Derived Brackets
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 Vinogradov, and we prove that the Vinogradov bracket is the skewsymmetrization 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., parityreversed 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".
 Publication:

Letters in Mathematical Physics
 Pub Date:
 July 2004
 DOI:
 10.1007/s1100500406088
 arXiv:
 arXiv:math/0312524
 Bibcode:
 2004LMaPh..69...61K
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Symplectic Geometry;
 17A32;
 17B63;
 53D17;
 17B70;
 58A50
 EPrint:
 Revised version of the lecture given at PQR 2003, Brussels, June 2003