Author： Kosmann-Schwarzbach Yvette
Publisher： Springer Publishing Company
ISSN： 0377-9017
Source： Letters in Mathematical Physics, Vol.69, Iss.1, 2004-07, pp. : 61-87
Disclaimer: Any content in publications that violate the sovereignty, the constitution or regulations of the PRC is not accepted or approved by CNPIEC.
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’.
Related content
Hydrodynamic Nambu brackets derived by geometric constraints
Journal of Physics A: Mathematical and Theoretical, Vol. 48, Iss. 10, 2015-03 ,pp. :
Access to resources
Recommend
Linearization of Poisson Brackets
By Fernandes Rui Monnier Philippe
Letters in Mathematical Physics, Vol. 69, Iss. 1, 2004-07 ,pp. :
Quantum and Classical Brackets
By Kisil V.V.
International Journal of Theoretical Physics, Vol. 41, Iss. 1, 2002-01 ,pp. :
Wronski Brackets and the Ferris Wheel
By Martin Keye
International Journal of Theoretical Physics, Vol. 44, Iss. 11, 2005-11 ,pp. :
Generalized Transvectants-Rankin–Cohen Brackets
Letters in Mathematical Physics, Vol. 63, Iss. 1, 2003-01 ,pp. :