Derived brackets

Details Derived brackets Derived brackets

2003-12-31

arxiv.org/abs/math/0312524v1

Lett.Math.Phys. 69 (2004) 61-87

Mathematics

Differential Geometry

Revised version of the lecture given at PQR 2003, Brussels, June 2003

Scientific paper

10.1007/s11005-004-0608-8

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 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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