Twisting of graded quantum groups and solutions to the quantum Yang-Baxter equation

TL;DR

This paper explores how certain algebraic twists of graded quantum groups relate to solutions of the quantum Yang-Baxter equation, introducing new concepts like twisting pairs to unify different twisting methods.

Contribution

It introduces the notion of twisting pairs for graded Hopf algebras, showing they can produce 2-cocycle twists and relate to Zhang twists, with applications to quantum groups and the Yang-Baxter equation.

Findings

Twisting pairs can generate Zhang twists from 2-cocycle twists.

The paper describes twists of Manin's universal quantum groups.

Solutions to the quantum Yang-Baxter equation are obtained via these twists.

Abstract

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 Zhang twist of $H$ by such a pair is a 2-cocycle twist. We use twisting pairs to describe twists of Manin's universal quantum groups associated to quadratic algebras and provide twisting of solutions to the quantum Yang-Baxter equation via the Faddeev-Reshetikhin-Takhtajan construction.