Lie algebroids $({A},{{A}^*})$ is a Lie bialgebroid  [@MR1362125; @MR1746902; @MR1262213] if ${{d}_*}$ is a derivation of the Gerstenhaber algebra $({\Gamma(\wedge{^{\scriptscriptstyle\bullet}}{A})},\wedge,{[\cdot,\cdot]})$ or, equivalently, if ${d}$ is a derivation of the Gerstenhaber algebra $({\Gamma(\wedge{^{\scriptscriptstyle\bullet}}{{A}^*})},\wedge,{[\cdot,\cdot]_*})$. Since the bracket ${[\cdot,\cdot]_*}$ (resp. ${[\cdot,\cdot]}$) can be recovered from the derivation ${{d}_*}$ (resp. ${d}$), one is led to the following alternative definition.

A Lie bialgebroid $({A},{{A}^*})$ is equivalent to a differential Gerstenhaber algebra structure on $({\Gamma(\wedge{^{\scriptscriptstyle\bullet}}{A})},\wedge,{[\cdot,\cdot]},{{d}_*})$ (or, equivalently, on $({\Gamma(\wedge{^{\scriptscriptstyle\bullet}}{{A}^*})},\wedge,{[\cdot,\cdot]}_*,{d})$).

Hamiltonian operators
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Let $({A},{{A}^*})$ be a complex Lie bialgebroid, and $H\in{\Gamma(\wedge^2{A})}$. We now replace the differential ${{d}_*}:{\Gamma(\wedge{^{\scriptscriptstyle\bullet}}{A})}\to{\Gamma(\wedge^{\bullet+1}{A})}$ by a twist by $H$: $$\label{eq:dH}
{{{d}_*}^H}:{\Gamma(\wedge^\bullet{A})}\to{\Gamma(\wedge^{\bullet+1}{A})}, \qquad {{{d}_*}^H}u={{d}_*}u+{[H,u]}.$$ It follows from a simple verification that if $H$ satisfies the Maurer-Cartan equation: $$\label{eq:Maure} d_{*}H +{\tfrac{1}{2}}[H,H]=0,$$ then $({{{d}_*}^H})^2=0$ and $({\Gamma(\wedge{^{\scriptscriptstyle\bullet}}{A})},\wedge,{[\cdot,\cdot]},{{{d}_*}^H})$ is again a differential Gerstenhaber algebra. Thus one obtains a Lie bialgebroid $({A},{{A}^*}_H)$. A solution $H\in{\Gamma(\wedge^{2}{A})}$ to Eq.  is called a **Hamiltonian operator** [@MR1472888]. The Lie algebroid structure on ${{A}^*}_H$ can be described explicitly: the anchor and the Lie bracket are given, respectively, by $${{{a}_*}^H}={{a}_*}+{a}\circ{H{^{\sharp}}}$$ and $${[\alpha,\beta]^H_*} ={[\alpha,\beta]_*}+[\alpha,\beta]_{H} .$$ Here $$[\alpha,\beta]_{H} ={L }_{{H{^{\sharp}}}(\alpha)}\beta
-{L }_{{H{^{\sharp}}}(\beta)}\alpha-{{d}_*}{\langle {H{^{\sharp}}}(\alpha ) | \beta\rangle} ,$$ for all $\alpha,\beta\in{\Gamma({{A}^*})}$. We shall use ${{A}^*}_H$ to denote such a Lie algebroid and call it the $H$-twisted Lie algebroid 