# nLab Schouten bracket

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

infinitesimal cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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.

## Applications

Suppose $P\in\Gamma(\Lambda^2 TM)$. The the bracket $\{f,g\}=\langle d f\wedge d g,P\rangle$ satisfies the Jacobi identity (and hence is a Poisson bracket) if and only if $[P,P]=0$.

## 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)50054-050054-0)]

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.

Last revised on May 4, 2023 at 08:58:51. See the history of this page for a list of all contributions to it.