Related papers: Quiver-theoretical approach to dynamical Yang-Baxt…

We introduce non-degenerate solutions of the Yang-Baxter equation in the setting of symmetric monoidal categories. Our theory includes non-degenerate set-theoretical solutions as basic examples. However, infinite families of non-degenerate…

We introduce the notion of Leibniz conformal bialgebras, presenting a bialgebra theory for Leibniz conformal algebras as well as the conformal analogues of Leibniz bialgebras. They are equivalently characterized in terms of matched pairs…

We propose a new generalization of the Yang-Baxter equation, where the R-matrix depends on cluster $y$-variables in addition to the spectral parameters. We point out that we can construct solutions to this new equation from the…

We study simple set-theoretic solutions of the Yang-Baxter equation that are finite and non-degenerate. Such retractable solutions are fully described and to investigate the irretracble solutions we give a new algebraic method. Our approach…

We introduce the notion of an anti-Leibniz bialgebra which is equivalent to a Manin triple of anti-Leibniz algebras, is equivalent to a matched pair of anti-Leibniz algebras. The study of some special anti-Leibniz bialgebras leads to the…

We discuss the canonical structure of a class of integrable quantum mappings, i.e. iterative canonical transformations that can be interpreted as a discrete dynamical system. As particular examples we consider quantum mappings associated…

We construct rational and piecewise-linear Yang-Baxter maps for a general N-reduction of the discrete BKP equation.

We present the explicit form of a family of Liouville integrable maps in 3 variables, the so-called triad family of maps and we propose a multi-field generalisation of the latter. We show that by imposing separability of variables to the…

In the classification of solutions of the Yang--Baxter equation, there are solutions that are not deformations of the trivial solution (essentially the identity). We consider the algebras defined by these solutions, and the corresponding…

The notion of matching Rota-Baxter algebras was recently introduced by Gao, Guo and Zhang [{\em J. Algebra} 552 (2020) 134-170] motivated by the study of algebraic renormalization of regularity structures. The concept of matching…

The quantum alcove model associated to a dominant weight plays an important role in many branches of mathematics, such as combinatorial representation theory, the theory of Macdonald polynomials, and Schubert calculus. For a dominant…

The local Yang-Baxter equation (YBE), introduced by Maillet and Nijhoff, is a proper generalization to 3 dimensions of the zero curvature relation. Recently, Korepanov has constructed an infinite set of integrable 3-dimensional lattice…

In this paper we introduce the notion of a geometric associative r-matrix attached to a genus one fibration with a section and irreducible fibres. It allows us to study degenerations of solutions of the classical Yang-Baxter equation using…

Left-Alia algebras are a class of algebras with symmetric Jacobi identities. They contain several typical types of algebras as subclasses, and are closely related to the invariant theory. In this paper, we study the construction theory of…

We survey the matrix product solutions of the Yang-Baxter equation obtained recently from the tetrahedron equation. They form a family of quantum $R$ matrices of generalized quantum groups interpolating the symmetric tensor representations…

We construct an analogue of Yang--Baxter deformations defined by a single Killing vector, that is a solution generating transformation in Einstein--Maxwell dilaton theory. We show that these are nothing but a coordinate transformation in a…

In 1992 Drinfeld posed the question of finding the set theoretic solutions of the Yang-Baxter equation. Recently, Gateva-Ivanova and Van den Bergh and Etingof, Schedler and Soloviev have shown a group theoretical interpretation of…

This paper aims to deepen the theory of bijective non-degenerate set-theoretic solutions of the Yang-Baxter equation, not necessarily involutive, by means of q-cycle sets. We entirely focus on the finite indecomposable ones among which we…

We construct solutions to the set-theoretic Yang-Baxter equation using braid group representations in free group automorphisms and their Fox differentials. The method resembles the extensions of groups and quandles.

The focus of the paper is on constructing new solutions of the generalized classical Yang-Baxter equation (GCYBE) that are not skew-symmetric. Using regular decompositions of finite-dimensional simple Lie algebras, we construct Lie algebra…