Reflections and Drinfeld twists for set-theoretic Yang-Baxter maps

TL;DR

This work links reflections for set-theoretic Yang–Baxter maps to Drinfeld twists in both set-theoretic and braided-group contexts. It introduces the $k$-guitar map and proves that reflections yield Drinfeld twists, yielding $r^{(k)} = J r J^{-1}$ in the set-theoretic case and, analogously, a group Drinfeld twist via the guitar map for braided groups; it also establishes optimality results for the reflection axioms. The paper develops the braided-action framework of De Commer to define group reflections and shows that these induce group Drinfeld twists, yielding a braided group ${G}^{(k)}$ with braiding $r^{(k)}$. Finally, it characterizes when a reflection on $(X,r)$ extends to a group reflection on the structure group $G(X,r)$, and it clarifies the structure of the groupoid of group Drinfeld twists, including extensions to rack-type braidings. Together, these results unify reflection theory with Drinfeld twists, provide constructive methods to generate new Yang–Baxter solutions, and illuminate representations of braid groups in both set-theoretic and group-thickened frameworks.

Abstract

We prove that reflections, for a solution to the set-theoretic Yang-Baxter equation, provide Drinfeld twists in the sense of Kulish and Mudrov. Using De Commer's notion of a braided action, we then define group reflections for a braided group, and prove that they provide group Drinfeld twists in the sense of Ghobadi. Finally, we characterise when a reflection on a solution (X,r) can be extended to a group reflection on the structure group G(X,r).

Figures (9)

• Figure 1: Pictorial representation of the braid relation. Each strand represents an element of $X$, and each crossing represents an application of $r$.
• Figure 2: Pictorial representation of the reflection equation. Each crossing represents an application of $r$, and each bouncing on a lateral wall represents an application of $k$. By the ybe and the re, these diagrams can be considered up to homotopies that never drive the strands beyond the wall.
• Figure 3: Extending $r_{V,V}$ (here depicted as a crossing) to a map $r_{V^{\otimes n}, V^{\otimes m}}$ (in this case $n = 4$ and $m = 3$). Notice that, in the above diagram, only interactions of the form $r_{i, i+1}$ appear: thus the extension is well-defined in every monoidal category.
• Figure 4: If $X$ generates $G$, and $k|_X$ satisfies \ref{['eq:weird']} for all $a,b\in X$, then $k$ satisfies \ref{['eq:weird']} for all $a,b\in G$. The proof is by double induction on the number of generators in the expressions of $a$ and $b$. In the picture, we see the case when $a$ is the product of two generators, and $b$ is the product of three generators.
• Figure 5: Graphic depiction of how $r$ and $k$ are extended to the free group $\mathop{\mathrm{\mathrm{Free}}}\nolimits(X)$ and to the structure group $G(X,r)$.

Theorems & Definitions (60)

• Definition 2.1: kulish2000twisting
• Lemma 2.2
• proof
• Remark 2.3
• Definition 2.5: DoikouQuandlesAsPreLie
• Definition 2.6
• Remark 2.8
• Proposition 2.9
• proof
• Theorem 3.2