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*, and a line bundle ℒ such that ℒ ⊗ ℒ = (∧^top A* ⊗ ∧^top T*M)^1/2 exists, there exist two canonically defined differential operators d* and ∂ on (∧A ⊗ ℒ). We prove that the pair (A, A*) constitutes a Lie bialgebroid if and only if the square of D = d* + ∂ is the multiplication by a function on M. As a consequence, we obtain that the pair (A, A*) is a Lie bialgebroid if and only if D is a Dirac generating operator as defined by Alekseev and Xu. Our approach is to establish a list of new identities relating the Lie algebroid structures on A and A*.