# nLab Gerstenhaber algebra

### Context

#### Higher algebra

higher algebra

universal algebra

# Contents

## Idea

A Gerstenhaber algebra is a Poisson 2-algebra, a Poisson algebra in graded vector spaces with Poisson bracket of degree -1.

## Definition

###### Definition

A Gerstenhaber algebra is a chain complex $A$ equipped with

• a symmetric product $\cdot : A \otimes A \to A$;

• a skew-symmetric bracket $[-,-] : A \otimes A \to A[1]$;

• such that associativity of $\cdot$ and the Jacobi identity for $[-,-]$ holds and such that $[a,-]$ is a derivation of $\cdot$.

## Properties

###### Theorem

The homology of the operad for Gerstenhaber algebras in chain complexes is the operad for Gerstenhaber algebras.

Accordingly the homology of an E2-algebra is a Gerstenhaber algebra.

This is due to Cohen (1976).

###### Remark

A Gerstenhaber algebra equipped in addition with a certain morphism $\Delta : A \to A$ is a BV-algebra. This is the homology of an algebra over the framed little 2-disk operad.

## References

• Cohen (1976)

Revised on January 24, 2013 12:47:45 by Urs Schreiber (82.113.99.233)