Idempotent solutions of the Yang-Baxter equation and twisted group division

Quasi-idempotent Rota-Baxter operators arising from quasi-idempotent elements

In this short note, we construct quasi-idempotent Rota-Baxter operators by quasi-idempotent elements and show that every finite dimensional Hopf algebra admits nontrivial Rota-Baxter algebra structures and tridendriform algebra structures. Several concrete examples are provided, including finite quantum groups and Iwahori-Hecke algebras.

Construction of Rota-Baxter algebras via Hopf module algebras

We propose the notion of Hopf module algebras and show that the projection onto the subspace of coinvariants is an idempotent Rota-Baxter operator of weight -1. We also provide a construction of Hopf module algebras by using Yetter-Drinfeld module algebras. As an application, we prove that the positive part of a quantum group admits idempotent Rota-Baxter algebra structures.

二次代数与幂等辫子集合

Quadratic algebras and idempotent braided sets

本文研究有限集合论辫子解对应的 Yang-Baxter 代数，证明左非退化幂等情形下该代数为 PBW 并分析其整体维数、Veronese 子代数及 Segre 积，同时构造 Koszul 对偶与非交换微分。

The matched product of the solutions to the Yang-Baxter equation of finite order

In this work, we focus on the set-theoretical solutions of the Yang-Baxter equation which are of finite order and not necessarily bijective. We use the matched product of solutions as a unifying tool for treating these solutions of finite order, that also include involutive and idempotent solutions. In particular, we prove that the matched product of two solutions $r_S$ and $r_T$ is of finite order if and only if $r_S$ and $r_T$ are. Furthermore, we show that with sufficient information on $r_S$ and $r_T$ we can precisely establish the order of the matched product. Finally, we prove that if $B…

Left semi-braces and solutions to the Yang-Baxter equation

Let $r:X^{2}\rightarrow X^{2}$ be a set-theoretic solution of the Yang-Baxter equation on a finite set $X$. It was proven by Gateva-Ivanova and Van den Bergh that if $r$ is non-degenerate and involutive then the algebra $K\langle x \in X \mid xy =uv \mbox{ if } r(x,y)=(u,v)\rangle$ shares many properties with commutative polynomial algebras in finitely many variables; in particular this algebra is Noetherian, satisfies a polynomial identity and has Gelfand-Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. S…

广义digroup、di-skew brace与集合论Yang-Baxter方程的解

Generalized digroups, di-skew braces, and solutions of the set-theoretic Yang-Baxter equation

本文提出di-skew brace结构，证明广义digroup可生成集合论Yang-Baxter方程的非退化解，并通过半直积分解和平均算子提供具体实例。