A Lie Algebroid on the Wiener Space

Infinite dimensional Poisson structures play a big role in the theory of infinite dimensional Lie algebras 1 , in the theory of integrable system 2 , and in field theory 3 . But for instance, in 2 , the test functional space where the hydrodynamic Poisson structure acts continuously is not conveniently defined. In 4, 5 we have defined such a test functional space in the case of a linear Poisson bracket of hydrodynamic type. On the other hand, it is very well known 6 that the theories of Lie groupoids and Lie algebroids play a key role in Poisson geometry. It is interesting to study a Lie algebroid for the Poisson structure 4 defined analytically in the framework of 4 . We postpone until later the study the Lie groupoid associated to the same Poisson structure but in the algebraic framework of 5 . The definition of this Lie groupoid in the framework of 4 presents, namely, some difficulties. Moreover some deformation quantizations for symplectic structures in infinite dimensional analysis were recently performed see the review of Léandre 7 on that . The theory of groupoids is related 8 to Kontsevich deformation quantization 9 . Let us recall what a Lie algebroid is 6, 10–13 . We consider a bundle E on a smooth finite dimensional manifold M. TM is the tangent bundle of M. Γ∞ E and Γ∞ TM denote the space of smooth section of E and TM. A Lie algebroid on E is given by the following data.