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

Abstract

A Rota-Baxter Lie algebra g T is a Lie algebra g equipped with a Rota-Baxter operator T : g → g.In this paper, we consider non-abelian extensions of a Rota-Baxter Lie algebra g T by another Rota-Baxter Lie algebra h S . We define the non-abelian cohomology H 2 nab (g T , h S ) which classifies equivalence classes of such extensions. Given a non-abelian extension 0of Rota-Baxter Lie algebras, we also show that the obstruction for a pair of Rota-Baxter automorphisms in Aut(h S )×Aut(g T ) to be induced by an automorphism in Aut(e U ) lies in the cohomology group H 2 nab (g T , h S ). As a byproduct, we obtain the Wells short-exact sequence in the context of Rota-Baxter Lie algebras. Finally, we show how these results fit with abelian extensions of Rota-Baxter Lie algebras. Contents 1. Introduction 1 2. Rota-Baxter Lie algebras 2 3. Non-abelian Extensions of Rota-Baxter Lie algebras 4 4. Inducibility of pair of automorphisms 8 5. Wells exact sequence for Rota-Baxter Lie algebras 11 6. Particular case: Abelian extensions of Rota-Baxter Lie algebras 14 References 18 i − → e p − → g → 0 be an (abelian)