A unified extension theory of Rota-Baxter algebras, dendriform algebras, and a fundamental sequence of Wells

A Rota-Baxter algebra $A_R$ is an algebra $A$ equipped with a distinguished Rota-Baxter operator $R$ on it. Rota-Baxter algebras are closely related to dendriform algebras introduced by Loday. In this paper, we first consider the non-abelian extension theory of Rota-Baxter algebras and classify them by introducing the non-abelian cohomology. Next, given a non-abelian extension $0\to B_S\to E_U\to A_R\to 0$ of Rota-Baxter algebras, we construct the Wells type exact sequences and find their role in extending a Rota-Baxter automorphism $\beta \in \mathrm{A}\mathrm{u}\mathrm{t}(B_S)$ and lifting a Rota-Baxter automorphism $\alpha \in \mathrm{A}\mathrm{u}\mathrm{t}(A_R)$ to an automorphism in $\mathrm{A}\mathrm{u}\mathrm{t}(E_U)$. We end this paper by considering a similar study for dendriform algebras.