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)
  • M. 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 October 14, 2014 17:59:53 by Zoran Škoda (161.53.130.104)