Let $M$ be an exact symplectic manifold with contact type boundary such that
$c_1(M)=0$. In this paper we show that the cyclic cohomology of the Fukaya
category of $M$ has the structure of an involutive Lie bialgebra.Inspired by a
work of Cieliebak-Latschev we show that there is a Lie bialgebra homomorphism
from the linearized contact homology of $M$ to the cyclic cohomology of the
Fukaya category. Our study is also motivated by string topology and
2-dimensional topological conformal field theory.