Relative Rota-Baxter operators and crossed homomorphisms on Lie 2-groups
Honglei Lang, Shining Wang
TL;DR
This work categorifies relative Rota-Baxter operators from Lie groups to Lie 2-groups, defining a pair $(\mathfrak{B},\mathfrak{B}_0)$ that is simultaneously a relative RBO on the underlying groups and a Lie 2-group morphism. It develops a factorization theorem via graphs and Cayley transforms, and constructs categorical Yang-Baxter equation solutions from these operators. A central result is the established equivalence between relative RBOs on Lie 2-groups and those on Lie group crossed modules, together with an infinitesimal theory on Lie algebra crossed modules. The paper also develops the dual notion of crossed homomorphisms as inverses of relative RBOs and studies their derived actions, providing a coherent higher-categorical framework for RBOs, 2-group actions, and YBE phenomena with potential applications in categorified algebra and higher gauge theory.
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.