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)

• Murray Gerstenhaber, The cohomology structure of an associative ring, Ann. of Math., 78 (1963), 267-288 MR28:5102

• Ping Xu, Gerstenhaber algebras and BV-algebras in Poisson geometry, Commun. Math. Phys. 200, No.3, 545-560 (1999).

Revised on December 21, 2016 05:01:21 by Urs Schreiber (188.1.230.74)