Kupershmidt-(dual-)Nijenhuis structures on a Lie algebra with a representation

Abstract

In this paper, first we study infinitesimal deformations of a Lie algebra with a representation and introduce the notion of a Nijenhuis pair, which gives a trivial deformation of a Lie algebra with a representation. Then we introduce the notion of a Kupershmidt-(dual-)Nijenhuis structure on a Lie algebra with a representation, which is a generalization of the $r$-$n$ structure ($r$-matrix-Nijenhuis structure) introduced by Ravanpak, Rezaei-Aghdam and Haghighatdoost. We show that a Kupershmidt-(dual-)Nijenhuis structure gives rise to a hierarchy of Kupershmidt operators. Finally, we define a Rota-Baxter-Nijenhuis structure to be a Kupershmidt-Nijenhuis structure on a Lie algebra with respect to the adjoint representation, and study the relation between Rota-Baxter-Nijenhuis structures and $r$-matrix-Nijenhuis structures.

arXiv comment16 pages

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Cited by in corpus (5)

• Kupershmidt operators on Hom-Malcev algebras and their deformation
• Conformal $r$-matrix-Nijenhuis structures, symplectic-Nijenhuis structures and $\mathcal{O}N$-structures
• $\mathcal{O}$-operators and related structures on Leibniz algebras
• Cohomology and relative Rota-Baxter-Nijenhuis structures on LieYRep pairs
• Kupershmidt-Nijenhuis structures on pre-Malcev algebras