of ${{A}^*}$. Thus we obtain the following theorem, which was first proved in [@MR1472888] by a different method.

If $({A},{{A}^*})$ constitutes a Lie bialgebroid, and $H\in{\Gamma(\wedge^2{A})}$ is a Hamiltonian operator, then $({A},{{A}^*}_H)$ is a Lie bialgebroid.

Maurer-Cartan elements
======================

The Lie bialgebroid stemming from a complex manifold
----------------------------------------------------

We fix a complex manifold ${\boldsymbol{X}}$ of complex dimension $n$ with almost complex structure $J$. We regard the tangent bundle ${T{\boldsymbol{X}}}$ as a real vector bundle over ${\boldsymbol{X}}$. The complexification of ${T{\boldsymbol{X}}}$ is denoted ${T_{{\mathbb{C}}}{\boldsymbol{X}}}$, namely: ${T_{{\mathbb{C}}}{\boldsymbol{X}}}={T{\boldsymbol{X}}}\otimes{\mathbb{C}}$. Similarly, ${T^*_{{\mathbb{C}}}{\boldsymbol{X}}}={T^*{\boldsymbol{X}}}\otimes{\mathbb{C}}$. Let ${\mathbb{J}}:{T_{{\mathbb{C}}}{\boldsymbol{X}}}\to{T_{{\mathbb{C}}}{\boldsymbol{X}}}$ be the ${\mathbb{C}}$-linear extension of the almost complex structure $J$, and ${{T^{1,0}{\boldsymbol{X}}}}$ and ${{T^{0,1}{\boldsymbol{X}}}}$ its $+{i}$ and $-{i}$ eigenbundles, respectively. We adopt the following notations: $${T^{p,q}{\boldsymbol{X}}}=\wedge^p{{T^{1,0}{\boldsymbol{X}}}}\otimes\wedge^q{{T^{0,1}{\boldsymbol{X}}}},$$ $${(T^{p,q}{\boldsymbol{X}})^*}=\wedge^p{{(T^{1,0}{\boldsymbol{X}})^*}}\otimes\wedge^q{{(T^{0,1}{\boldsymbol{X}})^*}}.$$

Consider the following two vector bundles which are obviously mutually dual: $$\label{Eqt:defnAbdAbds}
{A}={{T^{1,0}{\boldsymbol{X}}}}\oplus{{(T^{0,1}{\boldsymbol{X}})^*}},\quad {{A}^*}={{T^{0,1}{\boldsymbol{X}}}}\oplus{{(T^{1,0}{\boldsymbol{X}})^*}}.$$ We can endow $A$ with a complex Lie algebroid structure. The anchor is the projection onto the first component: $$a({{\tfrac{\partial}{\partial{z^{i}}}}})={{\tfrac{\partial}{\partial{z^{i}}}}} \qquad a(d{\overline{z_j}})=0 .$$ The bracket of two sections of ${{T^{1,0}{\boldsymbol{X}}}}$ is their bracket as vector fields; the bracket of any pair of sections of ${{(T^{0,1}{\boldsymbol{X}})^*}}$ is zero; and the bracket of a holomorphic vector field (i.e. a holomorphic section of the holomorphic vector bundle ${{T^{1,0}{\boldsymbol{X}}}}$) and an anti-holomorphic 1-form (i.e. 