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In [[differential geometry]], the '''Schouten–Nijenhuis bracket''', also known as the '''Schouten bracket''', is a type of [[graded Lie algebra|graded Lie bracket]] defined on [[multivector]] [[vector field|fields]] on a [[smooth manifold]] extending the [[Lie bracket of vector fields]]. There are two different versions, both rather confusingly called by the same name. The most common version is defined on alternating multivector fields and makes them into a [[Gerstenhaber algebra]], but there is also another version defined on symmetric multivector fields, which is more or less the same as the [[Poisson bracket]] on the [[cotangent bundle]]. It was discovered by [[Jan Arnoldus Schouten]] (1940, 1953) and its properties were investigated by his student [[Albert Nijenhuis]] (1955). It is related to but not the same as the [[Nijenhuis–Richardson bracket]] and the [[Frölicher–Nijenhuis bracket]].

==Definition and properties==
An alternating multivector field is a section of the [[exterior algebra]] ∧∗T''M'' over the [[tangent bundle]] of a manifold ''M''. The alternating multivector fields form a graded supercommutative ring with the product of ''a'' and ''b'' written as ''ab'' (some authors use ''a''∧''b''). This is dual to the usual algebra of [[differential form]]s Ω∗''M'' by the pairing on homogeneous elements:
: <math>\omega(a_1a_2 \dots a_p)=\left\{
\begin{matrix}
\omega(a_1,\dots,a_p)&(\omega\in \Omega^pM)\\
0&(\omega\not\in\Omega^pM)
\end{matrix}\right.
</math>
The '''degree''' of a multivector ''A'' in ∧''p''T''M'' is defined to be |''A''| = ''p''.

The skew symmetric Schouten–Nijenhuis bracket is the unique extension of the [[Lie bracket of vector fields]] to a graded bracket on the space of alternating multivector fields that makes the alternating multivector fields into a [[Gerstenhaber algebra]].
It is given in terms of the Lie bracket of vector fields by
:<math>[a_1\cdots a_m,b_1\cdots b_n]=\sum_{i,j}(-1)^{i+j}[a_i,b_j]a_1\cdots a_{i-1}a_{i+1}\cdots a_mb_1\cdots b_{j-1}b_{j+1}\cdots b_n</math>
for vector fields ''a''''i'', ''b''''j'' and
:<math>[f,a_1\cdots a_m] = -i_{df}(a_1 \cdots a_m)</math>
for vector fields ''a''''i'' and smooth function ''f'', where ''i''''df'' is the common inner product operator.
It has the following properties.
*|''ab''| = |''a''| + |''b''| (The product has degree 0)
*|[''a'',''b'']| = |''a''| + |''b''| − 1 (The Schouten–Nijenhuis bracket has degree −1)
*(''ab'')''c'' = ''a''(''bc''), ''ab'' = (−1)|''a''||''b''|''ba'' (the product is associative and (super) commutative)
*[''a'', ''bc''] = [''a'', ''b'']''c'' + (−1)|''b''|(|''a''| − 1)''b''[''a'', ''c''] (Poisson identity)
*[''a'',''b''] = −(−1)(|''a''| − 1)(|''b''| − 1) [''b'',''a''] (Antisymmetry of Schouten–Nijenhuis bracket)
*[[''a'',''b''],''c''] = [''a'',[''b'',''c'']] − (−1)(|''a''| − 1)(|''b''| − 1)[''b'',[''a'',''c'']] (Jacobi identity for Schouten–Nijenhuis bracket)
* If ''f'' and ''g'' are functions (multivectors homogeneous of degree 0), then [''f'',''g''] = 0.
* If ''a'' is a vector field, then [''a'',''b''] = '''L'''''a''''b'' is the usual [[Lie derivative]] of the multivector field ''b'' along ''a'', and in particular if ''a'' and ''b'' are vector fields then the Schouten–Nijenhuis bracket is the usual Lie bracket of vector fields.

The Schouten–Nijenhuis bracket makes the multivector fields into a Lie superalgebra if the grading
is changed to the one of opposite parity (so that the even and odd subspaces are switched), though
with this new grading it is no longer a supercommutative ring. Accordingly, the Jacobi identity may also be expressed in the symmetrical form
:<math>(-1)^{(|a|-1)(|c|-1)}[a,[b,c]]+(-1)^{(|b|-1)(|a|-1)}[b,[c,a]]+(-1)^{(|c|-1)(|b|-1)}[c,[a,b]] = 0.\,</math>

==Generalizations==
There is a common generalization of the Schouten–Nijenhuis bracket for alternating multivector fields and the [[Frölicher–Nijenhuis bracket]] due to Vinogradov (1990).

A version of the Schouten–Nijenhuis bracket can also be defined for symmetric multivector fields in a similar way. The symmetric multivector fields can be identified with functions on the cotangent space ''T''*(''M'') of ''M'' that are polynomial in the fiber, and under this identification the symmetric Schouten–Nijenhuis bracket corresponds to the [[Poisson bracket]] of functions on the [[symplectic manifold]] ''T''*(''M'').
There is a common generalization of the Schouten–Nijenhuis bracket for symmetric multivector fields and the [[Frölicher–Nijenhuis bracket]] due to Dubois-Violette and [[Peter W. Michor]] (1995).

==References==
*{{Cite journal |first=Michel |last=Dubois-Violette |first2=Peter W. |last2=Michor |title=A common generalization of the Frölicher–Nijenhuis bracket and the Schouten bracket for symmetric multi vector fields |arxiv=alg-geom/9401006 |journal=Indag. Mathem. |volume=6 |issue=1 |year=1995 |pages=51–66 |doi= }}
*{{Cite journal |first=Charles-Michel |last=Marle |url=http://perso.orange.fr/Charles-Michel.Marle/pdffiles/schouten.pdf |title=The Schouten-Nijenhuis bracket and interior products |journal=Journal of Geometry and Physics |volume=23 |issue= |pages=350–359 |year=1997 |doi= }}
*{{Cite journal |first=A. |last=Nijenhuis |title=Jacobi-type identities for bilinear differential concomitants of certain tensor fields I |journal=Indagationes Math. |volume=17 |issue= |year=1955 |pages=390–403 |doi= }}
*{{Cite journal |first=J. A. |last=Schouten |title={{lang|de|Über Differentialkonkomitanten zweier kontravarianten Grössen}} |journal=Indag. Math. |volume=2 |issue= |year=1940 |pages=449–452 }}
*{{Cite book |first=J. A. |last=Schouten |chapter=On the differential operators of the first order in tensor calculus |title=Convegno Int. Geom. Diff. Italia |year=1953 |editor=Cremonese |pages=1–7 }}
*{{Cite journal |first=A. M. |last=Vinogradov |title=Unification of Schouten–Nijenhuis and Frölicher–Nijenhuis brackets, cohomology and super differential operators |journal=Sov. Math. Zametki |volume=47 |issue= |year=1990 |pages= |doi= }}

==External links==
*Nicola Ciccoli [http://toknotes.mimuw.edu.pl/sem4/online/node9.html ''Schouten–Nijenhuis bracket''] in notes on [http://toknotes.mimuw.edu.pl/sem4/online/fpqg.html ''From Poisson to Quantum Geometry'']

{{DEFAULTSORT:Schouten-Nijenhuis bracket}}
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[[Category:Bilinear operators]]
[[Category:Differential geometry]]

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Alveolar–arterial gradient