Non-abelian extensions of Rota-Baxter Lie algebras and inducibility of automorphisms

TL;DR

This paper develops a cohomological framework for classifying non-abelian extensions of Rota-Baxter Lie algebras and investigates automorphism inducibility within these extensions.

Contribution

It introduces a non-abelian cohomology theory for Rota-Baxter Lie algebra extensions and analyzes automorphism inducibility via this cohomology.

Findings

Defined the non-abelian cohomology $H^2_{nab}$ for Rota-Baxter Lie algebra extensions.

Established the obstruction class for automorphism inducibility in $H^2_{nab}$.

Derived a Wells short-exact sequence in the context of Rota-Baxter Lie algebras.

Abstract

A Rota-Baxter Lie algebra $\mathfrak{g}_T$ is a Lie algebra $\mathfrak{g}$ equipped with a Rota-Baxter operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. In this paper, we consider non-abelian extensions of a Rota-Baxter Lie algebra $\mathfrak{g}_T$ by another Rota-Baxter Lie algebra $\mathfrak{h}_S.$ We define the non-abelian cohomology $H^2_{nab} (\mathfrak{g}_T, \mathfrak{h}_S)$ which classifies {equivalence classes of} such extensions. Given a non-abelian extension $$ 0 \rightarrow \mathfrak{h}_S \xrightarrow{i} \mathfrak{e}_U \xrightarrow{p} \mathfrak{g}_T \rightarrow 0$$ of Rota-Baxter Lie algebras, we also show that the obstruction for a pair of Rota-Baxter automorphisms in $\mathrm{Aut}(\mathfrak{h}_S ) \times \mathrm{Aut}(\mathfrak{g}_T)$ to be induced by an automorphism in $\mathrm{Aut}(\mathfrak{e}_U)$ lies in the cohomology group $H^2_{{nab}} (\mathfrak{g}_T, \mathfrak{h}_S)$. As a byproduct, we obtain the Wells short-exact sequence in the context of Rota-Baxter Lie algebras.