Nijenhuis operators on Hom-Lie algebras

Abstract

In this article, we study Nijenhuis operators on Hom-Lie algebras. We construct a graded Lie algebra (via the Hom-analog of the Frölicher-Nijenhuis bracket) whose Maurer-Cartan elements are given by Nijenhuis operators. This allows us to define the cohomology associated to a Nijenhuis operator. As an application, we study formal deformations of Nijenhuis operators that are generated by the above-defined cohomology. Finally, we introduce Hom-NS-Lie algebras as an algebraic structure behind Nijenhuis operators on Hom-Lie algebras. We provide various examples of Hom-NS-Lie algebras.

Keywords:

• Frölicher-Nijenhuis bracket
• formal deformation
• Hom-Lie algebra
• Hom-NS-Lie algebra
• Nijenhuis operator

2020 Mathematics Subject Classification:

• 17B61
• 17B40
• 17B70
• 16S80

Acknowledgments

The authors thank the referee for his/her pertinent suggestions. The second author acknowledges HRI, Allahabad for their postdoctoral fellowship.

Additional information

Funding

The research of the first author is supported by the postdoctoral fellowship of Indian Institute of Technology (IIT) Kanpur.