Post groups, post-groupoids and the Yang–Baxter equation

In this talk, first we recall the notion of a post-group. By differentiation of a post-Lie group, one can obtain a post-Lie algebra, which was introduced by Vallette in 2007 and have important applications in numerical integration on manifolds and Martin Hairer's regularity structures. There are close relationships between post-groups, Rota–Baxter groups, skew-left braces and Lie–Butcher groups. In particular, post-groups give rise to matched pairs of groups, and can be used to construct set-theoretical solutions of the Yang–Baxter equation. Then we introduce the notion of a post-groupoid. A post-groupoid is a group bundle equipped with another binary operation. The section space of a post-groupoid is a weak post group. A post-groupoid gives rise to a groupoid and an action on the original group bundle. By differentiation of a post-Lie groupoid, one can obtain a post-Lie algebroid, which was introduced by Munthe–Kaas and Lundervold in the study of geometric numerical analysis. Finally, we show that post-groupoids provide solutions of the Yang–Baxter equation on quivers. This is a joint work with Chengming Bai, Li Guo, Rong Tang and Chenchang Zhu.