Derivation extensions on Leibniz triple systems

Abstract

In this paper, we first define a concrete representation on an abelian extension of a Leibniz triple system L by a Leibniz triple system A. Using this new representation we construct the third-order cohomology classes by derivations of A and L, which characterize the splitting property of above abelian extensions. Then we study the obstruction for extensibility of derivation pairs. We prove that the set of compatible derivation pairs can define a Lie algebra, whose representation can also characterize the extensibility of the compatible derivation pairs.