Related papers: Reflections and Drinfeld twists for set-theoretic …
Sufficient conditions for an invertible two-tensor $F$ to relate two solutions to the Yang-Baxter equation via the transformation $R\to F^{-1}_{21} R F$ are formulated. Those conditions include relations arising from twisting of certain…
The Donald--Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and for the classical Weyl groups (whose rational group…
We construct universal Drinfel'd twists defining deformations of Hopf algebra structures based upon simple Lie algebras and contragredient simple Lie superalgebras. In particular, we obtain deformed and dynamical double Yangians. Some…
We study set-theoretic solutions $(X,r)$ of the Yang-Baxter equations on a set $X$ in terms of the induced left and right actions of $X$ on itself. We give a characterization of involutive square-free solutions in terms of cyclicity…
We investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang-Baxter equation. Specifically, a weak (left) brace is a non-empty set $S$ endowed with two binary operations $+$ and $\circ$ such that…
Solutions to the Yang-Baxter equation - an important equation in mathematics and physics - and their afforded braid group representations have applications in fields such as knot theory, statistical mechanics, and, most recently, quantum…
The quantum Yang-Baxter equation is a braiding condition on vector spaces which is of high relevance in several fields of mathematics, such as knot theory and quantum group theory. Their combinatorial counterpart are set-theoretic solutions…
New trigonometric and rational solutions of the quantum Yang-Baxter equation (QYBE) are obtained by applying some singular gauge transformations to the known Belavin-Drinfeld elliptic R-matrix for $sl(2,\mathbb{C})$. These solutions are…
We study tensor structures on (Rep G)-module categories defined by actions of a compact quantum group G on unital C*-algebras. We show that having a tensor product which defines the module structure is equivalent to enriching the action of…
We introduce a braid group action on $l$-tuple of rational functions for the finite-dimensional representations of Yangians $Y(\mathfrak{g})$, where $\mathfrak{g}$ is a complex simple Lie algebra. It provides an efficient way to compute…
Let $H$ be a Hopf algebra that is $\mathbb Z$-graded as an algebra. We provide sufficient conditions for a 2-cocycle twist of $H$ to be a Zhang twist of $H$. In particular, we introduce the notion of a twisting pair for $H$ such that the…
Skew braces have recently attracted attention as a method to study set-theoretical solutions of the Yang-Baxter equation. Here, we present a new approach to these solutions by studying Hopf algebras in the category, $\mathrm{SupLat}$, of…
In the first part of this paper, we investigate the retraction of finite uniconnected involutive non-degenerate set-theoretic solutions of the Yang-Baxter equation by means of left braces, giving a precise description in some cases. In the…
The Yang-Baxter Equation (YBE) plays a crucial role for studying integrable many-body quantum systems. Many known YBE solutions provide various examples ranging from quantum spin chains to superconducting systems. Models of solvable…
We show that the braid group associated to the complex reflection group $G(d,d,n)$ is an index $d$ subgroup of the braid group of the orbifold quotient of the complex numbers by a cyclic group of order $d$. We also give a compatible…
We find the general solution to the twisting equation in the tensor bialgebra $T({\bf R})$ of an associative unital ring ${\bf R}$ viewed as that of fundamental representation for a universal enveloping Lie algebra and its quantum…
We connect properties of set-theoretic solutions to the Yang--Baxter equation to properties of their permutation skew brace. In particular, a variation of the multipermutation level of a solution is presented and we show that it coincides…
Every unitary solution of the Yang-Baxter equation (R-matrix) in dimension $d$ can be viewed as a unitary element of the Cuntz algebra ${\mathcal O}_d$ and as such defines an endomorphism of ${\mathcal O}_d$. These Yang-Baxter endomorphisms…
Relative Rota--Baxter groups, a generalization of Rota--Baxter groups, are closely connected to skew left braces, which play a fundamental role in understanding non-degenerate set-theoretical solutions to the Yang-Baxter equation. In this…
The aim of this paper is to define and study Drinfeld twists for monoidal Hom-bialgebras. We show that a new Hom-bialgebra could be constructed by changing the coproduct of a monoidal Hom-bialgebra via a Drinfeld twist, and this…