Cartier–Gabriel–Kostant theorem for relative Rota–Baxter operators

Abstract

Let H and K be two cocommutative Hopf algebras over an algebraically closed field $F$ of characteristic 0. In this paper, we first prove any relative Rota–Baxter operator $T:K⟶H$ to be isomorphic to a relative Rota–Baxter operator $O_T:U\left(P\right(K\left)\right)♯F\left[G\right(K\left)\right]⟶U\left(P\right(H\left)\right)♯F\left[G\right(H\left)\right]$, which is determined by a relative Rota–Baxter operator $R:P\left(K\right)⟶P\left(H\right)$ on the Lie algebra $P\left(H\right)$, and a relative Rota–Baxter operator $B:G\left(K\right)⟶G\left(H\right)$ on the group $G\left(H\right)$. Next we prove that Rota–Baxter operators on H are in one to one correspondence with Rota–Baxter pairs on the Lie algebra $P\left(H\right)$ and the group $G\left(H\right)$. These results should be viewed as the Rota–Baxter operator version of classical Cartier–Gabriel–Kostant structure theorem. Finally, we prove that this well-known structure theorem also holds for post-Hopf algebras and Hopf braces.