---
tag: TB3Q55Q0G4
kind: definition
name: Gerstenhaber algebra
msc: [17B70, "16E40"]
defines: GJPPS5N213
uses: [04C5ZBG1WT, 8KB4SQGC0C, 3CTW1P4HBP]
sources:
  - url: https://en.wikipedia.org/wiki/Gerstenhaber_algebra
---

## Definition

Let $k$ be a [[field|04C5ZBG1WT]].
Let $A = \bigoplus_{n \in \Z} A^n$ be a [[graded|8KB4SQGC0C]] [[associative algebra|3CTW1P4HBP]] over $k$ that is graded commutative: $ab = (-1)^{|a||b|} ba$ for homogeneous $a, b$ of degrees $\abs{a}, \abs{b}$.
A *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 $a, b, c$:

1. $[a, b] = -(-1)^{(|a|-1)(|b|-1)} [b, a]$;
2. $[a, [b, c]\,] = [\,[a, b], c] + (-1)^{(|a|-1)(|b|-1)} [b, [a, c]\,]$;
3. $[a, bc] = [a, b]\, c + (-1)^{(|a|-1)|b|}\, b\, [a, c]$.

## 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|FJ1MDBZ680]].
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.