nLab Schouten bracket

Definition

The Schouten bracket on multivector fields is [Michor (1987)] the unique (up to a multiplication by a constant) natural operation on multivector fields

$$\Gamma(\Lambda^k T M) \otimes \Gamma(\Lambda^l T M) \longrightarrow \Gamma(\Lambda^{k+l-1} T M) \,.$$

Concretely, it is given in terms of the Lie bracket of vector fields by:

$$[X_1\wedge\cdots\wedge X_k,Y_1\wedge\cdots\wedge Y_l] \;=\; \sum_{i,j}(-1)^{i+j} [X_i, Y_j] \wedge X_1\wedge\cdots\hat X_i\cdots\wedge X_k\wedge Y_1\wedge\cdots \hat Y_j\cdots \wedge Y_l .$$

For multivector fields regarded as “antifields” in BV-BRST formalism, the Schouten bracket is called the antibracket.

Examples

Example

(Poisson structures)
Given a smooth manifold $X$ and a bivector $\pi \in\Gamma(\Lambda^2 T X)$, then the binary operation on smooth functions $f,g \in C^\inft(X)$ given by tensor contraction of $\pi$ with the wedge product of their de Rham differentials

$$\{f,g\} \,\coloneqq\, \langle \mathrm{d}f \wedge \mathrm{d}g,\, \pi \rangle$$

satisfies the Jacobi identity and hence is a Poisson bracket if and only if $\pi$ has vanishing Schouten bracket with itself, $[\pi, \pi] = 0$.

Related concepts

• Lie bracket of vector fields

• Frölicher–Nijenhuis bracket

• Nijenhuis–Richardson bracket

References

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

• W. M. Tulczyjew, The Graded Lie Algebra of Multivector Fields and the Generalized Lie Derivative of Forms. Bulletin de l’Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 22:9 (1974), 937–942. PDF.

Characterization as a natural operation is due to:

• Peter W. Michor, Remarks on the Schouten-Nijenhuis bracket, In: J. Bureš, V. Souček (eds.): Proceedings of the Winter School “Geometry and Physics” Circolo Matematico di Palermo, Palermo (1987) 207-215 [dml:701423, pdf, pdf]

Textbook account: Chapter 33.2 of

• Peter W. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics 93 (2008). PDF.

A generalization is via so called Vinogradov bracket

• A. M. Vinogradov, Объединение скобок Схоутена и Нийенхейса, когомологии и супердифференциальные операторы (Unification of the Schouten and Nijenhuis brackets, cohomology, and superdifferential operators), Mat. Zametki 47(6), 138 (1990) mathnet.ru:mzm3270

which is in turn the antisymmetrization of the derived bracket,

• Yvette Kosmann-Schwarzbach: Derived brackets, Lett. Math. Phys. 69 (2004) 61–87 [doi:10.1007/s11005-004-0608-8]