Extended Rota-Baxter Operators on Leibniz Algebras

Abstract

In this paper, we obtain the notion of an extended Rota-Baxter operator that generalizes the Rota-Baxter operator and the modified Rota-Baxter operator on Leibniz algebras, and give examples and general properties of extended Rota-Baxter operators. Next, we define a representation theory of extended Rota-Baxter Leibniz algebras and construct a cohomology theory. Furthermore, we justify this cohomology theory by interpreting lower degree cohomology groups as formal deformations of extended Rota-Baxter Leibniz algebras. Finally, we study non-abelian extensions of an extended Rota-Baxter Leibniz algebra by another extended Rota-Baxter Leibniz algebra, and we consider abelian extensions of extended Rota-Baxter Leibniz algebras as a particular case of non-abelian extensions.

Keywords

Leibniz algebra / extended Rota-Baxter operator / cohomology / formal deformation / non-abelian extension / 17A32 / 17B99

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Peking University