Central extension of the reflection equations and an analog of Miki's formula

Two different types of centrally extended quantum reflection algebras are introduced. Realizations in terms of the elements of the central extension of the Yang-Baxter algebra are exhibited. A coaction map is identified. For the special case of $U_q(\hat{sl_2})$, a realization in terms of elements satisfying the Zamolodchikov-Faddeev algebra - a `boundary' analog of Miki's formula - is also proposed, providing a free field realization of $O_q(\hat{sl_2})$ (q-Onsager) currents.Comment: 11 pages; two references added; to appear in J. Phys.

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

Algebraic Bethe ansatz for Q-operators of the open XXX Heisenberg chain with arbitrary spin

In this note we construct Q-operators for the spin s open Heisenberg XXX chain with diagonal boundaries in the framework of the quantum inverse scattering method. Following the algebraic Bethe ansatz we diagonalise the introduced Q-operators using the fundamental commutation relations. By acting on Bethe off-shell states and explicitly evaluating the trace in the auxiliary space we compute the eigenvalues of the Q-operators in terms of Bethe roots and show that the unwanted terms vanish if the Bethe equations are satisfied.Comment: 17 page

Non-compact quantum spin chains as integrable stochastic particle processes

In this paper we discuss a family of models of particle and energy diffusion on a one-dimensional lattice, related to those studied previously in Sasamoto and Wadati (Phys Rev E 58:4181\u20134190, 1998), Barraquand and Corwin (Probab Theory Relat Fields 167(3\u20134):1057\u20131116, 2017) and Povolotsky (J Phys A 46(46):465205, 2013) in the context of KPZ universality class. We show that they may be mapped onto an integrable (2) Heisenberg spin chain whose Hamiltonian density in the bulk has been already studied in the AdS/CFT and the integrable system literature. Using the quantum inverse scattering method, we study various new aspects, in particular we identify boundary terms, modeling reservoirs in non-equilibrium statistical mechanics models, for which the spin chain (and thus also the stochastic process) continues to be integrable. We also show how the construction of a \u201cdual model\u201d of probability theory is possible and useful. The fluctuating hydrodynamics of our stochastic model corresponds to the semiclassical evolution of a string that derives from correlation functions of local gauge invariant operators of \ue23a=4 super Yang\u2013Mills theory (SYM), in imaginary-time. As any stochastic system, it has a supersymmetric completion that encodes for the thermal equilibrium theorems: we show that in this case it is equivalent to the (2|1) superstring that has been derived directly from \ue23a=4 SYM

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