Derived Brackets
By Y. Kosmann-Schwarzbach. 2003
252 citations recorded by the source corpus.
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’.
Subfields: Algebra and Number Theory, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/e7279212cc59fd9a2508f22b5dc23ce9410aa816
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Citations in the corpus
- Maximally homogeneous para-CR manifolds of semisimple type (2006, unknown): “7) shows that the bracket of a Lie algebroid is a socalled derived bracket [24], generalizing the wellknown formulas of Cartan: Eq.”
- T-duality, generalized geometry and non-geometric backgrounds (2008, background): “This combination of V and X is actually none other than the Dorfman bracket [18,24] whose antisymmetrization gives the Courant bracket (2.”
- Deformations of generalized complex and generalized Kahler structures (2007, background): “This is known as the derived bracket construction [19].”
- A scan for new N = 1 vacua on twisted tori (2006, background): “A discussion of the Courant bracket as an extension of the Lie bracket can be found in [18], here we will introduce it in an alternative way, as a particular case of so–called derived brackets (see for example [44]).”
- Quasi, twisted, and all that... in Poisson geometry and Lie algebroid theory (2003, background): “2 can also be written as a derived bracket [34] [36].”
- Supergravity as generalised geometry I: type II theories (2011, background): “which is know as the Courant bracket [2, 42].”
- Generalized Complex Geometry (2004, unknown): “3) of T ⊕T ∗ on forms, we may define the Courant bracket of sections ei ∈ C∞(T ⊕ T ∗) in the same way, following [22]:”
- Generalized structures of N=1 vacua (2005, background): “12This definition is more generally known as derived bracket [33].”