  In this paper, we construct a homotopy Poisson algebra of degree 3 associated
to a split Lie 2-algebroid, by which we give a new approach to characterize a
split Lie 2-bialgebroid. We develop the differential calculus associated to a
split Lie 2-algebroid and establish the Manin triple theory for split Lie
2-algebroids. More precisely, we give the notion of a strict Dirac structure
and define a Manin triple for split Lie 2-algebroids to be a CLWX 2-algebroid
with two transversal strict Dirac structures. We show that there is a
one-to-one correspondence between Manin triples for split Lie 2-algebroids and
split Lie 2-bialgebroids. We further introduce the notion of a weak Dirac
structure of a CLWX 2-algebroid and show that the graph of a Maurer-Cartan
element of the homotopy Poisson algebra of degree 3 associated to a split Lie
2-bialgebroid is a weak Dirac structure. Various examples including the string
Lie 2-algebra, split Lie 2-algebroids constructed from integrable distributions
and left-symmetric algebroids are given.
