nLab
Gerstenhaber 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 AA equipped with

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

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

  • such that associativity of \cdot and the Jacobi identity for [,][-,-] holds and such that [a,][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 Δ:AA\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)