The Equivalent Characterization Of Li 2-double Algebra

In Lie algebra theory,a Lie bialgebra structure owns at the same time a Lie algebra structure and a Lie coalgebra structure and satisfies a certain compatibility condition.A Lie 2-bialgebra,as a generalized concept of Lie bialgebra,or a cate-gorification of Lie bialgebra,seems to have the same result,which is the main ideal of this paper.In section 1,we remind the certain concepts of Lie algebra,Lie coalgebra and Lie bialgebra.As we all know,if a vector space g,carries a Lie coalgebra structure,then the dual of g is a Lie algebra.Moreover,if a vector space g is a Lie bialgebra,then the double of g is a Lie algebra.In section 2,we give the concepts of Lie 2-algebra first.By using the big bracket,we give a equivalent description of Lie 2-algebra.Besides,by using the big bracket,,we give definition of Lie 2-coalgebra and Lie 2-bialgebra.We go through a inverse,a method of using big bracket,we find a equivalent definition of using structure maps and compatibility conditions for Lie 2-coalgebra and Lie 2-bialgebra.