{c}

The left Schouten-Nijenhuis bracket on <polynomial polyvector fields> is the degree-minus-one <Gerstenhaber bracket> extending the <commutator> of <derivations> and $[D,f]=D(f)$ by graded antisymmetry and the left <graded Leibniz rule>. For $\Pi=\partial_X\wedge\partial_Y$, this convention gives $[h\Pi,f]=h(f_Y\partial_X-f_X\partial_Y)$ and $[F\partial_X+G\partial_Y,h\Pi]=(D(h)-h(F_X+G_Y))\Pi$. A right insertion convention reverses the degree-two-with-function formula.


Schouten-Nijenhuis bracket

{title2=$\bigwedge_A^p\operatorname{Der}_k(A)$}

For a <polynomial ring> $A$, a degree-$p$ polynomial polyvector field is an element of $\bigwedge_A^p\operatorname{Der}_k(A)$. Its product is the <exterior product>. Together with the <Schouten-Nijenhuis bracket> these spaces form a <Gerstenhaber algebra>.


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 Incoming links: Polynomial polyvector field