Deformation of Kupershmidt operators and Kupershmidt-Nijenhuis structures of a Malcev algebra

Abstract

The aim of this paper is to study infinitesimal deformations of a Malcev algebra with a representation and introduce the notion of Nijenhuis pair, which gives a trivial deformation of a Malcev algebra with a representation. We introduce the notion of Kupershmidt-(dual-)Nijenhuis structure on a Malcev algebra with a representation. Furthermore, we show that a Kupershmidt-(dual-)Nijenhuis structure gives rise to a hierarchy of Kupershmidt operators. Finally, we establish a deformation theory of Kupershmidt operators in consistence with the general principles of deformation theories and introduce the notion of Nijenhuis elements.