  In this paper, we develop the bialgebra theory for Lie-Yamaguti algebras. For
this purpose, we exploit two types of compatibility conditions: local cocycle
condition and double construction. We define the classical Yang-Baxter equation
in Lie-Yamaguti algebras and show that a solution to the classical Yang-Baxter
equation corresponds to a relative Rota-Baxter operator with respect to the
coadjoint representation. Furthermore, we generalize some results by Bai in [1]
and Semonov-Tian-Shansky in [19] to the context of Lie-Yamaguti algebras. Then
we introduce the notion of matched pairs of Lie-Yamaguti algebras, which leads
us to the concept of double construction Lie-Yamaguti bialgebras following the
Manin triple approach to Lie bialgebras. We prove that matched pairs, Manin
triples of Lie-Yamaguti algebras, and double construction Lie-Yamaguti
bialgebras are equivalent. Finally, we clarify that a local cocycle condition
is a special case of a double construction for Lie-Yamaguti bialgebras.
