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

Abstract

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 → B S → E U → A R → 0 of Rota-Baxter algebras, we construct the Wells type exact sequences and find their role in extending a Rota-Baxter automorphism β ∈ Aut(B S ) and lifting a Rota-Baxter automorphism α ∈ Aut(A R ) to an automorphism in Aut(E U ). We end this paper by considering a similar study for dendriform algebras.