Related papers: On the dynamical Yang-Baxter equation

We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld…

We reformulate the method recently proposed for constructing quasitriangular Hopf algebras of the quantum-double type from the R-matrices obeying the Yang-Baxter equations. Underlying algebraic structures of the method are elucidated and an…

This paper is a continuation of [KS]. We develop the results of [KS] principally in two directions. First, we generalize the main result of [KS], the connection between the solutions of the classical dynamical Yang-Baxter equation and…

Let $ L $ be a Lie algebra over arbitrary field $ k $ with dim $ L $ =3 and dim $ L' $ =2. All solutions of constant classical Yang- Baxter equation (CYBE) in Lie algebra $ L $ are obtained and the necessary conditions which $ (L,[\…

We show explicitly a generalised Lie algebra embedded in the positive and negative parts of the Drinfeld-Jimbo quantum groups of type A_n. Such a generalised Lie algebra satisfy axioms closely related to the ones found by S.L. Woronowicz.…

In these introductory lectures we discuss the topic of Yangian symmetry from various perspectives. Forming the classical counterpart of the Yangian and an extension of ordinary Noether symmetries, first the concept of nonlocal charges in…

A fundamental construction of Poisson algebras is to derive them as the quasiclassical limits (QCLs) of associative algebra deformations of commutative associative algebras. This paper lifts this process to the level of classical…

A new quantization of groupoids under the name of \times-Hopf coalgebras is introduced. We develop a Hopf cyclic theory with coefficients in stable-anti-Yetter-Drinfeld modules for \times-Hopf coalgebras. We use \times-Hopf coalgebras to…

We introduce the notion of a braided dynamical group which is a matched pair of dynamical groups satisfying extra conditions. It is shown to give a solution of the dynamical Yang-Baxter equation and at the same time a braided groupoid,…

This paper studies operator forms of the nonhomogeneous associative classical Yang-Baxter equation (nhacYBe), extending and generalizing such studies for the classical Yang-Baxter equation and associative Yang-Baxter equation that can be…

In this paper, by making use of category theory, we construct dynamical reflection maps, solutions to a version of the reflection equation associated with suitable dynamical Yang-Baxter maps, set-theoretic solutions to the braid relation…

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…

Set-theoretic solutions to the Yang-Baxter equation have been studied extensively by means of related algebraic systems such as cycle sets and braces, dynamical versions of which have also been developed. No work focuses on set-theoretic…

The kinematical foundations of Schwinger's algebra of selective measurements were discussed in a previous paper (arXiv:1905.12274) and, as a consequence of this, a new picture of quantum mechanics based on groupoids was proposed. In this…

Let $k$ be a field and $X$ be a set of $n$ elements. We introduce and study a class of quadratic $k$-algebras called \emph{quantum binomial algebras}. Our main result shows that such an algebra $A$ defines a solution of the classical…

We establish a one-to-one correspondence between a class of Garside groups admitting a certain presentation and the structure groups of non-degenerate, involutive and braided set-theoretical solutions of the quantum Yang-Baxter equation. We…

A completely integrable dynamical system in discrete time is studied by means of algebraic geometry. The system is associated with factorization of a linear operator acting in a direct sum of three linear spaces into a product of three…

We consider the modified (or twisted) Yang-Baxter equations for the $SL_{q}(N)$ groups and $SL_{q}(N|M)$ supergroups. The general solutions for these equations are presented in the case of the linear quantum (super)groups. The introduction…

The (G, \theta)-Lie algebras are structures which unify the Lie algebras and Lie superalgebras. We use them to produce solutions for the quantum Yang-Baxter equation. The constant and the spectral-parameter Yang-Baxter equations and…

In this paper, we extend results connecting quantum groups to spherical Whittaker functions on metaplectic covers of $GL_r(F)$, for $F$ a nonarchimedean local field. Brubaker, Buciumas, and Bump showed that for a certain metaplectic…