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  "body": "\n## Definition\n\nLet $k$ be a [[field|04C5ZBG1WT]].\nLet $A = \\bigoplus_{n \\in \\Z} A^n$ be a [[graded|8KB4SQGC0C]] [[associative algebra|3CTW1P4HBP]] over $k$ that is graded commutative: $ab = (-1)^{|a||b|} ba$ for homogeneous $a, b$ of degrees $\\abs{a}, \\abs{b}$.\nA *Gerstenhaber algebra* is such an $A$ with a bilinear bracket $[\\cdot, \\cdot] \\colon A^p \\times A^q \\to A^{p+q-1}$ such that, for homogeneous $a, b, c$:\n\n1. $[a, b] = -(-1)^{(|a|-1)(|b|-1)} [b, a]$;\n2. $[a, [b, c]\\,] = [\\,[a, b], c] + (-1)^{(|a|-1)(|b|-1)} [b, [a, c]\\,]$;\n3. $[a, bc] = [a, b]\\, c + (-1)^{(|a|-1)|b|}\\, b\\, [a, c]$.\n\n## Remarks\n\nClauses 1 and 2 say that $A$ becomes a graded Lie algebra when each element of $A^n$ is given the degree $n - 1$.\nClause 3 says that $[a, \\cdot]$ is a derivation of the product of degree $\\abs{a} - 1$.\nExample: the polyvector fields $\\bigoplus_n \\Gamma(\\Lambda^n TM)$ on a smooth manifold $M$, with the wedge product and the [[Schouten–Nijenhuis bracket|FJ1MDBZ680]].\nExample: 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.\nNon-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.\nThen $[e, e \\wedge f] = 0$, while clause 3 asks for $[e, e] \\wedge f + e \\wedge [e, f] = e \\wedge f \\ne 0$, so clause 3 fails.\n",
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