Derived Brackets |
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Authors: | Yvette Kosmann-Schwarzbach |
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Institution: | (1) Centre de Mathématiques, École Polytechnique, UMR 7640 du CNRS, France) |
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Abstract: | We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and of Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp., parity-reversed cotangent bundle). Lie algebras have analogous properties, and the theory of Lie algebroids unifies the results valid for manifolds on the one hand, and for Lie algebras on the other. We outline the role of derived brackets in the theory of Poisson structures with background'. |
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Keywords: | Courant algebroid Courant bracket derived bracket Lie algebroid Loday– Leibniz algebra Poisson structure with background Odd Poisson supermanifold Vinogradov bracket |
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