The untwisted skew-symmetric Courant bracket on sections of the generalized tangent bundle isHere is the Lie bracket of vector fields, is the Lie derivative, and is the exterior derivative. It extends the Lie bracket of vector fields, but its Jacobi identity has an exact-form defect; it is not simply another Lie algebra bracket.[ref]
For pure vector fields, , and two pure one-forms have zero bracket. On the real line, . This mixed example shows how the bracket couples tangent and cotangent information through differentiation. Omitting the final exact term gives the nonskew Dorfman operation, a related convention with different identities.
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The Courant bracket is a mathematical operation that arises in the context of differential geometry and the theory of Dirac structures. It is named after the mathematician Richard Courant and plays a significant role in the study of symplectic geometry and Poisson geometry, as well as in the theory of integrable systems. In a more formal context, the Courant bracket is defined on sections of a specific vector bundle called the Courant algebroid.