Rota-Baxter operators on braces, post-braces and the Yang-Baxter equation
Li Guo, Yan Jiang, Yunhe Sheng, You Wang
TL;DR
This work develops a brace-theoretic generalization of Rota-Baxter theory by introducing relative Rota-Baxter operators on braces and their associated post-braces, enabling construction of set-theoretical Yang-Baxter equation solutions via Drinfel'd isomorphisms. It then introduces enhanced relative RBO operators and shows they yield matched pairs of braces, culminating in a factorization theorem for enhanced RBO two-sided braces that generalizes Lie group factorizations. The framework unifies braces, post-braces, and YBE within a single algebraic paradigm and is illustrated through concrete examples from the 3-dimensional Heisenberg Lie algebra. The results illuminate how brace-based RBOs can generate multiple, interrelated YBE solutions and provide structural decompositions akin to classical factorization results in integrable systems.
Abstract
Combining the notions of braces and relative Rota-Baxter operators on groups in connection with the Yang-Baxter equation and a factorization theorem of Lie groups from integrable systems, relative Rota-Baxter operators on braces and post-braces are introduced. A relative Rota-Baxter operator on a brace naturally induces a post-brace, and conversely, every post-brace determines a relative Rota-Baxter operator on its sub-adjacent brace. Furthermore, a post-brace yields two Drinfel'd-isomorphic solutions to the Yang-Baxter equation. As a special case, {\it enhanced} relative Rota-Baxter operators give rise to matched pairs of braces. Focusing on enhanced Rota-Baxter operators on two-sided braces, a corresponding factorization theorem is established. Examples are provided from the two-sided brace associated with the three-dimensional Heisenberg Lie algebra.