synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
The Schouten bracket on multivector fields is [Michor (1987)] the unique (up to a multiplication by a constant) natural operation on multivector fields
Concretely, it is given in terms of the Lie bracket of vector fields by:
For multivector fields regarded as “antifields” in BV-BRST formalism, the Schouten bracket is called the antibracket.
Suppose . The the bracket satisfies the Jacobi identity (and hence is a Poisson bracket) if and only if .
The notion is due to:
Jan Schouten, Über Differentialkonkomitanten zweier kontravarianten Grössen, Indagationes Mathematicae 2 (1940) 449–452
Jan Schouten, On the differential operators of the first order in tensor calculus, In: Convegno Int. Geom. Diff. Italia. (1953) 1–7
Albert Nijenhuis, Jacobi-type identities for bilinear differential concomitants of certain tensor fields I, Indagationes Mathematicae 17 (1955) 390–403 [doi:10.1016/S1385-7258(55)50054-0]
A coordinate-free treatment is given in
Characterization as a natural operation is due to:
Textbook account: Chapter 33.2 of
A generalization is via so called Vinogradov bracket
which is in turn the antisymmetrization of the derived bracket,
Last revised on October 15, 2024 at 20:55:48. See the history of this page for a list of all contributions to it.