In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector 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 invented 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 {\displaystyle \wedge ^{\bullet }TM} over the tangent bundle of a manifold M {\displaystyle M} . The alternating multivector fields form a graded supercommutative ring with the product of a {\displaystyle a} and b {\displaystyle b} written as a b {\displaystyle ab} (some authors use a ∧ b {\displaystyle a\wedge b} ). This is dual to the usual algebra of differential forms Ω ∙ ( M ) {\displaystyle \Omega ^{\bullet }(M)} by the pairing on homogeneous elements:
ω ( a 1 a 2 … a p ) = { ω ( a 1 , … , a p ) ( ω ∈ Ω p M ) 0 ( ω ∉ Ω p M ) {\displaystyle \omega (a_{1}a_{2}\dots a_{p})=\left\{{\begin{matrix}\omega (a_{1},\dots ,a_{p})&(\omega \in \Omega ^{p}M)\\0&(\omega \not \in \Omega ^{p}M)\end{matrix}}\right.}
The degree of a multivector A {\displaystyle A} in Λ p T M {\displaystyle \Lambda ^{p}TM} is defined to be | A | = p {\displaystyle |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
[ a 1 ⋯ a m , b 1 ⋯ b n ] = ∑ i , j ( − 1 ) i + j [ a i , b j ] a 1 ⋯ a i − 1 a i + 1 ⋯ a m b 1 ⋯ b j − 1 b j + 1 ⋯ b n {\displaystyle [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_{m}b_{1}\cdots b_{j-1}b_{j+1}\cdots b_{n}}
for vector fields a i {\displaystyle a_{i}} , b j {\displaystyle b_{j}} and
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