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Exact Gerstenhaber algebras and Lie bialgebroids. (English) Zbl 0837.17014
This work stresses the growing meaning of the theory of Lie algebroids in the sense of J. Pradines [C. R. Acad. Sci., Paris, Sér. A 264, 245-248 (1967; Zbl 0154.21704)] and Lie bialgebroids in the sense of K. Mackenzie and Ping Xu [Duke Math. J. 73, 415-452 (1994)]. The problems considered are important, for example, in string theory which, lately, makes extensive use of these algebraic structures. Lie algebroids appear in many domains of differential geometry. The theory of Poisson manifolds is one of them; it is important from the point of view of some applications to theoretical physics.
This work concerns triangular Lie bialgebroids generalizing both the Lie bialgebroids of Poisson manifolds and the triangular Lie bialgebras. To any such bialgebroid the author assigns two differential Gerstenhaber algebras in duality, one of which is canonically equipped with an operator generating the graded Lie algebra bracket, i.e. with the structure of a Batalin-Vilkovsky algebra.

17B70 Graded Lie (super)algebras
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
17B81 Applications of Lie (super)algebras to physics, etc.
17B66 Lie algebras of vector fields and related (super) algebras
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
Full Text: DOI
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