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

TL;DR

This paper develops a unified extension theory for Rota-Baxter and dendriform algebras, introducing non-abelian cohomology and Wells type sequences to classify and analyze their automorphisms.

Contribution

It introduces a non-abelian extension framework and Wells type sequences for Rota-Baxter algebras, connecting them with dendriform algebras and automorphism lifting.

Findings

Classified non-abelian extensions via cohomology.

Constructed Wells type exact sequences for Rota-Baxter algebras.

Extended the theory to dendriform algebras.

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 $0 \rightarrow B_S \rightarrow E_U \rightarrow A_R \rightarrow 0$ of Rota-Baxter algebras, we construct the Wells type exact sequences and find their role in extending a Rota-Baxter automorphism $\beta \in \mathrm{Aut}(B_S)$ and lifting a Rota-Baxter automorphism $\alpha \in \mathrm{Aut}(A_R)$ to an automorphism in $\mathrm{Aut}(E_U)$. We end this paper by considering a similar study for dendriform algebras.