Polynomial polyvector field (${\bigwedge }_A^p{\mathrm{Der}}_k(A)$)

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

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.