Non-abelian Omni-Lie 2-algebras

In this paper, we study the structure of non-abelian omni-Lie 2-algebras. Firstly, we define a \textG -valued pairing and a bracket operation on the direct sum space gl(\textG) \oplus \textG such that a non-abelian omni-Lie 2-algebra is constructed. At the same time, we prove that it is a strict Leibniz 2-algebra. Secondly, we prove that the bracket is compatible with the symmetric pairing and their properties are similar to the properties of omni-Lie 2-algebras. Lastly, a Nijenhuis operator on Leibniz 2-algebras is constructed, and it is shown that a non-abelian omni-Lie 2-algebra can be considered as a trivial deformation of an omni-Lie 2-algebra.