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

Abstract

A Rota-Baxter Lie algebra ${\mathfrak{g}}_T$ is a Lie algebra $\mathfrak{g}$ equipped with a Rota-Baxter operator $T:\mathfrak{g}\to \mathfrak{g}$. 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 $H_{nab}^2({\mathfrak{g}}_T,{\mathfrak{h}}_S)$ which classifies {equivalence classes of} such extensions. Given a non-abelian extension $$0\to {\mathfrak{h}}_S\stackrel{i}{\to }{\mathfrak{e}}_U\stackrel{p}{\to }{\mathfrak{g}}_T\to 0$$ 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 $H_{nab}^2({\mathfrak{g}}_T,{\mathfrak{h}}_S)$. As a byproduct, we obtain the Wells short-exact sequence in the context of Rota-Baxter Lie algebras.

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Cited by in corpus (1)

• Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras