On compatible Leibniz algebras

Abstract

In this paper, we study compatible Leibniz algebras. We explore the classification of complex Leibniz algebras in dimensions 2 and 3 to provide compatible pairs. We characterize compatible Leibniz algebras in terms of Maurer–Cartan elements of a suitable differential graded Lie algebra. Moreover, we define a cohomology theory of compatible Leibniz algebras, which in particular controls one-parameter formal deformations of this algebraic structure. Motivated by a classical application of cohomology, we moreover study abelian extensions of compatible Leibniz algebras.