---
tag: GJPPS5N213
kind: concept
name: Gerstenhaber algebra
msc: [17B70, "16E40"]
sources:
  - url: https://en.wikipedia.org/wiki/Gerstenhaber_algebra
same_as:
  wikidata: Q5553164
  wikipedia: Gerstenhaber algebra
---

## Summary

A Gerstenhaber algebra is a graded commutative algebra with a Lie bracket of degree $-1$ that acts on products by the Leibniz rule.
Murray Gerstenhaber found this structure on the Hochschild cohomology of an associative algebra.
Polyvector fields on a manifold with the [[Schouten–Nijenhuis bracket|FJ1MDBZ680]] form another basic example.
It is the graded analogue of a Poisson algebra, with the bracket shifted by one degree.

## Explanations

### L1 human

A Gerstenhaber algebra carries two operations: a product that is commutative up to signs, and a bracket like the commutator of vector fields.
The bracket lowers the total degree by one.
The two operations fit together by a product rule, just as the derivative of a product follows the Leibniz rule.
For polyvector fields on a manifold, the product is the wedge product and the bracket extends the Lie bracket of vector fields.
The same structure on Hochschild cohomology governs the deformations of an associative algebra.