Bialgebroid And Jacobi Algebroid Deformation Theory

This thesis mainly studies the deformation of Lie bialgebroids and Jacobi bial-gebroids, especially their structure problems, on the basis of known results. We first generalize the concept of the deformation of Poisson manifold and Jacobi manifold to that of Lie bialgebroids and Jacobi bialgebroids, and get some conclusions concerning the pairs.In this paper, we introduced the definition of Nijenhuis tensor on Lie algebroid firstly,especially for the definition of Poisson-Nijenhuis manifolds which is first introduced by Magri and Morosi in their paper [18] and further studied in [9],play a central role in the study of integrable systems.Moreover,based on the conclusions and theorems given by them we defined the fundamental vector fields on Poisson-Nijenhuis manifolds. Considered the characterization of PN manifold in terms of Lie algebroid gaven by Y.Kosmann,we related the Nijenhuis structures and Lie bialge-broid then get a necessary and sufficient conditions for them to define a new Lie bialgebroid-deformed Lie bialgebroid.A contact manifold is known to be the analogue of a symplectic manifold for the odd-dimensional case,but a natural framework for a unified study of both contact manifold and local conformal pre-symplectic manifolds is given by the Jacobi structures which was introduced by A.Lichnerowicz and study by him and his collaborators[34],[38]( see also [7]).Just like the case of Poisson-Nijenhuis struc-tures.in section 4.2.1,we gave necessary and sufficient conditions for a (l,l)-tensor field N and a Jacobi structure (A, E) to define,in a natural way,a new Jacobi sreucture which is compatible with (A, E) in the sense of [9].And we compared it with Jacobi-Nijehuis structures which is considered by Marrero,etc.Furthermore,we also consider the deformation of Jacobi bialgebroid and triangular Jacobi bialge-broid.And the relationship of Jacobi-Nijehuis manifold with deformed Jacobi bialgebroid also be considered.With the notion of characteristic pair(or dual characteristic pair )of Dirac structures and the relative theorems, the discussion of Poisson reduction seems to more direct and easier to understand.Moreover,its geometric meaning is more...