[论文解读] Relative Rota-Baxter operators and crossed homomorphisms on Lie 2-groups

Honglei Lang, Shining Wang|arXiv (Cornell University)|Feb 2, 2026

Advanced Topics in Algebra被引用 0

一句话总结

简要结论：论文在 Lie 2-群上定义相对 Rota-Baxter 运算符，证明其与 Lie 群交叉模上的相对 Rota-Baxter 运算符等价，建立因式分解定理与范畴杨–贝特征方程的解，并研究在此设定下的交叉同态作为相对 Rota-Baxter 运算符的逆。

ABSTRACT

A relative Rota-Baxter operator on Lie 2-groups is introduced as a pair of relative Rota-Baxter operators on the underlying Lie groups which is also a Lie groupoid morphism. Such an operator induces a factorization theorem for Lie 2-groups and gives rise to a categorical solution of the Yang-Baxter equation. We further define relative Rota-Baxter operators on Lie group crossed modules. The well-known one-to-one correspondence between Lie 2-groups and crossed modules is extended to an equivalence between the respective relative Rota-Baxter operators on these two structures. Finally, as the formal inverse of relative Rota-Baxter operators, crossed homomorphisms on Lie 2-groups are also studied.

研究动机与目标

• Motivate the study of categorified Rota-Baxter structures on Lie 2-groups and crossed modules.
• Define relative Rota-Baxter operators on Lie 2-groups and relate them to actions on Lie 2-groups.
• Establish an equivalence between relative Rota-Baxter operators on Lie 2-groups and on Lie group crossed modules.
• Develop a factorization theorem for Rota-Baxter Lie 2-groups and construct categorical solutions to the Yang–Baxter equation.
• Study crossed homomorphisms on Lie 2-groups as the formal inverses of relative Rota-Baxter operators.]
• method1:Introduce a notion of relative Rota-Baxter operators on Lie 2-groups via a pair of relative Rota-Baxter operators on underlying Lie groups and a Lie groupoid morphism.
• method2:Characterize such operators through the semi-direct product Lie 2-group and prove an equivalence with graphs forming Lie 2-subgroups (Theorem 3.9).
• method3:Show that a relative Rota-Baxter operator yields a descendant Lie 2-group and an induced action (Theorem 3.10).
• method4:Derive a factorization theorem for Rota-Baxter Lie 2-groups (Theorem 3.17) and construct categorical solutions of the Yang–Baxter equation (Theorem 3.18).
• method5:Extend the theory to relative Rota-Baxter operators on Lie group crossed modules and establish a bijection with the operator theory on Lie 2-groups (Theorem 4.6).
• method6:Discuss the infinitesimal counterpart and connections to Lie algebra crossed modules (Theorem 4.18).
• method7:Introduce crossed homomorphisms on Lie 2-groups as formal inverses of relative Rota-Baxter operators (Theorem 5.6).

实验结果

研究问题

• RQ1 How can relative Rota-Baxter operators be defined for Lie 2-groups and what are their main structural properties?
• RQ2 What is the relationship between relative Rota-Baxter operators on Lie 2-groups and those on Lie group crossed modules?
• RQ3 Can we obtain a factorization theorem and categorical Yang–Baxter solutions from these operators?
• RQ4 What is the role of crossed homomorphisms as inverses of relative Rota-Baxter operators in this higher-categorical setting?
• RQ5 What are the infinitesimal (Lie algebra) analogues of the relative Rota-Baxter theory for crossed modules?

主要发现

• A precise definition of relative Rota-Baxter operators on Lie 2-groups is given via a pair of operators on the underlying Lie groups together with a Lie groupoid morphism.
• A semi-direct product construction provides a characterization: the operator corresponds to a Lie 2-subgroup of the semi-direct product (Gr(B) is a Lie 2-subgroup).
• The theory yields a descendant Lie 2-group and an induced action, linking the operator to new Lie 2-group structures (Theorem 3.10).
• A factorization theorem for Rota-Baxter Lie 2-groups is established, paralleling Lie group factorization results (Theorem 3.17).
• Categorical solutions to the Yang–Baxter equation are constructed from relative Rota-Baxter operators (Theorem 3.18).
• Relative Rota-Baxter operators on Lie 2-groups correspond bijectively to relative Rota-Baxter operators on Lie group crossed modules, with an infinitesimal counterpart explored (Theorem 4.6 and 4.18).
• Crossed homomorphisms on Lie 2-groups are studied as the formal inverse of relative Rota-Baxter operators (Theorem 5.6).