nLab
Gerstenhaber algebra

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.

Related concepts

• Poisson n-algebra

• BV-algebra

References

• Cohen (1976)