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This paper presents a matrix formulation of the scalar laws of radiative transfer. The method applies to coupled mixed boundary condition problems on general domains. Participating media can range from transparent to absorbing, emitting,…
Recently V.Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions of the quantum Yang-Baxter equation, i.e. solutions given by a permutation $R$ of the set…
We present a generalization of the master solution to the quantum Yang-Baxter equation (obtained recently in arXiv:1006.0651) to the case of multi-component continuous spin variables taking values on a circle. The Boltzmann weights are…
We construct a noncommutative (Grassmann) extension of the well known Adler Yang-Baxter map. It satisfies the Yang-Baxter equation, it is reversible and birational. Our extension preserves all the properties of the original map except the…
We study the problem of existence of solutions for generalized backward stochastic differential equation with two reflecting barriers (GRBSDE for short) under weaker assumptions on the data. Roughly speaking we show the existence of a…
We solve the gl(1|2) generalized model by means of the algebraic Bethe ansatz. The resulting eigenvalue of the transfer matrix and the Bethe ansatz equations depend on three complex functions, called the parameters of the generalized model.…
The calculable $R$-matrix theory has been formulated successfully for regular boundary conditions with vanishing radial wave functions at the coordinate origins [P. Descouvemont and D. Baye, Rept. Prog. Phys. 73, 036301 (2010)]. We…
The unitarily invariant probability measures on infinite Hermitian matrices have been classified by Pickrell, and by Olshanski and Vershik. This classification is equivalent to determining the boundary of a certain inhomogeneous Markov…
In this paper, we introduce a class of rings which is a generalization of reflexive rings and $J$-reversible rings. Let $R$ be a ring with identity and $J(R)$ denote the Jacobson radical of $R$. A ring $R$ is called {\it $J$-reflexive} if…
This mainly tutorial paper is intended as a somewhat larger example for parts of the theory exposed in the Lectures on the dynamical Yang-Baxter equations by P. Etingof and O. Schiffmann, math.QA/9908064. We explicitly compute the matrix…
We present a Baxterization of a two-colour generalization of the Birman--Wenzl--Murakami (BWM) algebra. Appropriately combining two RSOS-type representations of the ordinary BWM algebra, we construct representations of the two-colour…
All solutions of constant classical Yang-Baxter equation (CYBE) in Lie algebra $L$ with dim $L \le 3$ are obtained and the sufficient and necessary conditions which $(L, \hbox {[ ]}, \Delta_r, r)$ is a coboundary (or triangular) Lie…
A trick to obtain a systematic solution to the set-theoretical reflection equation is presented from a known one to the Yang-Baxter equation. Examples are given from crystals and geometric crystals associated to the quantum affine algebra…
In this paper, we introduce the notion of Rota-Baxter Lie $2$-algebras, which is a categorification of Rota-Baxter Lie algebras. We prove that the category of Rota-Baxter Lie $2$-algebras and the category of $2$-term Rota-Baxter…
For every pair of positive coprime integers, m and n, with m<n, there is an associated generalized Cremmer-Gervais r-matrix r_{m,n} for the Lie algebra sl_n which provides a nonstandard quasitriangular solution to the classical Yang-Baxter…
The general solutions for the factorization equations of the reflection matrices $K^{\pm}(\theta)$ for the eight vertex and six vertex models (XYZ, XXZ and XXX chains) are found. The associated integrable magnetic Hamiltonians are…
We construct sets of structure matrices for the semi-dynamical reflection algebra, solving the Yang-Baxter type consistency equations extended by the action of an automorphism of the auxiliary space. These solutions are parametrized by…
A general functional definition of the infinite dimensional quantum $R$-matrix satisfying the Yang-Baxter equation is given. A procedure for the extracting a finite dimensional $R$-matrix from the general definition is demonstrated in a…
A general graded reflection equation algebra is proposed and the corresponding boundary quantum inverse scattering method is formulated. The formalism is applicable to all boundary lattice systems where an invertible R-matrix exists. As an…
In this paper, the different operator forms of classical Yang-Baxter equation are given in the tensor expression through a unified algebraic method. It is closely related to left-symmetric algebras which play an important role in many…