Dirac generating operators and Manin triples

Abstract

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 $\module$ such that $\module\otimes\module=(\wedge^{\TOP} A^*\otimes\wedge^{\TOP} T^*M)$, there exist two canonically defined differential operators $\bdees$ and $\bdel$ on $\sections{\wedge A\otimes\module}$. We prove that the pair $(A,A^{\ast })$ constitutes a Lie bialgebroid if, and only if, the square of $\bdirac =\bdees+\bdel$ is the multiplication by a function on $M$. As a consequence, we obtain that the pair $(A,A^{\ast })$ is a Lie bialgebroid if, and only if, $\bdirac$ 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}).

26 pages, introduction rewritten, minor corrections