalized to the non-symmetric case.Comment: 31 pages; added some references; minor corrections throughou

Boundary transfer matrices and boundary quantum KZ equations

A simple relation between inhomogeneous transfer matrices and boundary quantum KZ equations is exhibited for quantum integrable systems with reflecting boundary conditions, analogous to an observation by Gaudin for periodic systems. Thus the boundary quantum KZ equations receive a new motivation. We also derive the commutativity of Sklyanin's boundary transfer matrices by merely imposing appropriate reflection equations, i.e. without using the conditions of crossing symmetry and unitarity of the R-matrix