On Hom-Lie antialgebra

Abstract

In this paper, we introduced the notion of Hom-Lie antialgebras. The representations and cohomology theory of Hom-Lie antialgebras are investigated. We prove that the equivalent classes of abelian extensions of Hom-Lie antialgebras are in one-to-one correspondence to elements of the second cohomology group. We also prove that 1-parameter infinitesimal deformation of a Hom-Lie antialgebra are characterized by 2-cocycles of this Hom-Lie antialgebra with adjoint representation in itself. The notion of Nijenhuis operators of Hom-Lie antialgebra is introduced to describe trivial deformations.

Communicated by Dr. Pavel Kolesnikov

Keywords:

• Abelian extensions
• cohomology
• deformations
• Hom-Lie antialgebra
• Nijenhuis operators

2010 Mathematics Subject Classification:

• Primary 17D99
• Secondary 18G60

Acknowledgments

The author would like to thank the referee for careful reading of the manuscript and for valuable suggestions which helped us both in English and in depth to improve the quality of the paper.

Additional information

Funding

This research was supported by NSFC (11501179, 11961049) and a doctoral research program of Henan Normal University.