Related papers: Rota-Baxter groups, skew left braces, and the Yang…
Several years ago, it was proposed that the usual solutions of the Yang-Baxter equation associated to Lie groups can be deduced in a systematic way from four-dimensional gauge theory. In the present paper, we extend this picture, fill in…
An augmented rack is a set with a self-distributive binary operation induced by a group action, and has been extensively used in knot theory. Solutions to the Yang-Baxter equation (YBE) have been also used for knots, since the discovery of…
Skew braces are intensively studied owing to their wide ranging connections and applications. We generalize the definition of a skew brace to give a new algebraic object, which we term a skew bracoid. Our construction involves two groups…
Any solution to the Yang-Baxter equation yields a family of representations of braid groups. Under certain conditions, identified by Turaev, the appropriately normalized trace of these representations yields a link invariant. Any…
In 1992 V$.$Drinfeld formulated a number of problems in quantum group theory. In particular, he suggested to consider ``set-theoretical'' solutions to the quantum Yang-Baxter equation, i.e. solutions given by a permutation R of the set…
In this paper we determine all the Rota-Baxter operators of weight zero on semigroup algebras of order two and three with the help of computer algebra. We determine the matrices for these Rota-Baxter operators by directly solving the…
We establish a bialgebra structure on Rota-Baxter Lie algebras following the Manin triple approach to Lie bialgebras. Explicitly, Rota-Baxter Lie bialgebras are characterized by generalizing matched pairs of Lie algebras and Manin triples…
The notion of a modified Rota-Baxter algebra comes from the combination of those of a Rota-Baxter algebra and a modified Yang-Baxter equation. In this paper, we first construct free modified Rota-Baxter algebras. We then equip a free…
The Yang-Baxter and pentagon equations are two well-known equations of Mathematical Physic. If $S$ is a set, a map $s:S\times S\to S\times S$ is said to be a set theoretical solution of the Yang-Baxter equation if $$ s_{23}\, s_{13}\,…
Recently, Li, Sheng and Tang introduced post-Hopf algebras and relative Rota-Baxter operators (on cocommutative Hopf algebras), providing an adjunction between the respective categories under the assumption that the structures involved are…
We introduce skein theoretic techniques to compute the Yang-Baxter (YB) homology and cohomology groups of the R-matrix corresponding to the Jones polynomial. More specifically, we show that the YB operator $R$ for Jones, normalized for…
Conformal classical Yang-Baxter equation and $S$-equation naturally appear in the study of Lie conformal bialgebras and left-symmetric conformal bialgebras. In this paper, they are interpreted in terms of a kind of operators, namely,…
The correspondence of the braid group on a handlebody of arbitrary genus to the algebra of Yang-Baxter and extended reflection equation operators is shown. Representations of the infinite dimensional extended reflection equation algebra in…
In 2004, Carter, Elhamdadi and Saito defined a homology theory for set-theoretic Yang-Baxter operators(we will call it the "algebraic" version in this article). In 2012, Przytycki defined another homology theory for pre-Yang-Baxter…
As Hopf truss analogues of Rota-Baxter Hopf algebras, the notion of Rota-Baxter systems of Hopf algebras is proposed. We study the relatiohship between Rota-Baxter systems of Hopf algebras and Rota-Baxter Hopf algebras, show that there is a…
A complete description of all possible multiplicative groups of finite skew left braces whose additive group has trivial centre is shown. As a consequence, some earlier results of Tsang can be improved and an answer to an open question set…
Set-theoretic solutions of the Yang--Baxter equation form a meeting-ground of mathematical physics, algebra and combinatorics. Such a solution consists of a set $X$ and a function r:X x X --> X x X which satisfies the braid relation. We…
This talk is inspired by two previous ICM talks, by V.Drinfeld (1986) and G.Felder (1994). Namely, one of the main ideas of Drinfeld's talk is that the quantum Yang-Baxter equation (QYBE), which is an important equation arising in quantum…
In the paper, we introduce the notion of a Rota-Baxter operator of a non-scalar weight. As a motivation, we show that there is a natural connection between Rota-Baxter operators of this type and structures of quasitriangular Lie bialgebras…
We study Rota--Baxter operators on vertex algebras using the integrated $\lambda$-bracket formalism. A Rota--Baxter operator produces a deformed vertex algebra structure, and the difference between the deformed and original brackets yields…