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arXiv:1904.05653 · Journal of Physics A Mathematical and Theoretical

Reflection K matrices associated with an Onsager coideal of U p ( A n − 1 ( 1 ) ) , U p ( B n ( 1 ) ) , U p ( D n ( 1 ) ) and U p ( D n + 1 ( 2 ) )

Atsuo Kuniba, Masato Okado, Akihito Yoneyama

Published Jul 30, 2019• 4 citations
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Abstract

Abstract We determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra in the fundamental representations and in the spin representations. They reproduce the reflection K matrices obtained recently by the matrix product construction connected to the three dimensional integrability. In particular the present approach provides the first proof of the reflection equation for the non type A cases.

DOI
10.1088/1751-8121/ab3715

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Abstract We determine the intertwiners of a family of Onsager coideal subalgebras of the quantum affine algebra in the fundamental representations and in the spin representations. They reproduce the reflection K matrices obtained recently by the matrix product construction connected to the three dimensional integrability. In particular the present approach provides the first proof of the reflection equation for the non type A cases.
We present a representation of the generalized p-Onsager algebras Op(An−1(1)), Op(Dn+1(2)), Op(Bn(1)), Op(B˜n(1)) and Op(Dn(1)) in which the generators are expressed as local Hamiltonians of XXZ type spin chains with various boundary terms reflecting the Dynkin diagrams. Their symmetry is described by the reflection K matrices which are obtained recently by a q-boson matrix product construction related to the 3D integrability and characterized by Onsager coideals of quantum affine algebras. The spectral decomposition of the K matrices with respect to the classical part of the Onsager algebra is described conjecturally. We also include a proof of a certain invariance property of boundary vectors in the q-boson Fock space playing a key role in the matrix product construction.
A procedure to construct K -matrices from the generalized q -Onsager algebra is proposed. This procedure extends the intertwiner techniques used to obtain scalar ( c -number) solutions of the reflection equation to dynamical (non- c -number) solutions. It shows the relation between soliton non-preserving reflection equations or twisted reflection equations and the generalized q -Onsager algebras. These dynamical K -matrices are important to quantum integrable models with extra degrees of freedom located at the boundaries; for instance, in the quantum affine Toda field theories on the half-line, they yield the boundary amplitudes. As examples, the cases of and are treated in detail.
Abstract We formulate a quantized reflection equation in which q -boson valued L and K matrices satisfy the reflection equation up to conjugation by a solution to the Isaev–Kulish 3D reflection equation. By forming its n -concatenation along the q -boson Fock space followed by suitable reductions, we construct families of solutions to the reflection equation in a matrix product form connected to the 3D integrability. They involve the quantum R matrices of the antisymmetric tensor representations of and the spin representations of , and .
Abstract We study solutions of the reflection equation related to the quantum affine algebra U q ( s l n ^ ) . First, we explain how to construct a family of stochastic integrable vertex models with fixed boundary conditions. Then, we construct upper- and lower-triangular solutions of the reflection equation related to symmetric tensor representations of U q ( s l n ^ ) with arbitrary spin. We also prove the star–star relation for the Boltzmann weights of the Ising-type model, conjectured by Bazhanov and Sergeev, and use it to verify certain properties of the solutions obtained.
We derive the non-local conserved charges in the sine-Gordon model and affine Toda field theories on the half-line.They generate new kinds of symmetry algebras that are coideals of the usual quantum groups.We show how intertwiners of tensor product representations of these algebras lead to solutions of the reflection equation.We describe how this method for finding solutions to the reflection equation parallels the previously know method of using intertwiners of quantum groups to find solutions to the Yang-Baxter equation. * Speaker.1 Earlier calculations for the sine-Gordon model at the free fermion point were done in [16]