Gerstenhaber algebra

Remarks

Clauses 1 and 2 say that $A$ becomes a graded Lie algebra when each element of $A^n$ is given the degree $n - 1$. Clause 3 says that $[a, \cdot]$ is a derivation of the product of degree $\abs{a} - 1$. Example: the polyvector fields $\bigoplus_n \Gamma(\Lambda^n TM)$ on a smooth manifold $M$, with the wedge product and the Schouten–Nijenhuis bracket. Example: Gerstenhaber proved that the Hochschild cohomology $HH^\bullet(B, B)$ of an associative algebra $B$ is a Gerstenhaber algebra, with the cup product and the Gerstenhaber bracket. Non-example: let $\mathfrak{g}$ have basis $e, f$ with $[e, f] = f$, put it in degree $1$ of $\Lambda(\mathfrak{g})$, and set the bracket to zero on all other degrees. Then $[e, e \wedge f] = 0$, while clause 3 asks for $[e, e] \wedge f + e \wedge [e, f] = e \wedge f \ne 0$, so clause 3 fails.