nLab Schouten bracket



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 }


Lie theory, ∞-Lie theory

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

Γ(Λ kTM)Γ(Λ lTM)Γ(Λ k+l1TM). \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 1X k,Y 1Y l]= i,j(1) i+j[X i,Y j]X 1X^ iX kY 1Y^ jY l. [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.


Suppose PΓ(Λ 2TM)P\in\Gamma(\Lambda^2 TM). The the bracket {f,g}=dfdg,P\{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[P,P]=0.


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.