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Generalized Lie Bialgebroids and Strong Jacobi-Nijenhuis Structures

Roughly speaking, a Lie bialgebroid is a Lie algebroid A whose dual A is also equipped with a Lie algebroid structure which is compatible in a certain way with that on A (see [15]). An important example of a Lie bialgebroid is the associated one with a Poisson structure. More precisely, if M is a Poisson manifold with Poisson 2-vector Λ and on TM (respectively, T M) we consider the trivial Lie algebroid structure (resp. the cotangent Lie algebroid structure associated with Λ) then the pair (TM,T M) is a Lie bialgebroid. Other interesting examples of Lie bialgebroids are Lie bialgebras in the sense of Drinfeld [2] and the Lie bialgebroids associated with Poisson-Nijenhuis structures (see [11]). On the other hand, as it is well known, a Jacobi structure on a manifold M is a 2-vector Λ and a vector field E on M such that [Λ,Λ] = 2E ∧ Λ and [E,Λ] = 0, where [ , ] is the Schouten-Nijenhuis bracket [13]. If (M,Λ, E) is a Jacobi manifold one can define a bracket of functions, the Jacobi bracket, in such a way that the space C(M,R) endowed with the Jacobi bracket is a local Lie algebra in the sense of Kirillov [9]. Conversely, a local Lie algebra structure on C(M,R) induces a Jacobi structure on M [5, 9]. Jacobi manifolds are natural generalizations of Poisson manifolds and of other interesting manifolds: contact and locally conformal symplectic (l.c.s.) manifolds. If M is an arbitrary manifold, the vector bundle TM ×R →M possesses a natural Lie algebroid structure. Moreover, if M is a Jacobi manifold then the 1-jet bundle T M × R → M admits a Lie algebroid structure [8]. Since the pair