Kupershmidt operators and related structures on Leibniz algebras

Abstract

Kupershmidt operator is a key to extend a Leibniz algebra by its representation. In this paper, we investigate several structures related to Kupershmidt operators on Leibniz algebras and introduce (dual) KN-structures on a Leibniz algebra associated to a representation. It is proved that Kupershmidt operators and dual KN-structures can generate each other with certain conditions. It is also shown that a solution of the strong Maurer-Cartan equation on the twilled Leibniz algebra gives rise to a dual KN-structure. Finally, the notions of r-n structures,RBN-structures and BN-structures on Leibniz algebras are thoroughly studied and shown to bear interesting interrelations.Comment: 23 page