Kupershmidt-(dual-)Nijenhuis structure on the alternative algebra with a representation

Abstract

The aim of this paper is to study Kupershmidt-(dual-)Nijenhuis structures on alternative algebras with representations. The notion of a (dual-)Nijenhuis pair is introduced and it can generate a trivial deformation of an alternative algebra with a representation. We introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on an alternative algebra with a representation. Furthermore, we verify that Kupershmidt operators and Kupershmidt-(dual-)Nijenhuis structures can give rise to each other under some conditions. Finally, we study the notions of Rota–Baxter–Nijenhuis structures and alternative [Formula: see text]-matrix-Nijenhuis structures. Meanwhile, we investigate the relation between them.