Dirac generating operators and Manin triples

Given a pair of (real or complex) Lie algebroid structures on a vector bundle $A$ (over $M$) and its dual $A^{\ast }$, and a line bundle $\text{module}$ such that $\text{module}\otimes \text{module}=({\wedge }^{\text{TOP}}{A}^{\ast }\otimes {\wedge }^{\text{TOP}}{T}^{\ast }M)$, there exist two canonically defined differential operators $\text{bdees}$ and $\text{bdel}$ on $\text{sections}\wedge A\otimes \text{module}$. We prove that the pair $(A,A^{\ast })$ constitutes a Lie bialgebroid if, and only if, the square of $\bdirac =\bdees+\bdel$isthemultiplicationbyafunctionon$M$. As a consequence, we obtain that the pair $(A,A^{\ast })$ is a Lie bialgebroid if, and only if, $\text{bdirac}$ is a Dirac generating operator as defined by Alekseev & Xu \cite{AlekseevXu}. Our approach is to establish a list of new identities relating the Lie algebroid structures on $A$ and $A^{\ast }$ (Theorem \ref{Thm:C}).